The concept of spaser as the coherent near-field generator and nanolaser based on nanoscale plasmonic resonators has been successfully demonstrated in number of experiments. Here we have developed the theoretical framework for the basic linewidth description of these active plasmonic structures and, in particular, linewidth enhancement – additional line broadening due to the resonator noise. In order to achieve this, we have introduced explicitly the time dependence in the quasistatic description of localized surface plasmon resonances via inclusion of the dispersion of a spectral parameter defining the resonant frequency. Linewidth enhancement factor was estimated for semiconductor gain medium and was found to be of order of 3 to 6, strongly depending on carrier density in the active layer, and resulting in more than order of magnitude broader linewidth compared to that, predicted by the Schawlow-Townes theory.
©2013 Optical Society of America
Coherent light and surface plasmon generation on the nanoscale has recently attracted significant attention due to numerous potential applications in the fields ranging from biosensing to optical interconnects. The physical dimensions of a laser are defined by its cavity which generates optical feedback based on constructive interference of propagating waves. Since the interference requires phase accumulation of at least 2π, the resonator dimensions cannot be much smaller than half a wavelength in each direction and, in fact, are limited by the classical diffraction. The smallest lasers approaching this limit are VCSELs .
Conceptually different approaches are required for creation of sub-diffraction coherent light generating systems. One of the promising solutions is based on incorporation of nanometric structures made of noble metals supporting plasmonic excitations . Such structures can confine light far-beyond the natural limit of diffraction by coupling it into coherent oscillations of free conduction band electrons near metal surfaces . The plasmonic effect can confine optical mode in one or two dimensions , while the facet reflections in third dimension can be used to generate sufficient feedback for lasing. Several concepts of surface plasmon polariton lasers have been recently demonstrated [5, 6]. The third dimension in laser cavities may be further reduced by employing the idea of negative phase accumulation on the reflection from negative permittivity objects, proposed in  and recently demonstrated in coaxial plasmonic cavities .
Ideologically new approach to laser resonators was proposed by D. J. Bergman and M. I. Stockman , who suggested replacing the interference phenomenon by the near-field feedback, introducing the concept of spaser as a coherent source of plasmonic excitations. In a case of efficient coupling of localized surface plasmons (LSPs) in far-field photons, the spaser can be considered as a nanolaser. This system was successfully demonstrated, employing a core metal nanoparticle and surrounding active dyes .
One of the most basic properties of a laser is its linewidth which represents the measure of temporal coherence of the output beam. The distinctive property is the narrowing of the output spectrum when the laser passes from the spontaneous emission regime through a threshold to the predominant stimulated emission and lasing. The Schawlow-Townes theory predicts the linewidth to be proportional to the square of the resonator bandwidth divided by the output power . However, almost all realistic laser systems, in particular those based on semiconductor structures, cannot reach this limit due to additional noise in a resonator caused by inhomogeneities, mechanical instability and other factors. The important parameter, especially in semiconductor lasers, is a linewidth enhancement (LWE) factor (α) resulting from fluctuations of the refractive index of the laser cavity . The actual line width is α2 + 1 times larger than the predicted by the Schawlow-Townes theory.
The basic theoretical description of nanolasers and spasers is based on the notion of quasistatic localised surface plasmon (LSP) resonances. Unfortunately, the rigorous problem of their emission linewidth cannot be considered with the same formalism as for conventional lasers which involves time-dependent treatment of the laser cavity and absent in quasistatic approximation. The quasistatic description of LSP resonances defining the cavity of a spaser or nanolaser does not include the radiation losses, at the same time, material losses are predominant for particle diameter smaller than 20 nm ; moreover, the quasistatic approach does not explicitly include the time derivatives.
Here we present a theoretical framework to treat the linewidth and, in particular, its enhancement in nanolasers and spasers (Fig. 1 ). In order to investigate the linewidth properties, we have introduced the time dependence in the quasistatic treatment via the dispersion of spectral parameter which defines the LSP frequency. The LWE was estimated for bulk GaAs material as an active medium of a spaser to be of order of 3-6, strongly depending on carrier density in the active layer.
2. Theoretical description
The basic rate equations describing spaser action have been derived in . The description of a plasmonic mode in the quasistatic regime used for this purpose is given by:14], in line with the Schawlow-Townes theory, shows that the linewidth in the stimulated emission regime is inversely proportional to the energy stored in the resonator (i.e., number of surface plasmons,):. This treatment, however, neglects the effect of the cavity noise, originating from the refractive index fluctuations in the active media which leads to LWE and the line broadening.
Equation (1) does not include time evolution of the field in an explicit way, and this is its fundamental difference from the wave equation, which is used in the description of conventional lasers. In order to introduce the times dependence in the nanolaser action description, we will introduce the time derivatives using the dispersion properties of the material permittivities present in Eq. (1). Note that both real and imaginary parts of permettivities of metal and active dielectric have to be considered dispersive in order to satisfy Kramers-Kronig relations.
We consider a single cavity mode to be of the following form in a time domain representation:Eq. (1), in vicinity of the resonance, we obtain:Eq. (1) defines the actual location of the LSP resonance (via the material dispersion) and the eigen vector of Eq. (1) corresponds to the spatial distribution of the mode field. The term with angular frequency multiplierin Eq. (3) introduces the nontrivial time-dependence in Eq. (1), manifesting the significance of the spectral parameter dispersion. Consideration of higher order terms in the expansion (Eq. (3)) will lead to higher order time derivatives and, as the result, more complicated stochastic differential equations, as will be clear after Eq. (6). However, the dispersion of spectral parameter is rather smooth, since the material responses are not sharp functions of frequency. Similar approach of called ‘Dispersion relation method’ is used to obtain dynamical equations, introducing additional term of the resonance wavelength shift [15,16]. Here, however, this correction could be neglected, since the LPR is initially very broadband. Substituting the Fourier transform of the Eq. (2) (the frequency domain expression of the spaser mode) into Eq. (1) and using Eq. (3), we arrive back to the time domain description by applying the inverse Fourier transform:Eq. (4) in the following form, considering the gain fluctuations outside the metallic resonator, where :
Using the conventional relations between mode amplitude, intensityand phase, namely, and, the resulting set of differential equations, describing a spaser operation, can be obtained:
These equations, derived for a spaser, have similar structure to those, describing conventional lasers , but with the coefficients specific for any fluctuations in the spaser cavity.
Equations (6) may be integrated, and the phase may be shown to have a Gaussian probability distribution, which determines the spaser linewidth via the phase-dependent autocorrelation function . From the autocorrelation, the coherence time can be determined by. The spaser linewidth can then be found as. The total linewidth taking into account fluctuations in the cavity can be related to the Schawlow-Townes quantity as:12]. Specific values of may be derived from the gain characteristics by applying the Kramers-Kronig relations to the pump-dependent permittivity of an active region , evaluating its real part, or directly calculating both components of this complex quantity .Eq. (8), if we assume the linear dependence between the carriers’ density and power (number of surface plasmons) in the nano-resonator. It should be noted that Eqs. (1) and (5) describe only spasing/nanolasing at the threshold conditions, but the estimated LWE value is approximately valid also above the threshold; the intuitive explanation of this is the fast carrier population clamping which prevents the increase of the carrier density fluctuations in the gain material above the threshold.
To estimate the influence of the LWE on spaser operation, we have investigated the simplest model where bulk GaAs was taken to be the active medium and plasmonic resonator has been considered to be silver sphere ( for material parameters. In particular, εsilver ≈-40 + i0.5). The bandgap of the active layer dictates the resonant frequency of the spaser and the LSP resonance of the metal nanostructure should be tuned to this frequency. Fortunately, plasmonic resonances may be engineered to cover entire visible and infrared frequencies using approaches such as particle-particle coupling , particle elongation , concavity tuning  or ultimate evolutionary methods . The gain characteristics were estimated using free-carrier theory ; many-body effects, such as bandgap shrinkage, electron-electron scattering, and lineshape broadening may be included by more advanced modeling . Here, we used the former approach, taking the material parameters of GaAs  (In particular, Egap = 1.424 eV, nbackground ≈3.65) and neglecting the minor free carries contribution to the index change .
Figure 2(a) represents the spaser central wavelength as the function of injected carrier density above the threshold. It may be seen that the increase of the carrier injection shifts the central wavelength in the short wavelength range where the gain has a maximum value, similar to the experimental observations in . It should be noted that conventional lasers also exhibit short-wavelength shift of the emission peak which, however, corresponds to the decrease of the refractive index of the cavity material and, thus, the shorter wavelength in order to conserve the 2π phase accumulation per round trip .
Figure 2(b) shows the LWE as the function of injected carrier density taking into account the shift of the spasing wavelength as discussed above. The figure shows the characteristic decrease of LWE with the carrier density at low injection currents followed by a small increase and saturation with the increase of carrier density. Generally, this behavior is determined by several competing effects, the simplified intuitive explanation may involve the reduction of to the relative carrier’s noise with the increase of the overall carriers’ density, similarly to the situation in bulk semiconductors . The further increase to the carrier’s densities in the case of spaser leads to refractive index changes in the surrounding of the resonator resulting in the shift of the emission line wavelength and, subsequently, leading to a more complicated LWE behavior. The data were recast to show the spaser linewidth dependence ( (Fig. 2(c)). The carrier density () dependent gain profile of semiconductors and dispersive nature of LSP resonances result in the dispersion of the LWE, and, as the result, the deviation of the spaser linewidth from the classical ‘one over power’ proportionality. The spaser linewidth exhibits monotonic narrowing with the injection current, similarly to the recently reported results . Moreover, the shape of the curve on Fig. 2(c) has similar behavior to the linewidth, reported at .
The distinctive difference between plasmonic-based lasers and conventional lasers is that the quality factors of the resonators used in the former case, metal particles, are generally of order of 10-100 resulting in the very broadband spectral response (Fig. 3 ). Typical quality factors of the resonators of high-quality semiconductor laser is over 1000 with and much higher for other lasers . The mode selection in classical laser system results from the interplay of sharp-resonant cavity modes competing for the gain, while the spaser operation frequency is determined by the material gain properties. Therefore, we assumed that the LSP resonance is relatively flat in the vicinity of the gain line (Fig. 3). The gain bandwidth in bulk semiconductors could exceed 100nm and reach values of ~1000 cm−1. Complex plasmonic structures may, however, support number of the so-called dark modes obeying non-exponential decay laws  with much shaper resonances and higher quality factors, governed mainly by material loss; these will be of main importance for spasers, while conventional bright LSP modes are important for plasmonic-based nanolasers. The difference in LWE between dark and bright modes (if the far field radiation is neglected) will enter via the dispersion of the spectral parameter and will depend on specific spaser realization.
We have developed the theory to introduce the time dependence in the description of spaser action through the dispersion of the spectral parameter, describing the localized surface plasmon spectrum. This has allowed investigations of the spaser and plasmonic nanolaser linewidth properties. The developed theory predicts significant linewidth enhancement with an order of magnitude broadening of the output spectrum in the case of the active medium based on bulk GaAs, with the deviation of the spaser linewidth from the classical ‘one over spasing power’ proportionality. The developed theory could be of potential interest to rapidly developing nano-plasmonic devises with incorporated gain .
This work has been supported by EPSRC (UK). P. Ginzburg acknowledges The Royal Society for a Newton International Fellowship and Yad Hanadiv for Rothschild Fellowship.
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