## Abstract

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.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

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 [1], 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 [6]. 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 [10], 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 [13]. 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 [28], and an optical differentiator and integrator has been proposed [29]. 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. [30]. And similar results have been further illustrated by M. A. Noginov et al. [17]. However, gain saturation effect [31] 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 [32]. 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 [10] 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 [44] 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 *d _{m}* and permittivity

*∊*is sandwiched between a prism and a dielectric layer. A high permittivity

_{m}*∊*

_{0}prism is used here for ATR excitation of SPPs. The thickness of the adjacent dielectric layer with permittivity

*∊*is of the order of 10

_{d}*λ*, 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],

*k*is the wavevector of SPPs,

_{SPP}*ω*represents the angular frequency of the incident light, and

*c*is the speed of light in free space. Note that,

*k*usually is a complex number, and Eq. (1) is still valid even if

_{SPP}*∊*and

_{m}*∊*are both complex numbers.

_{d}SPPs will be resonantly excited if the wavevector component of the incident beam along the interface ${k}_{x}=\frac{\omega}{c}\sqrt{{\u220a}_{0}}\text{sin}\theta $, here *θ* is the angle of incidence, matches the wavevector of SPPs, i.e.,

*θ*is known as the surface plasmon resonance angle. At this angle, the energy of incident light is resonantly transferred to SPPs, yielding that the curve of the reflectivity

_{SPP}*R*versus

*θ*exhibits a dip, as shown in Fig. 1.

The reflectivity *R* for a TM-polarized light in this structure is given by [1,17,30]

*T*can be written as

*i*,

*j*= 0,

*m*,

*d*) are respectively the Fresnel factors for the reflection and the transmission of a TM-polarized light beam between two adjacent media, with

*k*(

_{iz}*i*= 0,

*m*,

*d*) represent the normal components of the wavevector in three different media. It should be pointed out that the sign of the square root in Eq. (5) must be selected to enforce the physical energy propagation [17].

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. [17]: the prism with permittivity *∊*_{0} = 3.18, silver film with permittivity *∊ _{m}* = −15.584 + 0.424

*i*at

*λ*= 594

*nm*, the dielectric with permittivity

*∊*= 2.25. The dielectric without optical gain (i.e., the imaginary part of the dielectric

_{d}*∊*= 0) is considered here, and the signs of the square root in Eq. (5) are all positive in three different media. In Fig. 1, ${\theta}_{c}=\text{arcsin}\sqrt{\frac{{\u220a}_{d}}{{\u220a}_{0}}}=57.27\xb0$ is the critical angle for total internal reflection, and

_{i}*θ*= 65.38°, which is determined by Eqs. (1) and (2), is the surface plasmon resonance angle. The reflectivity

_{SPP}*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

*d*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 [45] of silver film

_{m}*d*=

_{opt}*d*

_{m2}= 53.9

*nm*for SPPs excitation exists in this structure. The energy of the incident light is almost 100% transferred to SPPs at

*θ*=

*θ*and

_{SPP}*d*=

_{m}*d*. Actually, SPPs can be effectively excited in this structure when the incident angle

_{opt}*θ*is close to

*θ*and the thickness of metal film

_{SPP}*d*is in the vicinity of

_{m}*d*.

_{opt}## 3. The permittivity of the gain medium

Assume that the permittivity of the gain medium can be described by a classical Lorentzian oscillator model [4]. Within the usual dipole approximation, the electric polarization *P* can be expressed as

*N*/

*V*is the number density of dipole moments,

*e*and

*m*are respectively charge and mass of an electron,

*ω*

_{0}is the angular frequency of the harmonic oscillator and

*γ*describes damping, and

*E*=

*E*

_{0}exp(

*iωt*) describes the harmonic electric field.

The electric polarization depends on the electric field in a manner that can often be described by the relationship

where*ε*

_{0}= 8.85 × 10

^{−12}

*C*/(

*Vm*) is the permittivity in vacuum and

*χ*is the susceptibility. Thus the macroscopic electric susceptibility is,

*ω*≫

*γ*and close to resonance, we have,

*χ*describes absorption or gain of the dielectric, depending on whether

_{i}*χ*is positive or negative, respectively. And

_{i}*χ*is associated with

_{r}*χ*by the Kramers-Kronig relations [4,13,15] in linear optics.

_{i}The relative permittivity *∊ _{d}* of the gain medium is related to the susceptibility through,

*χ*is the background susceptibility [46]. The small value of absorption (or gain) of the background susceptibility can be neglected, then

_{b}*∊*is a complex number and can be written as, here

_{d}*∊*and

_{r}*∊*are the real and the imaginary parts of

_{i}*∊*, respectively.

_{d}Note that usually |*∊ _{r}*| ≫ |

*∊*|, then the gain coefficient

_{i}*g*is related to

*∊*through [16,46],

_{d}## 4. ETR in the case of small-signal gain

In an optical amplifying medium, small-signal gain coefficient *g*^{0} can be expressed as [31],

*N*represents population inversion, and $\sigma (v,{v}_{0})=\frac{{c}^{2}}{{\omega}^{2}}\frac{\pi {A}_{21}}{2{n}^{2}}f(v,{v}_{0})$ represents the stimulated emission cross section, with

*A*

_{21}is the spontaneous emission rate in the host medium,

*f*(

*v*,

*v*

_{0}) is the lineshape function,

*v*is the frequency of the incident light and

*ω*= 2

*πv*,

*v*

_{0}is the transition resonance frequency, and

*n*is the refractive index of the host medium at the frequency

*v*. Small-signal gain coefficient

*g*

^{0}can be modulated by the pump power and the density of active ions in the host medium [47,48].

For the light intensity in the gain medium *I _{d}* ≪

*I*(here

_{s}*I*is the saturation intensity of the gain medium), small-signal gain coefficient

_{s}*g*

^{0}is almost unchanged with the variations of

*I*. The permittivity

_{d}*∊*is independent of the light intensity

_{d}*I*.

_{d}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 [17], 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) *d _{m}* = 39

*nm*, (b)

*d*= 60

_{m}*nm*, and (c)

*d*= 70

_{m}*nm*. Other parameters are aforementioned. The color bar on the right of Fig. 2 represents log

_{10}(

*R*). Small-signal gain coefficients are 705

*cm*

^{−1}, 1410

*cm*

^{−1}and 2115

*cm*

^{−1}corresponding to

*∊*= −0.010,

_{i}*∊*= −0.020 and

_{i}*∊*= −0.030, respectively. Such high optical gain may exist in semiconductor quantum wells [47]. Figure 2 clearly demonstrates that ETR with the simultaneous excitation of SPPs exist in this structure when small-signal gain coefficient

_{i}*g*

^{0}above a certain value [30], which is not only a function of the incident angle

*θ*but also depends on the thickness of metal film

*d*. 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

_{m}*R*versus

*∊*near the surface plasmon resonance angle at a fixed

_{i}*d*, and that the singular point and the reflectivity minimum point are close to each other as

_{m}*d*increases. The singular point of

_{m}*R*is determined by the denominator of Eq. (3), i.e., 1 +

*r*

_{0m}

*r*exp(2

_{md}*ik*) = 0, which is also named as the characteristic equation [3]. Comparing subplots (a) and (c) whit subplot (b), it is found that the more

_{mz}d_{m}*d*close to

_{m}*d*= 53.9

_{opt}*nm*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 *I _{d}* further increases, gain coefficient

*g*becomes smaller, and this phenomenon is known as gain saturation [31]. According to Eq. (11),

*∊*is not independent of

_{d}*I*anymore under gain saturation.

_{d}*∊*and

_{r}*∊*are changed simultaneously with the variations of

_{is}*I*, here

_{d}*∊*, to distinguish

_{is}*∊*in the case of small signal gain, represents the imaginary part of

_{i}*∊*in the case of gain saturation. Combining Eqs. (7) and (8) with Eq. (10) provides that

_{d}*x*is a change of

*x*with

*x*=

*∊*,

_{r}*∊*,

_{is}*χ*,

_{r}*χ*. The value of $\frac{\mathrm{\Delta}{\u220a}_{r}}{\mathrm{\Delta}{\u220a}_{\mathit{is}}}$ will be quite different for different kinds of gain materials [15].

_{i}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* ≃ *ω*_{0}. Near resonance, *∊ _{r}* also can be viewed as independent of

*I*, while

_{d}*∊*is decreased as

_{is}*I*increases.

_{d}#### 5.1. The gain medium with a homogeneously broadened line

In a homogeneously broadened (HB) gain medium, the gain profile has a Lorentzian lineshape [31,49]. And the gain coefficient can be written as,

where*g*represents the gain coefficient in a HB gain medium,

_{L}*f*(

*v*) is a normalized Lorentzian function. At line center

*v*=

*v*

_{0}, the gain coefficient can be simplified as, Combining Eqs. (9)–(11) with Eq. (15), the relative permittivity

*∊*of a HB gain medium at line center can be expressed as,

_{d}Note that, the distribution of the light intensity *I _{d}* in the gain medium is inhomogenous in the Kretschmann geometry owing to the evanescent character of SPP.

*∊*changes gradually with increasing distance from the interface. An effective dielectric constant is proposed here, by utilizing the mean light intensity $\overline{{I}_{d}}$ instead of the gradient light intensity

_{is}*I*in the gain medium.

_{d}Assume that the magnetic field at the metal-dielectric interface is *H _{md}*, then the light intensity at the interface can be depicted by ${I}_{\mathit{md}}=\frac{1}{2}\frac{1}{\text{Re}\left(\sqrt{{\u220a}_{d}}\right){\epsilon}_{0}c}{\left|{H}_{\mathit{md}}\right|}^{2}$, here |

*H*| represents the amplitude of

_{md}*H*. And then the mean light intensity $\overline{{I}_{d}}$ in the gain medium can be approximated as,

_{md}*H*is known, the value of the input intensity

_{md}*I*can be written as,

*I*and the reflectivity

*R*can be constructed with

*H*.

_{md}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 *d _{m}* = 60

*nm*and

*∊*= 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

_{i}*I*= 1

_{s}*MW*/

*cm*

^{2}[50].

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 *θ _{SPP}* = 65.38°. The blue-dashed, red-dotted and black-solid curves in the inset (a) correspond to

*θ*

_{1}= 65.34°,

*θ*

_{2}= 65.30° and

*θ*

_{3}= 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

*θ*, and that the phenomenon of bistable ETR will disappear if the deviation between

_{SPP}*θ*and

*θ*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

_{SPP}*θ*tends to

*θ*. In addition, as

_{SPP}*θ*approaches

*θ*, 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

_{SPP}*θ*>

*θ*, as shown in the inset (b) of Fig. 3.

_{SPP}The thickness of metal film *d _{m}* plays an important role for the excitation of SPPs in the Kretschmann geometry, as shown in Fig. 1. Here, the effect of

*d*on bistable ETR is investigated when

_{m}*θ*= 65.46° and

*∊*= −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

_{i}*d*= 53.9

_{opt}*nm*. The red-dashed, blue-dashdotted, black-dotted and magenta-solid curves in Fig. 4 correspond to

*d*

_{m1}= 58

*nm*,

*d*

_{m2}= 60

*nm*,

*d*

_{m3}= 62

*nm*and

*d*

_{m4}= 64

*nm*, respectively. The red-dashed curve shows that, the phenomenon of bistable ETR will disappear if

*d*is below a certain value, which mainly relies on the incident angle

_{m}*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

*d*increases, both switch thresholds and the width of OB hysteresis will increase, while the peak value of bistable ETR will decrease.

_{m}Figure 5 illustrates the effect of the value of small-signal gain *g*^{0} associated with *∊ _{i}* by Eq. (11) on bistable ETR when

*d*= 60

_{m}*nm*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 (${g}_{1}^{0}=1269{\mathit{cm}}^{-1}$),

*∊*

_{i2}= −0.020 (${g}_{2}^{0}=1410{\mathit{cm}}^{-1}$),

*∊*

_{i3}= −0.022 (${g}_{3}^{0}=1551{\mathit{cm}}^{-1}$) and

*∊*

_{i4}= −0.024 (${g}_{1}^{0}=1692{\mathit{cm}}^{-1}$), respectively. The red-dashed curve shows that, if

*g*

^{0}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

*g*

^{0}increases, but the peak value of bistable ETR is independent of

*g*

^{0}.

#### 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 [31],

here*g*represents the gain coefficient in an inHB gain medium at

_{G}*v*=

*v*

_{0}. Then the relative permittivity

*∊*of an inHB gain medium at line center can be expressed as,

_{d}Figure 6 illustrates the effect of spectral broadening of the gain medium on bistable ETR when *d _{m}* = 56

*nm*,

*∊*= −0.020, and

_{i}*θ*= 65.40°. Here, assume that the saturation intensity in an inHB gain medium is the same as that in a HB gain medium,

*I*= 1

_{s}*MW*/

*cm*

^{2}. 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

*d*and

_{m}*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

*∊*in an inHB gain medium is littler than that in a HB gain medium under the same light intensity $\overline{{I}_{d}}$. At a fixed

_{is}*d*and

_{m}*θ*, the lower (or higher) switch threshold of OB hysteresis corresponds to a certain value of

*∊*. Comparing with a HB gain medium, an inHB gain medium need a higher light intensity $\overline{{I}_{d}}$ 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.

_{is}## 6. Conclusion

In the Kretschmann geometry for SPPs excitation, the dielectric with a variable real part of *∊ _{d}*, such as a nonlinear Kerr medium [36] and a refractive index thermosensitive dielectric [42], contributes to the formation of bistabe ATR; the dielectric with a negative and fixed imaginary part of

*∊*, 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.

_{d}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 *θ _{SPP}*; 2) the thickness of metal film is a little bigger than

*d*; 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 [51] 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.

_{opt}## Funding

National Natural Science Foundation of China (NSFC) (11304274, 11404186, 51661029, 61665014, 21661036); Foundation of Yunnan Educational Committee (2013Y430); Doctoral Scientific Research Foundation of YNNU.

## References and links

**1. **H. Raether, *Surface Plasmons on Smooth and Rough Surfaces and on Gratings* (Springer, 1988). [CrossRef]

**2. **P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. **1**(3), 484–588 (2009). [CrossRef]

**3. **L. Novotny and B. Hecht, *Principles of Nano-Optics* (2nd Ed., Cambridge, 2012), Chap. 12. [CrossRef]

**4. **P. Törmä and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. **78**(1), 013901 (2015). [CrossRef]

**5. **J. Bellessa, C. Bonnand, J. C. Plenet, and J. Mugnier, “Strong Coupling between Surface Plasmons and Excitons in an Organic Semiconductor,” Phys. Rev. Lett. **93**(3), 036404 (2004). [CrossRef] [PubMed]

**6. **A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**(3–4), 131–314 (2005). [CrossRef]

**7. **E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science **311**(5758), 189–193 (2006). [CrossRef] [PubMed]

**8. **N. C. Lindquist, P. Nagpal, K. M. McPeak, D. J. Norris, and Sang-Hyun Oh, “Engineering metallic nanostructures for plasmonics and nanophotonics,” Rep. Prog. Phys. **75**(3), 036501 (2012). [CrossRef] [PubMed]

**9. **S. Hayashi and T. Okamoto, “Plasmonics: visit the past to know the future,” J. Phys. D: Appl. Phys. **45**(43), 433001 (2012). [CrossRef]

**10. **M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nature Photon. **6**(11), 737–748 (2012). [CrossRef]

**11. **V. M. Shalaev and S. Kawata, *Nanophotoncis with surface plasmons* (Elsevier, 2007).

**12. **J. Zhang and L. Zhang, “Nanostructures for surface plasmons,” Adv. Opt. Photon. **4**(2), 157–321 (2012). [CrossRef]

**13. **S. V. Boriskina, T. A. Cooper, L. Zeng, G. Ni, J. K. Tong, Y. Tsurimaki, Y. Huang, L. Meroueh, G. Mahan, and G. Chen, “Losses in plasmonics: from mitigating energy dissipation to embracing loss-enabled functionalities,” Adv. Opt. Photon. **9**(4), 775–826 (2017). [CrossRef]

**14. **M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express **12**(17), 4072–4079 (2004). [CrossRef] [PubMed]

**15. **J. Seidel, S. Grafström, and L. Eng, “Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution,” Phys. Rev. Lett. **94**(17), 177401 (2005). [CrossRef] [PubMed]

**16. **M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Gain assisted surface plasmon polariton in quantum wells structures,” Opt. Express **15**(1), 176–182 (2007). [CrossRef] [PubMed]

**17. **M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express **16**(2), 1385–1392 (2008). [CrossRef] [PubMed]

**18. **I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. **4**(6), 382–387 (2010). [CrossRef]

**19. **P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. **35**(8), 1197–1199 (2010). [CrossRef] [PubMed]

**20. **X. Zhang, Y. Li, T. Li, S. Y. Lee, C. Feng, L. Wang, and T. Mei, “Gain-assisted propagation of surface plasmon polaritons via electrically pumped quantum wells,” Opt. Lett. **35**(18), 3075–3077 (2010). [CrossRef] [PubMed]

**21. **A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-Plasmonic Modulation via Stimulated Emission of Copropagating Surface Plasmon Polaritons on a Substrate with Gain,” Nano Lett. **11**(6), 2231–2235 (2011). [CrossRef] [PubMed]

**22. **P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. **6**(1), 16–24 (2012). [CrossRef]

**23. **T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi Splitting and Strong-Coupling Dynamics for Surface-Plasmon Polaritons and Rhodamine 6G Molecules,” Phys. Rev. Lett. **103**(5), 053602 (2009). [CrossRef] [PubMed]

**24. **F. Valmorra, M. Bröll, S. Schwaiger, N. Welzel, D. Heitmann, and S. Mendach, “Strong coupling between surface plasmon polariton and laser dye rhodamine 800,” Appl. Phys. Lett. **99**(5), 051110 (2011). [CrossRef]

**25. **T. U. Tumkur, G. Zhu, and M. A. Noginov, “Strong coupling of surface plasmon polaritons and ensembles of dye molecules,” Opt. Express **24**(4), 3921–3928 (2016). [CrossRef] [PubMed]

**26. **Y. Chen, J. Li, M. Ren, B. Wang, J. Fu, S. Liu, and Z. Li, “Direct observation of amplified spontaneous emission of surface plasmon polaritons at metal/dielectric interfaces,” Appl. Phys. Lett. **98**(26), 261912 (2011). [CrossRef]

**27. **M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated Emission of Surface Plasmon Polaritons,” Phys. Rev. Lett. **101**(22), 226806 (2008). [CrossRef] [PubMed]

**28. **T. Okamoto, T. Kamiyama, and I. Yamaguchi, “All-optical spatial light modulator with surface plasmon resonance,” Opt. Lett. **18**(18), 1570–1572 (1993). [CrossRef] [PubMed]

**29. **Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. **40**(4), 601–604 (2015). [CrossRef] [PubMed]

**30. **G. A. Plotz, H. J. Simon, and J. M. Tucciarone, “Enhanced total reflection with surface plasmons,” J. Opt. Soc. Am. **69**(3), 419–422 (1979). [CrossRef]

**31. **P. W. Milonni and J. H. Eberly, *Laser Physics* (Wiley, 2010), Chap. 3–4. [CrossRef]

**32. **R. Reinisch and G. Vitrant, “Optical Bistability,” Prog. Quant. Electr. **18**, 1–38 (1994). [CrossRef]

**33. **Z. Zhang, H. Wang, P. Ye, Y. Shen, and X. Fu, “Low-power and broadband optical bistability by excitation of surface plasmons in doped polymer film,” Appl. Opt. **32**(24), 4495–4500 (1993). [CrossRef] [PubMed]

**34. **P. T. Thanh, D. Tanaka, R. Fujimura, Y. Takanishi, and K. Kajikawa, “Low-Power All-Optical Bistable Device of Twisted-Nematic Liquid Crystal Based on Surface Plasmons in a Metal-Insulator-Metal Structure,” Appl. Phys. Express **6**(1), 011701 (2013). [CrossRef]

**35. **X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Scientific Reports **5**, 12271 (2015). [CrossRef] [PubMed]

**36. **G. M. Wysin, H. J. Simon, and R. T. Deck, “Optical bistability with surface plasmons,” Opt. Lett. **6**(1), 30–32 (1981). [CrossRef] [PubMed]

**37. **R. K. Hickernell and D. Sarid, “Optical bistability using prism-coupled, long-range surface plasmons,” J. Opt. Soc. Am. B **3**(8), 1059–1069 (1986). [CrossRef]

**38. **S. D. Gupta and G. S. Agarwal, “Optical bistability with surface plasmons beyond plane waves in a nonlinear dielectric,” J. Opt. Soc. Am. B **3**(2), 236–238 (1986). [CrossRef]

**39. **J. Chen, P. Wang, X. Wang, Y. Lu, R. Zheng, H. Ming, and Q. Zhan, “Optical bistability enhanced by highly localized bulk plasmon polariton modes in subwavelength metal-nonlinear dielectric multilayer structure,” Appl. Phys. Lett. **94**(8), 081117 (2009). [CrossRef]

**40. **F. Zhou, Y. Liu, Z. Li, and Y. Xia, “Analytical model for optical bistability in nonlinear metal nano-antennae involving Kerr materials,” Opt. Express **18**(13), 13337–13344 (2010). [CrossRef] [PubMed]

**41. **C. Argyropoulos, C. Ciracì, and D. R. Smith, “Enhanced optical bistability with film-coupled plasmonic nanocubes,” Appl. Phys. Lett. **104**(6), 063108 (2014). [CrossRef]

**42. **P. Martinot, A. Koster, and S. Laval, “Experimental Observation of Optical Bistability by Excitation of a Surface Plasmon Wave,” IEEE J. Quantum Electron. **21**(8), 1140–1143 (1985). [CrossRef]

**43. **G. A. Wurtz, R. Pollard, and A.V. Zayats, “Optical Bistability in Nonlinear Surface-Plasmon Polaritonic Crystals,” Phys. Rev. Lett. **97**(5), 057402 (2006). [CrossRef] [PubMed]

**44. **H. Zhou, X. Chen, P. Hou, and C. Li, “Giant bistable lateral shift owing to surface plasmon excitation in Kretschmann configuration with a Kerr nonlinear dielectric,” Opt. Lett. **33**(11), 1249–1251 (2008). [CrossRef] [PubMed]

**45. **E. Kretschmann, “The Determination of the Optical Constant of Metals by Excitation of Surface Plasmons,” Z. Physik **241**, 313–324 (1971). [CrossRef]

**46. **M. A. Parker, *Physics of Optoelectronics* (CRC, 2005), Chap. 7. [CrossRef]

**47. **D. Ahn and S. L. Chuang, “High optical gain of I–VII semiconductor quantum wells for efficient light-emitting devices,” Appl. Phys. Lett. **102**(12), 121114 (2013). [CrossRef]

**48. **C. Tzschaschel, M. Sudzius, A. Mischok, H. Fröb, and K. Leo, “Net gain in small mode volume organic microcavities,” Appl. Phys. Lett. **108**(2), 023304 (2016). [CrossRef]

**49. **K. F. Renk, *Basics of Laser Physics: For Students of Science and Engineering* (Springer, 2012), Chap. 4. [CrossRef]

**50. **N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. **85**(21), 5040–5042 (2004) [CrossRef]

**51. **A. V. Malyshev, “Condition for resonant optical bistability,” Phys. Rev. A **86**(6), 065804 (2012). [CrossRef]