A theoretical analysis of noise in high-gain surface plasmon-polariton amplifiers incorporating dipolar gain media is presented. An expression for the noise figure is obtained in terms of the spontaneous emission rate into the amplified surface plasmon-polariton taking into account the different energy decay channels experienced by dipoles in close proximity to the metallic surface. Two amplifier structures are examined: a single-interface between a metal and a gain medium and a thin metal film bounded by identical gain media on both sides. A realistic configuration is considered where the surface plasmon-polariton undergoing amplification has a Gaussian field profile in the plane of the metal and paraxial propagation along the amplifier’s length. The noise figure of these plasmonic amplifiers is studied considering three prototypical gain media with different permittivities. It is shown that the noise figure exhibits a strong dependance on the real part of the permittivities of the metal and gain medium, and that its minimum value is . The origin of this minimum value is discussed. It is also shown that amplifier configurations supporting strongly confined surface plasmon-polaritons suffer from a large noise figure, which follows from an enhanced spontaneous emission rate due to the Purcell effect.
© 2011 OSA
At optical frequencies, a simple metallic surface bounded by a dielectric can support transverse-magnetic polarized optical surface waves coupled to free electron oscillations called surface plasmon-polaritons (SPPs) . More complex metallodielectric structures can also support SPPs, including thin metal films and stripes , metallic nanoparticles, and patterned metallic surfaces [3, 4]. These surface waves exhibit useful properties, such as strong field enhancement and localization, large bulk sensitivities, and subwavelength confinement. During the past decade, researchers have exploited these properties finding numerous applications in fields such as spectroscopy , nanophotonics , imaging  and biosensing . However, the performance of such applications is often degraded by the large propagation losses of SPPs at visible and near-infrared wavelengths that result mainly from power dissipation in the metal.
Considerable efforts have been devoted to investigate the amplification of SPPs – aiming to mitigate or even eliminate their large intrinsic losses in order to enable applications that exploit the full potential of SPPs. A variety of active plasmonic structures have been recently studied, including planar structures –, nanoresonators [16, 17] and nanoparticles [18, 19] that incorporate gain media in the form of semiconductor nanowires, quantum dots and quantum wells, laser dyes in solution or in polymers, and erbium-doped glass. In such a vigorous research field, studying the noise characteristics of plasmonic amplifiers is an important task. Recently, Thylén and colleagues  studied the signal-to-noise ratio (SNR) degradation induced by signal amplification in plasmonic systems, and De Leon and Berini  measured the amplified spontaneous emission (ASE) in a long-range surface plasmon-polariton amplifier at a near infrared wavelength finding an effective input noise power of 3.3 photons per mode.
In this paper, we present a theoretical and numerical study of the noise properties of high gain planar SPP amplifiers incorporating dipolar gain media. We consider two important planar structures, namely, a single interface between a metal and a gain medium, and a thin metal film bounded by identical gain media on both sides. A realistic configuration is considered where the SPP undergoing amplification has a Gaussian field profile in the plane of the metal and paraxial propagation along the amplifier’s length. This represents an experimental situation where a three-dimensional Gaussian beam is coupled at the input of these structures. An expression for the noise figure of these SPP amplifiers is obtained in terms of the spontaneous emission rate into the amplified SPP taking into account the different energy decay channels experienced by dipoles in close proximity to the metallic surface [22, 23]. The theory is applied to three variations of each structure considering silver as the metal region and three prototypical dipolar gain media that exhibit different permittivities.
2. Amplifier geometries
The two SPP amplifier geometries considered for our analysis are illustrated in cross-sectional view over the (x,y)-plane in Figs. 1(a) and (b). The first structure consists of a planar interface between a metal and a gain medium, and a the second one consists of a metal film of thickness tm bounded by identical gain media on both sides. The structures are invariant along the x-axis. The metal and gain medium are characterized by the relative permittivities ɛm = ɛ′m + iɛ″m and ɛg = ɛ′g + iɛ″g respectively, where ɛ′m and ɛ″m are negative quantities and ɛ′g and ɛ″g are positive quantities (eiωt time-harmonic dependance implicit). The gain medium is dipolar (e.g., dye molecules, rare-earth ions, semiconductor quantum dots) and isotropic, with finite length along the z axis, bounded by passive regions for z < −la and z > 0 that define the amplifier’s input and output planes, respectively [see Fig. 1(c)]. The passive regions are taken to be identical to the active region except that the medium bounding the metal has ɛ″g = 0.
The single-interface structure supports SPPs with fields that peak at the interface and decay exponentially into both media . Provided that tm is small enough, the metal-film structure supports SPP modes formed through symmetric or asymmetric coupling of single-interface SPPs on each surface of the film. The symmetric mode is termed the long-range SPP (LR-SPP)  as it exhibits a low propagation loss, whereas the asymmetric mode exhibits a large propagation loss and hence it is termed the short-range SPP (SRSPP). The large propagation loss of the SRSPP renders impractical its use for amplification purposes. Hence, in what follows we shall study only the noise properties of LRSPP and single-interface SPP amplifiers.
Both single-interface SPPs and LRSPPs can propagate freely over the (x,z)-plane. However, we shall restrict our analysis to paraxial SPP “beams” with Gaussian field distribution in the plane of the metal, to which we shall refer as Gaussian SPPs. Inspired by the arrangement proposed by Kogelnik and Yariv  in their analysis of noise in Gaussian photon amplifiers, we assume that the Gaussian SPP propagates in +z direction having it waist, w0, located at the amplifier’s output plane, as depicted in Fig. 1(c). Thus, the main electric field component of the Gaussian SPP in the amplifier region is given byFigs. 1(a) and (b); κι = βι + iγι/2 is the SPP complex propagation constant with βι and γι being the phase and mode power gain coefficients, respectively; and q̃ι(z) = z + izR,ι is the complex beam parameter that defines the evolution of the Gaussian SPP along z, with being the Rayleigh range.
In practice, the amplifier architectures described above can be realized (for example) by using an optically pumped dipolar gain medium with the amplifier region (−la < z < 0) defined by the optical pump extension , and end-fire coupling a three-dimensional Gaussian beam to the input of the structure with its waist (focus) located at the amplifier output plane (z = 0).
3. Theoretical approach
The noise figure of an optical amplifier is defined as the ratio of the SNR measured at the input port to the SNR measured at the output port. For a single mode amplifier and for optical signals characterized by Poisson statistics (e.g., coherent light signals) it is given by Eq. (2) can be traced back to a particular noise contribution. The first term is associated with the signal’s beat and shot noise, while the second term is associated with the ASE’s beat and shot noise.
For a high gain amplifier (G ≫ 1) with a large enough input signal such that Gn0 ≫ nN, the noise characteristics are dominated by the signal’s beat noise term, 2nN/G. In this limit, the amplifier’s ASE output power takes the form  PN = Ahν0Gξγ−1, where h is Plank’s constant, ν0 is the optical frequency, ξ is the effective spontaneous emission rate per unit volume into the particular optical mode being amplified, and A is its effective transverse area. Then, using the relation nN = PN(Bνhν0)−1, with Bν being the optical detection bandwidth centred around ν0, the noise figure can be approximated as
For the structures in Fig. (1), the input SNR is degraded due to the SPP propagation losses in the passive section preceding the amplifier region, as well as additional losses caused by imperfect signal coupling into the structure. As a result, the amplifier’s noise figure is degraded. To account for this, the right hand side of Eq. (3) must be divided by the transmittance of the input passive section, T (0 ≤ T ≤ 1) . Here, we shall use Eq. (3) to describe the minimum noise figure of these amplifiers in the limit where T → 1 (efficient signal coupling and low-loss passive section). The results can then be scaled by the factor 1/T to represent other cases.
The remainder of this section is devoted to finding appropriate expressions for A and ξ in Eq. (3) to describe the noise figure of the SPP amplifiers under consideration. The analysis is restricted to single-mode amplifiers with high gain. The high-gain restriction, G ≫ 1, is in principle feasible for both amplifier architectures considered here . The single mode restriction is certainly satisfied for the single-interface SPP amplifier, as only one bound mode is supported by the structure; it is also satisfied to a very good approximation for the LRSPP amplifier because the SRSPP is rapidly (in a short length) absorbed by the metal.
3.1. Effective mode area
The effective mode area is the equivalent transverse area of the mode assuming a constant field distribution over the transverse plane. Different definitions of effective mode area exist (see for example Ref.  and references therein) because the choice of the constant field value is not unique. Here, we adopt the commonly used statistical measure Eq. (4) run over the entire (x,y) plane. Since both the single-interface SPP and the LRSPP have a very small fraction of their fields inside the metal, one can approximate Ey,ι (y) in Eq. (5) simply by the evanescent field in the gain region(s), which upon substitution in Eq. (4) yields
3.2. Effective spontaneous emission rate per unit volume
We regard the gain medium as an ensemble of isotropically oriented dipoles in an excited state. Dipoles located far from the metal surface relax through spontaneous emission into bulk modes; however, those located close to the metal surface experience additional energy decay channels, such as spontaneous emission into SPPs and radiative modes of the structure, and non-radiative energy transfer to electron-hole pairs in the metal . We are interested in the spontaneous emission into SPPs in order to describe the amplifier’s noise. To this end, we describe the radiative properties of individual dipoles close to the metal surface following the well-known Chance-Prock-Silbey model  as reformulated by Ford and Weber . This aspect of the present theory has reproduced measured results on a LRSPP amplifier with reasonable agreement .
For a dipole located at a distance y0 from the metal surface, the spontaneous emission rate into SPPs with arbitrary propagation directions can be written as (see Appendix A.1)Eq. (7) over dipole positions yields the effective spontaneous emission rate:
The excited state dipole density is also a position dependent quantity, as it is inversely proportional to the total spontaneous emission rate (in general it also depends on the pump intensity distribution  which is assumed uniform here). It is then convenient to define an effective excited state density in terms of the mode power gain coefficient asEq. (9) we have implicitly assumed that the gain medium is a four-level system and that σe is not affected by the presence of the metal surface.
It follows that over a narrow optical bandwidth Bν centred around ν0, such that Bνg(ν) ≈ Bνg(ν0), the total spontaneous emission rate per unit volume into the Gaussian SPP can be written as Fig. 1(c)].
3.3. Noise figure of a high-gain SPP amplifier30], namely, the ratio of stimulated emission not dissipated to the total stimulated emission. For the LRSPP amplifier, this equation must be solved numerically. However, for the single-interface SPP amplifier, one can obtain a simple analytical expression (see Appendix A.2) given by
4. Results and discussion
In this section we use Eqs. (11) and (12) to estimate the noise figure of the two plasmonic amplifiers of interest in the high gain limit (γι ≫ αι), where the amplifier quantum efficiency approaches unity (ηι = 1). The gain media are assumed to be constituted by excited dipoles embedded in a host material of permittivity ɛ′g. For the calculations we consider ɛ′g = 1 (a gas), ɛ′g = 2.25 (a polymer or a glass), and ɛ′g = 13.8 (similar to the permittivity of a semiconductor). The metal is silver in all cases. The calculations cover wavelengths in the range 400 nm ≤ λ0 ≤ 2 μm, over which the dispersion of ɛ′g is neglected and ɛm is approximated by the Drude model with the parameters for silver used in Ref. . Furthermore, the value of βι in Eq. (11) is calculated in the absence of gain. This is a reasonable approximation because Im(κι) is always much smaller than Re(κι) under realistic modal gains, and hence βι is not significantly different from its passive value.
4.1. Single-interface SPP amplifier
As indicated previously, the noise figure of the single-interface SPP amplifier can be obtained numerically and analytically. Figure 2(a) shows the values of NF1 as a function of the wavelength calculated analytically using Eq. (12) (dashed curves) and by solving numerically Eq. (11) (markers). For the latter, the integration limits with respect to u are obtained as the full-width at first minima of the pole in S(u,y) associated with the single-interface SPP. We observe a very good agreement between the two results.
Clearly, NF1 depends strongly on the material permittivities and operation wavelength. From Eq. (12) we note that the noise figure reaches a minimum value of when |ɛ′m| ≫ ɛ′g and η1 = 1. This is observed in Fig. 2(a), as NF1 approaches such a minimum value (indicated by the horizontal dashed line) at infrared wavelengths because ɛ′m acquires a large magnitude (∼ 100) in this range. The condition |ɛ′m| ≫ ɛ′g is more demanding for a gain medium with a large permittivity; hence, a noise figure degradation is observed as ɛ′g increases. Also, from Eq. (12) we note that NF1 = ∞ when ɛ′g = ɛ′m, which occurs at the energy asymptote of the SPP dispersion curve. This is observed in Fig. 2(a) for the case ɛ′g = 13.8, as NF1 increases sharply for wavelengths approaching the energy asymptote (indicated by the vertical dashed line). A similar trend is observed at shorter wavelengths for the cases with lower ɛ′g, for which the energy asymptote occurs at shorter wavelengths outside the simulation range.
It is worth mentioning that SPPs at wavelengths close to the energy asymptote suffer from extremely large propagation losses; thus, achieving a high gain with such SPPs is very challenging.
The noise figure degradation observed for increasing values of ɛ′g and for wavelengths approaching the energy asymptote follows from the Purcell effect , as the spontaneous emission rate is enhanced due to an increased confinement of the SPP. This is shown in Fig. 2(b), which plots the normalized spontaneous emission rate into single-interface SPPs as a function of the wavelength for the three gain media. Clearly, the rate increases with increasing ɛ′g. In particular for ɛ′g = 13.8, the spontaneous emission rate is enhanced by more than an order of magnitude in the visible wavelength region relative to that in the other two cases.
4.2. LRSPP amplifier
An analytical solution for Eq. (11) does not exists for the LRSPP amplifier; thus, NF2 is obtained by solving Eq. (11) numerically. In doing so, we calculate β2 using a transfer matrix method  and define the integration limits with respect to u as in the previous case but considering the pole in S(u,y) associated with the LRSPP.
Figure 2(c) shows the calculated values of NF2 as a function of the wavelength for a silver film of thickness tm = 20 nm. We note that for long wavelengths, NF2 approaches the same minimum value as that of NF1. However, in this case, the overall noise figure spectrum is significantly improved with respect to NF1. In particular, for the two cases with smaller ɛ′g, NF2 is weakly dependent on the wavelength and close to the minimum value over the visible and infrared range. On the other hand, for ɛ′g = 13.8, NF2 presents a slight increment over the infrared region as the wavelength shortens and increases sharply for wavelengths approaching the energy asymptote, as it was also observed for NF1. The comparatively small noise figure of the LRSPP amplifier is a direct consequence of a smaller spontaneous emission rate into LRSPPs, as indicated by the results in Figure 2(d), which plots the normalized spontaneous emission rate into LRSPPs as a function of the wavelength for the three different gain media.
Figure 3(a) shows the computed values of NF2 for silver films with various thicknesses assuming ɛ′g = 2.25; NF1 for the same gain medium is also shown for reference. Note that NF2 increases with the film thickness because the LRSPP confinement increases, which in turn increases the spontaneous emission rate due to the Purcell effect. For sufficiently thick metal films, NF2 approaches NF1 because the LRSPP evolves into isolated single-interface SPPs on each surface of the film. As the metal film becomes thinner, NF2 approaches the minimum value for all wavelengths in our calculations. Similar trends are observed for the other two gain media (not shown); however, their actual noise figure values are scaled according to Fig. 2(c).
4.3. Minimum noise figure
It can be shown that a high-gain Gaussian photon amplifier (i.e., a Gaussian beam propagating through a bulk gain medium) with unitary amplifier quantum efficiency has a noise figure of NF0 = 2 (3 dB), and that such a value corresponds to the theoretical minimum for high-gain phase-insensitive photon amplifiers . On the other hand, the Gaussian SPP amplifiers studied here have a minimum noise figure of . To understand this difference we consider a Gaussian photon amplifier and a Gaussian SPP amplifier, both having a unitary amplifier quantum efficiency, identical gain media, and the same optical bandwidth. From the theory in Section 3 (with proper adaptations for the photon amplifier case) one obtains an expression for NFι normalized to NF0, given by
The first term on the right hand side of Eq. (13), fa, relates the rates of spontaneous emission noise (emitted in all directions) in the SPP and photon amplifiers; the second term, fb, relates their ability to couple such noise into the amplified mode. Figure 3(b) plots fa and − fb for different values of tm assuming ɛ′g = 2.25. Consider the results in the visible wavelength range. We note that when tm is very small (ι = 2), fa is also very small (〈Γ2〉 ≪ Γ0/2) because the density of LRSPP modes is small; however, fb is very large (F2A2 ≫ F0A0) because A2 → ∞ as tm → 0. On the other hand, when tm = ∞ (ι = 1), fa is very large because the density of SPP modes is large, while fb is small because the SPP is well confined, causing A1 ≪ A0. A gradual transition between these two cases is observed as tm increases. Nonetheless, (NFι/NF0)dB is always positive across the spectrum with values that depend strongly on tm at short wavelengths and remain essentially constant (∼ 0.53 dB) at long wavelengths. From this analysis, we observe that the minimum noise figure of these Gaussian SPP amplifiers results from the interplay between the noise generated into SPPs propagating in all directions, given by 〈Γι〉, and the coupling efficiency of such noise into the Gaussian SPP, given by the product FιAι.
It is important to mention that fa is fundamental, as it depends only on the physical properties of the dipole ensemble and metallic structure. This is not the case for fb, as it depends partly on the properties of the Gaussian SPP, such as the relation between the divergence angle and the beam waist. Thus the minimum noise figure of may not be a fundamental limit.
Finally, we point out that the theory presented here cannot be applied directly to a one-dimensional SPP amplifier given our use of the mode area to determine the noise photon number [nN in Eq. (2)]. Indeed, we have shown that both the mode area and the mode divergence (which are related by a Fourier relation) play a crucial role in defining the amplifier noise figure. We emphasize that the arrangements investigated (Fig. 1) represent physically realizable systems.
In summary, we have presented a theoretical analysis of noise in high-gain single-mode SPP amplifiers incorporating a dipolar gain medium. Two active plasmonic structures were examined: an interface between a metal and a gain medium supporting the single-interface SPP, and a thin metal film bounded by identical gain media on both sides supporting the LRSPP. A realistic configuration was considered where the SPP undergoing amplification has a Gaussian field profile in the plane of the metal and paraxial propagation along the amplifier’s length (Gaussian SPP). An expression for the noise figure was obtained in terms of the spontaneous emission rate into the Gaussian SPP taking into account the different energy decay channels experienced by dipoles in close proximity to the metallic surface. It was shown that the noise figure of these amplifiers exhibits a strong dependance on the real part of the permittivities of the metal (ɛ′m) and gain medium (ɛ′g) and that the figure is degraded in amplifiers with large ɛ′g. From an analytical expression derived for the noise figure of the single-interface SPP amplifier, it was shown that the noise figure diverges at the SPP energy asymptote (ɛ′m = ɛ′g) and that in the high-gain limit it approaches a minimum value of when |ɛ′m| ≫ ɛ′g. Numerical calculations indicated that these extrema apply also to the LRSPP amplifier; however, the noise figure in this case is significantly smaller than that of the single-interface SPP amplifier over the same wavelength range. The origin of the minimum noise figure of these SPP amplifiers was discussed, showing that it results from the interplay between the rate of spontaneous emission noise into SPPs and the coupling efficiency of such noise into the Gaussian SPP.
A.1. Spontaneous emission rate for a dipole over a metal surface
The total power dissipated by an isotropically oriented dipole located at a distance y0 from the metal plane is given by Eqs. (A.1) and (A.2) are defined in the main text. The power dissipated into a particular SPP (propagating in arbitrary directions) can then be obtained by integrating Eq. (A.2) over the range of normalized in-plane wave-vectors, uι, associated with the SPP . Thus, one can represent the dipole’s spontaneous emission rate into a particular SPP (ι = 1,2) as
A.2. Analytical expression for NF1
The power dissipated by an isotropically oriented dipole located at a distance y0 from the metal surface into single-interface SPPs can be obtained analytically as Eq. (A.3) that Eq.(A.5) we have used a Taylor expansion to obtain the real part in Eq. (A.4), assuming that |ɛ′m| ≫ |ɛ″m|, |ɛ′m| > ɛ′g and ɛ′m < 0, as this is generally true for metals at visible and near infrared wavelengths. Then, neglecting the term O(ɛ″m) in Eq. (A.5) because it is small, and performing a straightforward integration over the space coordinate yields Eq. (11) yields Eq. (12).
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