## Abstract

We develop a semiclassical theory of passively mode-locked surface plasmon polariton (SPP) lasers based on a SPP Bragg resonator with a metal film deposited on a polymer host and adjacent layers of a slow saturable absorber and a slow saturable gain medium. The mode-locked laser dynamics is studied for the case that both the gain medium and the saturable absorber are solid-state dyes. The SPP laser pulse parameters are calculated in dependence on layer thicknesses of the metal film and pump parameters. We predict the possibility of SPP pulse generation with ∼ 100 fs pulse duration.

© 2011 OSA

## 1. Introduction

Surface plasmon polaritons (SPPs) allow the localization of light below the diffraction limit and promise progress in miniaturization of sophisticated compact optical devices with new functionalities. One of the main limitation for applications of SPPs is its short propagation length which is typically in the range of 30 ∼ 200 *μ*m due to the high SPP loss. Introducing gain to a dielectric adjacent of the metallic film has driven recent research to examine stimulated emission [1–6] and amplification of spontaneous emission [7] of SPPs. The next step in this development was the realization of lasers on the nanoscale by appropriate feedback [8–10]. The smallest laser reported to date has been achieved [10] by the realization of a concept developed by Bergman and Stockman [11, 12] by using a single gold core and dye doped silica shell structure. Additional theoretical studies of spasers [13–20] and SPP amplification [21–24] have been published. Especially, the possibility of ultrafast plasmon amplification has been proposed in Ref. [20].

The main purpose of the present paper is to investigate the possibility of ultrashort surface plasmon pulse generation which may find applications for ultrafast plasmonic applications including ultrafast surface spectroscopy, ultrafast surface nonlinear optics, highly integrated plasmonic information processors and others. We theoretically study, for the first time to our knowledge, passive mode-locking in SPP lasers by slow saturable absorbers and predict the possibility of femtosecond plasmon pulse generation in such lasers.

We consider a configuration (Fig. 1) with a dielectric layer *c* composed of a gain sublayer *g* and a sublayer with a saturable absorber *sa* adjacent to the metal film *b*. At the interface of the flat continuous metal film *b* SPPs can be excited which are confined to the proximity of the metal-dielectric interface and decay exponentially in both media. Optical pumping from below the dielectric layer *c* leads to a population inversion in the gain medium. Feedback in this scheme is realized by using a SPP grating at both end sides of the metal layer [25]. Such a grating with an appropriate period coherently couples the forward-propagating plasmon mode to the backward-propagating mode providing efficient reflection as well as laser outcoupling to the traveling waves. Passive mode-locking can be achieved by the combined action of slow saturable gain and slow saturable absorption in one round trip. Such a mode-locking mechanism is realized in standard dye lasers [26–28] where both media exhibit different saturation intensities and different focusing. In a SPP laser considered here, the saturation in the absorber layer *sa* is stronger than in the gain sublayer *g* because of the stronger field localization of the SPP modes as shown in Fig. 1. Everywhere in the paper below, we denote the quantities related to gain by subindex *g* and for saturable absorber by subindex *q*.

## 2. Surface plasmon polariton laser equation

In this section, we study the evolution of longitudinal SPP modes. First, we consider a passive SPP resonator. The dispersion relation for propagating SPPs is given by (see Ref. [29–31])

*d*is the sub-wavelength thickness of the metal film,

*ɛ*(

_{a}*ω*),

*ɛ*(

_{b}*ω*) and

*ɛ*(

_{c}*ω*) are the frequency-depending dielectric functions of layers

*a*,

*b*and

*c*taken from Ref. [32]. The constants

*α*,

_{a}*α*and

_{b}*α*are directly related with the wavenumber

_{c}*K*of propagating SPP by ${\alpha}_{j}^{2}={K}^{2}-{\varepsilon}_{j}{\omega}^{2}/{c}^{2}$, where

*j*=

*a*,

*b*and

*c*,

*ω*is the angular frequency of light and

*c*is the light velocity in vacuum. For the passive SPP resonator in Fig. 1, the counterpropagating fields interfere resulting in intracavity modes described as

**E**

*=*

_{n}**U**

_{n}e^{−(γn+iΩn)t}, where indices

*n*indicate longitudinal mode numbers and the mode function

**U**

*(*

_{m}*x, z*) is given by

*A*(

_{j}*j*=

*a*,

*b*and

*c*) are field amplitudes for each layers,

*x̂*and

*ẑ*are unit vectors along the x and z axes, respectively,

*K*=

_{n}*n*

_{eff,}

*Ω*

_{n}*/*

_{n}*c*,

*n*

_{eff,}

*is the linear mode index and Ω*

_{n}*is the angular frequency of the n-th longitudinal mode of the passive SPP resonator. From the field equations, we can see that SPP polarization direction is altered from normal to parallel at the interface in a half spatial period of the longitudinal interferometric fringe of each mode. Nevertheless, normal components are dominant in layers*

_{n}*a*and

*c*because |

*α*/

_{jn}*K*| ≪ 1,

_{n}*j*=

*a*, and

*c*. The relations between the field amplitudes are defined by boundary condition for the electromagnetic field and are given by

*K*, the field attenuation factors in x-direction

_{n}*α*and the resonator loss

_{jn}*γ*obey the following relations:

_{n}*j*=

*a*,

*b*and

*c*. In fact, additional loss in the resonator arises due to the imperfect coupling and reflectance of the resonator mirrors which is included into the parameter

*γ*. The additional modification for modal structure and loss is related to the finite width of the waveguide in

_{n}*y*direction. For a waveguide thickness much larger than

*d*but smaller than a few

*μ*m there exist only one mode in this direction and its properties can be calculated using the effective index approach.

Next, we consider the evolution of modes of an active SPP resonator with real wavenumbers and complex frequencies. The fields in layers *a* and *b* of the SPP resonator can be calculated from fields of the layer *c* by using the relations for the field amplitudes given by Eqs. (3, 4). The field in layer *c* can be presented as follows

*ω*is the angular frequency of n-th mode in the active resonator and we assume that

_{n}*ω*≈ Ω

_{n}*. The induced polarization is described by the same expression replacing the field amplitude*

_{n}*A*(

_{n}*t*) with

*P*(

_{n}*t*).

Substituting the above equation into Maxwell’s equations for TM waves and using the slowly varying envelope approximation (SVEA) and the rotating-wave approximation for the induced polarization [33], we obtain the master equation for the SPP laser

*ρ*is electric dipole moment of the gain, 𝒟

_{g}*= [Γ*

_{n}*+*

_{g}*i*(

*ω*–

_{n}*ω*)]

_{L}^{−1}, ${T}_{2g}={\Gamma}_{g}^{-1}$ is the dephasing time, ${\sigma}_{g}={\rho}_{g}^{2}{\omega}_{n}/\left(2{\varepsilon}_{0}{\varepsilon}_{c}\overline{h}{\Gamma}_{g}\right)$ is the gain cross-section,

*N*=

_{g}*N*(

_{g}*x, z,t*) is a space-time dependent population inversion, ${M}_{n}={\int}_{0}^{L}{\int}_{-\infty}^{\infty}dzdx{\left|{\mathbf{U}}_{n}\left(x,z\right)\right|}^{2}\text{Re}\left[n\left(x\right)\right]$ is a normalization factor,

*n*(

*x*,

*ω*) is the refractive index different at each layer,

_{n}*L*is the length of the SPP resonator,

*ɛ*

_{0}is the vacuum permittivity, and

*h̄*is Plank constant. In the above equation, the right hand determines the field source generated from the gain polarization induced by pumping.

We describe the right hand of Eq. (7) as *g _{n}A_{n}*, where

*g*is the transient nonlinear mode gain. Using Eq. (2) we obtain

_{n}*κ*= 2|Re[

*α*(

_{c}*ω*)]|,

_{L}*ω*is the lasing frequency, ${\overline{N}}_{g}\left(x,t\right)={\int}_{0}^{L}{N}_{g}\left(x,z,t\right)dz/L$. The analogous procedure can be applied for the slow saturable absorber which yields similar expressions as above. For the study of mode-locking, below we consider the modes as a continuum by replacing

_{L}*ω*,

_{n}*K*,

_{n}*γ*and

_{n}*n*

_{eff,n}with

*ω*,

*K*,

*γ*, and

*n*

_{eff}, respectively.

In difference to bulk mode-locked lasers, in the case of SPP lasers the mode field is confined in the vicinity of the metallic layer. Hence, the strength of gain and absorption saturation also depend on the position *x* because the SPP mode intensity is higher at the position nearer to the metal surface. In addition, the pump intensity distribution is modulated in space due to the absorption of the pump in the gain sublayer and standing wave formation by the reflection from the metal film. Therefore we can not simply apply the relations or the master equations for passive mode-locking with saturable absorbers in bulk lasers, in which all the above given parameters do not depend on the transverse spatial coordinate [27, 28].

## 3. Master equation for mode-locked SPP lasers

The sum in Eq. (6) is replaced by the forward propagating field *A*(*T*,*τ*) = ∫*A*(*t*,*k*)*e*^{−iδωt+ikz}*dk*, where *k* = *K* – *K _{L}* = (

*n*

_{eff}/

*c*)(

*ω*–

*ω*),

_{L}*δω*=

*ω*–

*ω*, and

_{L}*A*(

*t*,

*k*) is the continuous form of the mode fields

*A*(

_{n}*t*). We apply a coordinate transformation

*T*=

*t*,

*τ*=

*t*–

*z/v*, where

_{g}*T*is the laboratory time,

*τ*

*is*the local time,

*γ*(

*ω*)

*is*the frequency-dependent loss,

*T*

_{R}*is*the round trip time, and

*v*

_{g}*is*the group velocity of the SPPs. Here the subscripts

*i*for gain and loss represent the corresponding values just before the pulse. From Eqs. (7,8), we obtain the following master equation of mode-locked SPP lasers

*g*(

*τ*),

*q*(

*τ*) and

*γ*

_{0}are the total gain, the absorber loss and the resonator loss for a round trip,

*γ*

_{0}=

*γ*|

_{k}_{=0}, ${\delta}_{1}={-i{v}_{g}^{-1}\left(\partial \gamma /\partial k\right)|}_{k=0}+{g}_{i}{\Gamma}_{g}^{-1}-{q}_{i}{\Gamma}_{q}^{-1}$ and ${\delta}_{2}={g}_{i}{\Gamma}_{g}^{-2}-{q}_{i}{\Gamma}_{q}^{-2}+{\left(2{v}_{g}^{2}\right)}^{-1}{\left({\partial}^{2}\left[i\omega \left(k\right)\left({T}_{R}+{g}_{i}{\Gamma}_{g}^{-1}-{q}_{i}{\Gamma}_{q}^{-1}\right)+\gamma \left(k\right)\right]/\partial {k}^{2}\right)|}_{k=0}$, respectively. The detailed derivation can be found in the Appendix. Here the dispersion of the plasmonic modes is explicitly taken into account by the function

*ω*(

*k*) defined by the inverse function of

*k*= (

*n*

_{eff}/

*c*)(

*ω*–

*ω*). The dependence of

_{L}*δ*

_{1}and

*δ*

_{1}on the intensity can be neglected. Both the gain and loss in the above equation are dimensionless quantities corresponding to those per one resonator round trip.

For stable mode-locked operation, *g _{i}* must be partly recovered during one round trip. The linear parts of gain and loss rates are represented by
${g}_{i}={T}_{R}\beta {\int}_{-{D}_{g}-{D}_{q}-d}^{-{D}_{q}-d}{g}_{l}{e}^{\kappa \left(x+d\right)}dx$ and
${q}_{i}={T}_{R}\beta {\int}_{-{D}_{q}-d}^{-d}{q}_{l}{e}^{\kappa \left(x+d\right)}dx$, respectively, and

*β*is defined below Eq. (8) with the substitution

*ω*→

_{n}*ω*,

_{L}*g*and

_{l}*q*are the nonlinear local gain and loss rates dependent on the spatial and temporal variables,

_{l}*γ*(

*k*) is the resonator loss per round trip. The evolution of nonlinear local gain

*g*(

_{l}*x*,

*τ*) and

*q*(

_{l}*x*,

*τ*) are given by the equations [27]

*τ*

_{0g}and

*τ*

_{0q}are the upper-level lifetimes,

*A*and

_{sg}*A*are the saturation fields for gain and absorber dyes, respectively.

_{sg}The SPP field intensity distribution is nonuniform and the saturation at the position nearer to metal surface is stronger [20]. We do not use a power expansion for the gain with respect to the intensity [27] but self-consistently solve the combined Eqs. (9–11). For slow passive mode-locking the gain and absorber dyes exhibit a longer relaxation time compared with the pulse duration: *τ*_{0q}, *τ*_{0g} < *τ*_{0}. Besides, stable mode-locking operation is possible only if the conditions 0.1 ≤ *T _{R}*/

*τ*

_{0g}≤ 10 and

*τ*

_{0q}<

*T*are fulfilled [26–28, 34]. The master equation for mode-locking of SPP lasers Eq. (9) can not be analytically solved because both the local gain and nonlinear absorption Eq. (10) depend on time and space, therefore we apply the split-step-Fourier method [35] to solve Eq. (9).

_{R}## 4. Design of long range SPP lasers

In this section, we discuss parameters for appropriate gain and absorber medium and determine the main structural parameters supporting lasing of SPPs.

The dielectric layers *a* and *c* are assumed to be made of PMMA and we consider a metallic layer *b* made from silver. The permittivities of each layer are *ɛ _{a}* (

*λ*) =

_{L}*ɛ*(

_{c}*λ*) = 2.20 (PMMA) and

_{L}*ɛ*(

_{b}*λ*) = −35.99+ 2.20

_{L}*i*(silver), respectively [32]. We restrict ourselves to the symmetric SPP mode [36] which has the smallest propagation loss.

We assume that the gain and the saturable absorber sublayers are doped with dyes Styryl-9 (Ref. [37]) and IR-26 (Refs. [38–40]) in a PMMA polymer host. Pumping and lasing wavelengths are taken to be 532 nm and 900 nm, respectively. The main parameters of the dyes are as follows: For the gain medium using Styryl-9, cross-sections for stimulated emission at 900 nm and absorption at 532 nm are *σ _{s}* = 1.8 × 10

^{−16}cm

^{2}and

*σ*= 1.2 × 10

_{ag}^{−16}cm

^{2}, respectively, the upper-state relaxation time (or longitudinal relaxation time) is

*τ*

_{0g}= 400 ps and the dephasing rate is Γ

*= 3.3 × 10*

_{g}^{13}Hz (see Ref. [37]). For saturable absorber molecules IR-26, absorption cross-section

*σ*= 1.5 × 10

_{aq}^{−16}cm

^{2}(Ref. [38]), upper-state relaxation time (or absorber recovery time) is

*τ*

_{0q}= 22 ps (Ref. [39, 40]) and dephasing rate is Γ

*= 2.5 × 10*

_{q}^{13}Hz. All the SPP laser parameters can be calculated from the above quantities based on the formulas given in the last section. The concentrations of gain and saturable absorber molecules were taken to be 2.5 × 10

^{18}cm

^{−3}and 1 × 10

^{17}cm

^{−3}, respectively.

Under the condition of CW pump operation, a severe problem in the use of dyes is the long-lived transient triplet-triplet absorption that gradually reduce the net gain and ultimately terminates the lasing process. In liquid dye lasers [41, 42], this problem is solved by free-flying dye jets [43] and in solid dye lasers by a rotating disc [44–46]. Here we consider as an alternative a solid-state dye gain medium that is optically pumped with pulsed light sources with ns duration.

We choose a length of the SPP laser resonator of *L* = 1 cm and consider only the case of single mode guiding. From the model in section 2, we can see that the upper limit of the SPP waveguide width for single mode guiding is ∼ *W* = 2 *μ*m for a thickness 30 nm of the Ag layer *b* [29–31]. Here we take it to be 1.8 *μ*m and obtain for the effective index and propagation loss *n*_{eff} = 1.4988 and *γ*_{0} = 44.78, respectively. For the structure described above, we find *δ*_{1} = (2.9 – 3.0*i*) × 10^{−12} s and *δ*_{2} = (8.2 – 1.4*i*) × 10^{−26} s^{2}.

Below we determine the appropriate range of thicknesses for the saturable absorber sublayer and metal layer *b*. In Fig. 2 we show the SPP intensity profile (a) and the gain and loss quantities (b) for different metal layer thicknesses. In Fig. 2(a) we can see that the effective intensity confinement width is 227.8 nm (FWHM). This means that we must take the thickness of the absorber layer to be larger than this value if the saturation intensities for gain and absorber are nearly the same. Therefore, we take it to be 400 nm. In Fig. 2(b) the linear resonator loss and the unsaturated gain are shown in dependence on the film thickness *d* for pump intensities of 5, 10 and 15 MW/cm^{2}. With increasing metal film thickness the field energy becomes more concentrated towards the metal film. Therefore, we can expect that there is an upper limit of the metal film thickness for lasing. For a Ag film thicker than *d* ∼ 40 nm at a pump intensity 10 MW/cm^{2} for *D _{q}* = 400 nm and

*D*= 5

_{g}*μ*m, the resonator loss is greater than the linear gain. Taking into account this fact, we choose the thickness of the metal film as

*d*= 30 nm for the calculations below.

In the final part of the section, we consider the influence of the inhomogeneity of the pump. The intensity of the pump beam becomes inhomogeneous [see Fig. 3(a)] due to the absorption by the gain molecules and the formation of standing waves by the reflection from the metal film surface. The spatial modulation of the pump beam is calculated by using the matrix formulation [22,23,47] and is taken into account in all simulations for the mode-locking behavior. The pump absorption in the saturable absorber layer, for this case, is negligible (see [38]).

In addition the shortening of *τ*_{0g} due to fluorescence quenching of the gain molecules by the dipole energy transfer to the metal layer [48,49], is calculated by using the method represented in Refs. [22,23,49] [see Fig. 3(b)]. This effect is very weak because the gain sublayer is located 400 nm far from the metal film and the quantum yield of Styryl-9 molecules is relatively small (0.05, see Ref. [37]). As for the saturable absorber, the decrease of the upper level lifetime responsible for fluorescence increases the ground state recovery time and makes the absorption to be recovered faster. However, the calculation shows that this effect is valid only in a narrow region [below ∼ 10 nm from the metal surface, see Fig. 3(b)] because the fluorescence quantum yield of IR-26 dye is as small as ∼ 10^{−3} (Refs. [38–40]). Taking into account the very weak lifetime shortening, we neglected this effect for both gain and absorber.

## 5. Numerical results and discussion

In Fig. 4 we show an example of our numerical study of the pulse evolution in mode-locked long range SPP lasers. Figure 4 shows the evolutions of gain and total loss (a), and the generated pulse profile at the position adjacent to the metal film in the dielectric layers (b). For self-starting stable mode-locking, three conditions have to be satisfied [26]: absorption saturation has to be stronger than gain saturation, the unsaturated net gain has to take a positive value and the net gain has to be negative before and after the pulse. As mentioned above, for design parameters as discussed in section IV the absorber is stronger saturated than the gain because of the high confinement of the plasmon field. The unsaturated net gain per round trip is positive with a value of 59.6 (the contribution of the linear loss in the saturable absorber to this value is 15.6). However, in the steady state of pulse formation, net gain before and after the pulse is negative and its value just before the pulse is about −2.02 [see Fig. 4(a)]. The Figure 4(b) illustrates the dynamics of ultrashort pulse formation in the considered SPP laser: even though the response time of the gain and absorber medium is longer than the pulse duration in this regime a stable fs pulse regime is established because noise at the wings of the pulse is suppressed due to the negative net gain in this region. In the initial stage of pulse formation the evolving pulse is shortened due to the different saturation behavior of the gain and the absorber because the absorber recovery is faster than that of the gain. This general scenario of passive mode-locking is analogous as in bulk femtosecond dye lasers [26, 28, 34]. In the example given in Fig. 4 for a SPP laser with a pump intensity of 10.26 MW/cm^{2}, a stable pulse train is formed with ∼ 20 mJ/cm^{2} maximum pulse fluence per pulse [at the positions adjacent to metal in the layers a and c, see Fig. 2 (a)]. In this case the pulse duration is 128.15 fs and the maximum peak intensity in the dielectric layers is *I*_{max} = 143 GW/cm^{2}.

Note that from Fig. 4(b) we can see that for the considered SPP laser parameters both the gain and the loss of the absorber are high and both are strongly saturated. This means that a simplified approach based on the assumption of small net gain and weak saturation [27] can not be applied for this case.

In Fig. 5 we show the dependence of the output pulse fluence and the pulse duration on the pump intensity. Figure 5 shows that the pulse fluence linearly increases and the pulse duration decreases in a nonlinear way with increasing pump intensity. The limits of shortest pulse duration and largest output fluence is set by damage threshold at roughly 1 J/cm^{2} as well as by available pump sources.

## 6. Conclusion

In this paper we studied the possibility of femtosecond plasmon pulse generation by mode-locking of long range SPP lasers. We developed a theory of passive mode-locking of SPP lasers with a Bragg resonator consisting of a silver film, a saturable absorber layer adjacent to the metal film, and a gain medium. For the specific example of solid dyes acting both as slow saturable gain medium and a slow saturable absorber the characteristic laser pulse parameters are calculated numerically. The results show that SPP femtosecond pulses with maximum peak intensity in the range of ∼ 140 GW/cm^{2} and shortest duration down to ∼ 130 fs can be generated. We believe that mode-locked long range SPP lasers can find a variety of ultrafast plasmonic applications.

## 7. Appendix: Derivation of Eq. (9)

The SPP field in laser resonator given by Eq. (6) can be written by a sum of forward and backward waves:

where*K*is the wavenumber for the central lasing frequency

_{L}*ω*. We neglect the discrete spectral structure and consider the SPP field as a continuum: where

_{L}*δω*=

*ω*–

*ω*, and

_{L}*A*(

*t, k*) is the continuous form of the mode fields

*A*(

_{n}*t*) (slowly varying envelope).

From Eq. (A-5),

*δω*can be expanded as follows:

*γ*(

*k*) and

*g*(

*t*,

*k*) are the continuous form of

*γ*and

_{n}*g*(

_{n}*t*). The passive resonator loss

*γ*(

*k*) can be expanded as follows:

*β*is a value of

*β*given at

_{n}*ω*. From Eq. (A-11),

_{L}*T*on the both sides:

_{R}*g*′ =

*gT*,

_{R}*q*′ =

*qT*, and

_{R}*γ*′ =

*γT*. In the above equations the parameters for saturable absorber are defined in the same way as that for the gain. Now we introduce the coordinate transformation

_{R}*T*=

*t*, $\tau =t-{v}_{g}^{-1}z$, where

*T*is the laboratory time,

*τ*is the local time. In addition, from the property of the Fourier transformation, the terms containing the powers of

*v*can be changed by the derivative for local time

_{g}k*τ*and we obtain Eq. (9).

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