## Theory of noise in high-gain surface plasmon-polariton amplifiers incorporating dipolar gain media |

Optics Express, Vol. 19, Issue 21, pp. 20506-20517 (2011)

http://dx.doi.org/10.1364/OE.19.020506

Acrobat PDF (946 KB)

### Abstract

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

© 2011 OSA

## 1. Introduction

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

3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

4. S. Maier and H. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. **98**, 011101 (2005). [CrossRef]

5. T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. **12**, 575–586 (2002). [CrossRef]

6. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science **311**, 189–193 (2006). [CrossRef] [PubMed]

7. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. **3**, 388–394 (2009). [CrossRef]

8. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. **7**, 442–453 (2008). [CrossRef]

9. J. Seidel, S. Grafstrom, 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**, 177401 (2005). [CrossRef] [PubMed]

19. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature **460**, 1110–1113 (2009). [CrossRef] [PubMed]

9. J. Seidel, S. Grafstrom, 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**, 177401 (2005). [CrossRef] [PubMed]

15. I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. **18**, 18633–18641 (2010). [CrossRef] [PubMed]

16. M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. **1**, 589–594 (2007). [CrossRef]

17. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**, 629–632 (2009). [CrossRef] [PubMed]

18. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems.” Phys. Rev. Lett. **90**, 027402 (2003). [CrossRef] [PubMed]

19. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature **460**, 1110–1113 (2009). [CrossRef] [PubMed]

20. L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. **46**, 518–524 (2010). [CrossRef]

21. I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B **83**, 081414(R) (2011). [CrossRef]

22. R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. **37**, 1–65 (1978). [CrossRef]

23. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces.” Phys. Rep. **113**, 195–287 (1984). [CrossRef]

## 2. Amplifier geometries

*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

*t*bounded by identical gain media on both sides. The structures are invariant along the

_{m}*x*-axis. The metal and gain medium are characterized by the relative permittivities

*ɛ*=

_{m}*ɛ*′

*+*

_{m}*i*

*ɛ*″

*and*

_{m}*ɛ*=

_{g}*ɛ*′

*+*

_{g}*iɛ*″

*respectively, where*

_{g}*ɛ*′

*and*

_{m}*ɛ*″

*are negative quantities and*

_{m}*ɛ*′

*and*

_{g}*ɛ*″

*are positive quantities (*

_{g}*e*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

^{iωt}*z*axis, bounded by passive regions for

*z*< −

*l*and

_{a}*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*″

*= 0.*

_{g}*t*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) [2

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

*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 [24

24. H. Kogelnik and A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE **52**, 165–172 (1964). [CrossRef]

*z*direction having it waist,

*w*

_{0}, 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 by In this expression and throughout the manuscript, the subscript

*ι*indicates the number of interfaces along the

*y*axis of a particular structure and is employed to identify quantities associated with single-interface SPPs (

*ι*= 1) and LRSPPs (

*ι*= 2);

*E*

_{y,ι}(

*y*) is the transverse electric field distribution along the

*y*axis, whose magnitude is sketched on Figs. 1(a) and (b);

*κ*=

_{ι}*β*+

_{ι}*iγ*/2 is the SPP complex propagation constant with

_{ι}*β*and

_{ι}*γ*being the phase and mode power gain coefficients, respectively; and

_{ι}*q̃*(

_{ι}*z*) =

*z*+

*iz*

_{R,ι}is the complex beam parameter that defines the evolution of the Gaussian SPP along

*z*, with

*l*<

_{a}*z*< 0) defined by the optical pump extension [12

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

*z*= 0).

## 3. Theoretical approach

*G*is the optical gain of the amplifier,

*n*is the mean ASE photon number at the output port, and

_{N}*n*

_{0}is the mean signal photon number at the input port. Each term in 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.

*G*≫ 1) with a large enough input signal such that

*Gn*

_{0}≫

*n*, the noise characteristics are dominated by the signal’s beat noise term, 2

_{N}*n*/

_{N}*G*. In this limit, the amplifier’s ASE output power takes the form [21

21. I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B **83**, 081414(R) (2011). [CrossRef]

*P*=

_{N}*Ahν*

_{0}

*Gξγ*

^{−1}, where

*h*is Plank’s constant,

*ν*

_{0}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

*n*=

_{N}*P*(

_{N}*B*

_{ν}hν_{0})

^{−1}, with

*B*being the optical detection bandwidth centred around

_{ν}*ν*

_{0}, the noise figure can be approximated as

*T*(0 ≤

*T*≤ 1) [25]. 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.

*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 [26

26. I. De Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express **17**, 20191–2020 (2009). [CrossRef] [PubMed]

### 3.1. Effective mode area

27. R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. **10**, 105018 (2008). [CrossRef]

*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

*E*

_{y,ι}(

*y*) in Eq. (5) simply by the evanescent field in the gain region(s), which upon substitution in Eq. (4) yields with

*δ*being the

_{ι}*e*

^{−1}field penetration depth into the gain medium

### 3.2. Effective spontaneous emission rate per unit volume

29. I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B **78**, 161401(R) (2008). [CrossRef]

22. R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. **37**, 1–65 (1978). [CrossRef]

23. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces.” Phys. Rep. **113**, 195–287 (1984). [CrossRef]

21. I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B **83**, 081414(R) (2011). [CrossRef]

*y*

_{0}from the metal surface, the spontaneous emission rate into SPPs with arbitrary propagation directions can be written as (see Appendix A.1) Here, Γ

_{0}is the dipole’s spontaneous emission rate in the bulk (no metal present),

*S*(

*u*,

*y*) is the dipole’s power dissipation density normalized to its total dissipation in the bulk, and

*u*is the in-plane wave-vector normalized to the dipole’s far-field wavenumber. The integral runs over the range of wave-vectors,

*u*, associated with the single-interface SPP (

_{ι}*ι*= 1) or LRSPP (

*ι*= 2). Averaging Eq. (7) over dipole positions yields the effective spontaneous emission rate: where the integration range

*ℓ*is taken over the entire gain medium (or media).

_{g}29. I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B **78**, 161401(R) (2008). [CrossRef]

*α*is the mode power attenuation coefficient characterizing the optical losses of the structure (e.g., absorption in the metal, scattering loss and ground-state absorption of dipoles) such that the term (

_{ι}*γ*+

_{ι}*α*) represents the material gain of the gain medium,

_{ι}*σ*is the dipole’s emission cross-section,

_{e}*g*(

*ν*) is the dipole’s emission linewidth, and

*λ*

_{0}being the wavelength in vacuum. In defining Eq. (9) we have implicitly assumed that the gain medium is a four-level system and that

*σ*is not affected by the presence of the metal surface.

_{e}*B*centred around

_{ν}*ν*

_{0}, such that

*B*(

_{ν}g*ν*) ≈

*B*(

_{ν}g*ν*

_{0}), the total spontaneous emission rate per unit volume into the Gaussian SPP can be written as [21

**83**, 081414(R) (2011). [CrossRef]

*F*=

_{ι}*θ*/

_{ι}*π*is the geometrical spontaneous emission factor, with

*θ*= 2(

_{ι}*w*

_{0}

*β*)

_{ι}^{−1}. This factor is included to account only for the spontaneous emission with in-plane wave-vectors falling within the convergence angle of the Gaussian SPP in the propagation direction [see Fig. 1(c)].

### 3.3. Noise figure of a high-gain SPP amplifier

*η*=

_{ι}*γ*/(

_{ι}*γ*+

_{ι}*α*) is the amplifier quantum efficiency [30

_{ι}30. C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. **4**, 288–297 (1986). [CrossRef]

*ɛ*′

*| ≫ |*

_{m}*ɛ*″

*|, |*

_{m}*ɛ*′

*| >*

_{m}*ɛ*′

*and*

_{g}*ɛ*′

*< 0, which is generally the case for metals at visible and near infrared wavelengths.*

_{m}## 4. Results and discussion

*γ*≫

_{ι}*α*), where the amplifier quantum efficiency approaches unity (

_{ι}*η*= 1). The gain media are assumed to be constituted by excited dipoles embedded in a host material of permittivity

_{ι}*ɛ*′

*. For the calculations we consider*

_{g}*ɛ*′

*= 1 (a gas),*

_{g}*ɛ*′

*= 2.25 (a polymer or a glass), and*

_{g}*ɛ*′

*= 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 ≤*

_{g}*λ*

_{0}≤ 2

*μ*m, over which the dispersion of

*ɛ*′

*is neglected and*

_{g}*ɛ*is approximated by the Drude model with the parameters for silver used in Ref. [31

_{m}31. W. L. Barnes, “Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices,” J. Lightwave Technol. **17**, 2170–2182 (1999). [CrossRef]

*β*in Eq. (11) is calculated in the absence of gain. This is a reasonable approximation because Im(

_{ι}*κ*) is always much smaller than Re(

_{ι}*κ*) under realistic modal gains, and hence

_{ι}*β*is not significantly different from its passive value.

_{ι}### 4.1. Single-interface SPP amplifier

*NF*

_{1}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.

*NF*

_{1}depends strongly on the material permittivities and operation wavelength. From Eq. (12) we note that the noise figure reaches a minimum value of

*ɛ*′

*| ≫*

_{m}*ɛ*′

*and*

_{g}*η*

_{1}= 1. This is observed in Fig. 2(a), as

*NF*

_{1}approaches such a minimum value (indicated by the horizontal dashed line) at infrared wavelengths because

*ɛ*′

*acquires a large magnitude (∼ 100) in this range. The condition |*

_{m}*ɛ*′

*| ≫*

_{m}*ɛ*′

*is more demanding for a gain medium with a large permittivity; hence, a noise figure degradation is observed as*

_{g}*ɛ*′

*increases. Also, from Eq. (12) we note that*

_{g}*NF*

_{1}= ∞ when

*ɛ*′

*=*

_{g}*ɛ*′

*, which occurs at the energy asymptote of the SPP dispersion curve. This is observed in Fig. 2(a) for the case*

_{m}*ɛ*′

*= 13.8, as*

_{g}*NF*

_{1}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*′

*, for which the energy asymptote occurs at shorter wavelengths outside the simulation range.*

_{g}*ɛ*′

*and for wavelengths approaching the energy asymptote follows from the Purcell effect [32*

_{g}32. A. Hryciw, Y. Jun, and M. Brongersma, “Plasmon-enhanced emission from optically-doped MOS light sources,” Opt. Express **17**, 185–192 (2009). [CrossRef] [PubMed]

*ɛ*′

*. In particular for*

_{g}*ɛ*′

*= 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.*

_{g}### 4.2. LRSPP amplifier

*NF*

_{2}is obtained by solving Eq. (11) numerically. In doing so, we calculate

*β*

_{2}using a transfer matrix method [34

34. C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express **7**, 260–272 (2000). [CrossRef] [PubMed]

*u*as in the previous case but considering the pole in

*S*(

*u*,

*y*) associated with the LRSPP.

*NF*

_{2}as a function of the wavelength for a silver film of thickness

*t*= 20 nm. We note that for long wavelengths,

_{m}*NF*

_{2}approaches the same minimum value as that of

*NF*

_{1}. However, in this case, the overall noise figure spectrum is significantly improved with respect to

*NF*

_{1}. In particular, for the two cases with smaller

*ɛ*′

*,*

_{g}*NF*

_{2}is weakly dependent on the wavelength and close to the minimum value over the visible and infrared range. On the other hand, for

*ɛ*′

*= 13.8,*

_{g}*NF*

_{2}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

*NF*

_{1}. 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.

*NF*

_{2}for silver films with various thicknesses assuming

*ɛ*′

*= 2.25;*

_{g}*NF*

_{1}for the same gain medium is also shown for reference. Note that

*NF*

_{2}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,

*NF*

_{2}approaches

*NF*

_{1}because the LRSPP evolves into isolated single-interface SPPs on each surface of the film. As the metal film becomes thinner,

*NF*

_{2}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

*NF*

_{0}= 2 (3 dB), and that such a value corresponds to the theoretical minimum for high-gain phase-insensitive

*photon*amplifiers [25]. On the other hand, the Gaussian SPP amplifiers studied here have a minimum noise figure of

*NF*normalized to

_{ι}*NF*

_{0}, given by where (·)

_{dB}= 10log

_{10}(·) Γ

_{0}/; 2,

*F*

_{0}= (

*w*

_{0}

*β*)

_{g}^{−2}are parameters of the Gaussian photon amplifier, namely, the spontaneous emission rate (generated into one polarization state only), the effective mode area at the beam waist, and the geometrical spontaneous emission factor, respectively.

*f*, relates the rates of spontaneous emission noise (emitted in all directions) in the SPP and photon amplifiers; the second term,

_{a}*f*, relates their ability to couple such noise into the amplified mode. Figure 3(b) plots

_{b}*f*and −

_{a}*f*for different values of

_{b}*t*assuming

_{m}*ɛ*′

*= 2.25. Consider the results in the visible wavelength range. We note that when*

_{g}*t*is very small (

_{m}*ι*= 2),

*f*is also very small (〈Γ

_{a}_{2}〉 ≪ Γ

_{0}/2) because the density of LRSPP modes is small; however,

*f*is very large (

_{b}*F*

_{2}

*A*

_{2}≫

*F*

_{0}

*A*

_{0}) because

*A*

_{2}→ ∞ as

*t*→ 0. On the other hand, when

_{m}*t*= ∞ (

_{m}*ι*= 1),

*f*is very large because the density of SPP modes is large, while

_{a}*f*is small because the SPP is well confined, causing

_{b}*A*

_{1}≪

*A*

_{0}. A gradual transition between these two cases is observed as

*t*increases. Nonetheless, (

_{m}*NF*/

_{ι}*NF*

_{0})

_{dB}is always positive across the spectrum with values that depend strongly on

*t*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 〈Γ

_{m}*〉, and the coupling efficiency of such noise into the Gaussian SPP, given by the product*

_{ι}*F*.

_{ι}A_{ι}*f*is fundamental, as it depends only on the physical properties of the dipole ensemble and metallic structure. This is not the case for

_{a}*f*, 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

_{b}## 5. Summary

*ɛ*′

*) and gain medium (*

_{m}*ɛ*′

*) and that the figure is degraded in amplifiers with large*

_{g}*ɛ*′

*. 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 (*

_{g}*ɛ*′

*=*

_{m}*ɛ*′

*) and that in the high-gain limit it approaches a minimum value of*

_{g}*ɛ*′

*| ≫*

_{m}*ɛ*′

*. 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.*

_{g}## A.1. Spontaneous emission rate for a dipole over a metal surface

*y*

_{0}from the metal plane is given by [23

23. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces.” Phys. Rep. **113**, 195–287 (1984). [CrossRef]

*r*

_{TE}and

*r*

_{TM}being the Fresnel reflection coefficients for transverse-electric polarized waves (electric field parallel to the metal surface) and transverse-magnetic polarized waves (magnetic field parallel to the metal surface), respectively. The remaining parameters in 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,

*u*, associated with the SPP [23

_{ι}**113**, 195–287 (1984). [CrossRef]

*ι*= 1,2) as

## A.2. Analytical expression for NF_{1}

*y*

_{0}from the metal surface into single-interface SPPs can be obtained analytically as [23

**113**, 195–287 (1984). [CrossRef]

*κ*

_{1}=

*β*

_{0}[

*ɛ*′

*/(*

_{g}ɛ_{m}*ɛ*′

*+*

_{g}*ɛ*)]

_{m}^{1/2}is the SPP complex propagation constant, with

*β*

_{0}= 2

*π*/

*λ*

_{0}. It follows from Eq. (A.3) that where

*β*

_{1}=

*β*

_{0}[

*ɛ*′

*′*

_{g}ɛ*/(*

_{m}*ɛ*′

*+*

_{g}*ɛ*′

*)]*

_{m}^{1/2}is the SPP phase coefficient. In deriving 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*′

*and*

_{g}*ɛ*′

*< 0, as this is generally true for metals at visible and near infrared wavelengths. Then, neglecting the term*

_{m}*O*(

*ɛ*″

*) in Eq. (A.5) because it is small, and performing a straightforward integration over the space coordinate yields Using this result and the expression for*

_{m}*β*

_{1}into Eq. (11) yields Eq. (12).

## References and links

1. | H. Raether, |

2. | P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. |

3. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

4. | S. Maier and H. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

5. | T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. |

6. | E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science |

7. | S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. |

8. | J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. |

9. | J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution.” Phys. Rev. Lett. |

10. | M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. |

11. | J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. |

12. | I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. |

13. | M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer.” Nature Photon. |

14. | 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. |

15. | I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. |

16. | M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. |

17. | R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature |

18. | D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems.” Phys. Rev. Lett. |

19. | M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature |

20. | L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. |

21. | I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B |

22. | R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. |

23. | G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces.” Phys. Rep. |

24. | H. Kogelnik and A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE |

25. | E. Desurvire, |

26. | I. De Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express |

27. | R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. |

28. | A. E. Siegman, |

29. | I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B |

30. | C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. |

31. | W. L. Barnes, “Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices,” J. Lightwave Technol. |

32. | A. Hryciw, Y. Jun, and M. Brongersma, “Plasmon-enhanced emission from optically-doped MOS light sources,” Opt. Express |

33. | M. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express |

34. | C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(140.4480) Lasers and laser optics : Optical amplifiers

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 25, 2011

Revised Manuscript: September 3, 2011

Manuscript Accepted: September 3, 2011

Published: October 3, 2011

**Citation**

Israel De Leon and Pierre Berini, "Theory of noise in high-gain surface plasmon-polariton amplifiers incorporating dipolar gain media," Opt. Express **19**, 20506-20517 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20506

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### References

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