## Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides

Optics Express, Vol. 12, Issue 17, pp. 4072-4079 (2004)

http://dx.doi.org/10.1364/OPEX.12.004072

Acrobat PDF (327 KB)

### Abstract

The propagation of surface plasmon polaritons on metallic waveguides adjacent to a gain medium is considered. It is shown that the presence of the gain medium can compensate for the absorption losses in the metal. The conditions for existence of a surface plasmon polariton and its lossless propagation and wavefront behavior are derived analytically for a single infinite metal-gain boundary. In addition, the cases of thin slab and stripe geometries are also investigated using finite element simulations. The effect of a finite gain layer and its distance from the SPP waveguide is also investigated. The calculated gain requirements suggest that lossless gain-assisted surface plasmon polariton propagation can be achieved in practice for infrared wavelengths.

© 2004 Optical Society of America

## 1. Introduction

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

*ε′*, is accompanied by an energy dissipating imaginary part

*ε″*, causing lossy propagation of SPPs. Consequently, this energy dissipation limits the effective propagation length of SPPs to values in the micrometer to millimeter range, thereby creating an obvious obstacle in utilizing them in practical optical devices and circuits.

4. G. A. Plotz, H. J. Simon, and J. M. Tucciarone, “Enhanced total reflection with surface plasmons,” JOSA **69**, 419–421 (1979). [CrossRef]

## 2. SPP propagation on an infinite metal-gain medium boundary

*x*direction on a metal-dielectric boundary lying in the

*x-y*plane, with the fields tailing off into the positive (dielectric) and negative (metal)

*z*directions, described by the following TM waves (the indices 1 and 2 denote dielectric and metal regions, respectively):

*k*

_{0}=

*ω/c*is the free space wave vector of the incident excitation photon. From Eq. (2), the SPP dispersion relations can be derived [7]:

*k*

_{x}is responsible for lossy propagation of the SPP along the interface. As mentioned, this results from the non-zero imaginary component of the metal permittivity. Apart from choosing a metal with a high plasmonic resonance (i.e., (

*ε*

_{2}′)

^{2}/

*ε*″

_{2}>>1), there seems to be no other method to reduce the metallic losses so as to increase the SPP propagation length in the metal-dielectric configuration.

*ε*

_{1}=

*ε*′

_{1}+

*iε*″

_{1}(with negative

*ε*″

_{1}representing gain) and investigating the conditions for a bound wave to propagate at the interface. At this point our only assumption is that

*ε*′

_{2}is negative and its absolute value is much larger than the other three permittivity components. The equations governing the SPP propagation for this material configuration are:

*ε*″

_{1}for a bound solution (i.e., Im

*ε*′

_{2}guarantee that these bounds are real and have opposite signs. This places a limit on the allowable amount of absorption or gain in the dielectric for bound waves to exist. The longitudinal propagation characteristics of the SPP are given by

*k*

_{x}in Eq. (4-a), which simplifies to:

*ε*|

^{2}=(

*ε*′)

^{2}+(

*ε*″)

^{2}. Assuming

*ε*′

_{1},

*ε*′

_{2}and

*ε*″

_{2}are fixed by the choice of the metal and gain medium, we solve for

*ε*″

_{1}to find the roots of the imaginary part of Eq. (8), which has to be zero for lossless propagation of the SPP, yielding:

*ε*″

_{1}is opposite to that of

*ε*″

_{2}, which implies gain in region 1. The magnitude of the solution in Eq. (9-a) is very large and outside the bounds given in Eq. (7), so we will only consider Eq. (9-b). Note that for this solution, the stronger the plasmonic resonance, the less gain is required for lossless propagation. Note also that this result is in approximate agreement with the result derived in [4

4. G. A. Plotz, H. J. Simon, and J. M. Tucciarone, “Enhanced total reflection with surface plasmons,” JOSA **69**, 419–421 (1979). [CrossRef]

*ε*″

_{1}/

*ε*″

_{2}|(

*ε*′

_{2}/

*ε*′

_{1})

^{2}>1. (The reason for the slight difference is that the approximations used in [4

4. G. A. Plotz, H. J. Simon, and J. M. Tucciarone, “Enhanced total reflection with surface plasmons,” JOSA **69**, 419–421 (1979). [CrossRef]

*ε*

_{2}|

^{2}=

*ε*′

_{2}|>>

*ε*″

_{2}).

*ε*″

_{1}derived in Eq. (9-b) to the actual optical power gain, we use

*γ*=-

*k*

_{0}

*ε*″

_{1}/(

*ε*′

_{1})

^{½}where

*γ*is the power gain coefficient. This gives us the gain coefficient required for lossless SPP propagation:

*γ*

_{0}, the SPP propagation will still be lossy but the propagation length will increase accordingly. If the gain is increased past

*γ*

_{0}, the SPP amplitude (i.e., the amplitude of the fields in Eq. (1)) will increase as the SPP propagates along the interface. In practice, the gain medium will saturate at a specific amplitude level, which will inhibit further increase of the amplitude. An interesting point to note is that if the metal-gain medium interface is placed inside a longitudinal cavity and the gain is high enough to compensate for both cavity and SPP losses, sustained oscillations (i.e., lasing) will occur. The experimental realization of surface plasmon quantum cascade lasers in the middle to far infrared range [8

8. C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Long-wavelength (λ=8-11.5 *µ*m) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. **23**, 1366 (1998). [CrossRef]

*ε*″

_{1}=-

*ε*″

_{2}, i.e., as the gain grows larger than this amount, the wavefront tilt increases away from the interface, until at the limit derived in Eq. (7) the wave is no longer bound to the interface. It should be noted that this amount of gain can not be achieved in practice for the visible/infrared range, such that the wavefront will always be tilted towards the metal surface. Figure 1 shows the different operating regions of the SPP versus

*ε*″

_{1}.

*ε*

_{2}=-116.38+

*i*11.1 @ 1550nm) [9]. For the gain medium, we assume

*ε*′

_{1}to be 11.38, which approximates a InGaAsP-based gain medium. From Eqs. (9-b) and (10) we find

*ε*″

_{1}=-0.106 and

*γ*

_{0}=1275 cm

^{-1}which is within the limits of currently available semiconductor based optical gain media ([10

10. T. Saitoh and T Mukai, “1.5 µm GaInAsP traveling-wave semiconductor laser amplifier,” IEEE J. Quant. Elec. **QE-23**, 1010–1020 (1987). [CrossRef]

11. N. Hatori, M. Sugawara, K. Mukai, Y. Nakata, and H. Ishikawa, “Room-temperature gain and differential gain characteristics of self-assembled InGaAs/GaAs quantum dots for 1.1–1.3 *µm* semiconductor lasers,” Appl. Phys. Lett. **77**, 773–775 (2000). [CrossRef]

*γ*

_{0}decreases with decreasing

*ε*′

_{1}. Hence, we anticipate that using lower refractive index gain media will require lower gain compared to InGaAsP and similar high refractive index media. Examples of such gain media are quantum dots embedded in a glass matrix or polymer [12

12. K. Wundke, J. Auxier, A. Schulzgen, N. Peyghambarian, and N. F. Borrelli, “Room-temperature gain at 1.3 *µm* in PbS-doped glasses,” Appl. Phys. Lett. **75**, 3060–3062 (1999). [CrossRef]

13. P. Ramvall, Y. Aoyagi, A. Kuramata, P. Hacke, K. Domen, and K. Horino, “Doping-dependent optical gain in GaN,” Appl. Phys. Lett. **76**, 2994–2996 (2000). [CrossRef]

*k*

_{x}), the propagation length, Re(

^{-1}, 13341cm

^{-1}and 67045cm

^{-1}respectively. The slight discrepancies between these numbers and values predicted from Eqs. (7) and (10) are due to the first order Taylor approximations used throughout the derivations. Note that the propagation length becomes very sensitive to gain values near

*γ*

_{0}, which may be of interest for switching and modulation applications.

## 3. Gain-assisted propagation of SPPs in slab and stripe geometries

*s*

_{b}and

*a*

_{b}, respectively [14

14. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B **61**, 10484 (2000). [CrossRef]

*s*

_{b}mode decreases with decreasing slab thickness, thereby reducing absorption losses, we can expect that the gain requirement will also be reduced. Figures 3(a) and 3(b) show FEA mode calculation results (calculated using FEMLAB from COMSOL Inc.) for a 40nm thick silver slab embedded in the same dielectric/gain medium of the previous example. The propagation constant for the passive (no gain) case is found to be 14.06+

*i*0.0197

*µm*

^{-1}(Fig. 3(a)), the complex part corresponding to a loss of 0.17 dB

*µm*

^{-1}. Once the gain is set to 360.4 cm

^{-1}, the propagation constant becomes a real number and equal to 14.06

*µm*

^{-1}(Fig. 3(b)), corresponding to lossless SPP propagation.

*i*0.0094

*µm*-1 and 0.081 dB

*µm*

^{-1}, respectively. For this case, the loss is completely compensated by a gain equal to 180.24 cm

^{-1}, as shown in Fig. 3(d).

*γ*

_{0}originally predicted by Eq. (10), the stripe configuration shows a substantial (almost an order of magnitude) reduction in the gain requirement for the same material system and further supports the viability of experimental and practical implementation of such a waveguiding scheme. Note that the mode profiles of the uncompensated and compensated cases in Fig. 3 are nearly identical, as predicted by Eq. (5).

18. N. Kirstaedter, O. G. Schmidt, N. N. Ledentsov, D. Bimberg, V. M. Ustinov, A. Y. Egorov, A.E. Zhukov, M. V. Maximov, P.S. Kopev, and Z. I. Alferov, “Gain and differential gain of single layer InAs/GaAs quantum dot injection lasers,” Appl. Phys. Lett. **69**, 1226–1228 (1996). [CrossRef]

^{-1}, which are still within the values reported in the literature. For example, gain values of 1200cm

^{-1}, 2600cm

^{-1}and 6.8×10

^{4}cm

^{-1}are reported for a SOA with a 110nm thick active layer [10

10. T. Saitoh and T Mukai, “1.5 µm GaInAsP traveling-wave semiconductor laser amplifier,” IEEE J. Quant. Elec. **QE-23**, 1010–1020 (1987). [CrossRef]

17. S. Y. Hu, D. B. Young, S. W. Corzine, A.C. Gossard, and L. A. Coldren, “High-efficiency and low-threshold InGaAs/AlGaAs quantum-well lasers,” J. of Appl. Physics **76**, 3932–3934 (1994). [CrossRef]

18. N. Kirstaedter, O. G. Schmidt, N. N. Ledentsov, D. Bimberg, V. M. Ustinov, A. Y. Egorov, A.E. Zhukov, M. V. Maximov, P.S. Kopev, and Z. I. Alferov, “Gain and differential gain of single layer InAs/GaAs quantum dot injection lasers,” Appl. Phys. Lett. **69**, 1226–1228 (1996). [CrossRef]

19. J. R. Sambles, “Grain-boundary scattering and surface-plasmon attenuation in noble-metal films,” Solid State Comm. **49**, 343–345 (1984). [CrossRef]

## 4. Conclusion

## References and links

1. | C. Vassalo, |

2. | F. A. Fernández and Y. Lu, |

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

4. | G. A. Plotz, H. J. Simon, and J. M. Tucciarone, “Enhanced total reflection with surface plasmons,” JOSA |

5. | B. Ya Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. |

6. | A. N. Sudarkin and P. A. Demkovich, “Excitation of surface electromagnetic waves on the boundary of a metal with an amplifying medium,” Sov. Phys.Tech. Phys. |

7. | H. Raether, |

8. | C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Long-wavelength (λ=8-11.5 |

9. | E. D. Palik, |

10. | T. Saitoh and T Mukai, “1.5 µm GaInAsP traveling-wave semiconductor laser amplifier,” IEEE J. Quant. Elec. |

11. | N. Hatori, M. Sugawara, K. Mukai, Y. Nakata, and H. Ishikawa, “Room-temperature gain and differential gain characteristics of self-assembled InGaAs/GaAs quantum dots for 1.1–1.3 |

12. | K. Wundke, J. Auxier, A. Schulzgen, N. Peyghambarian, and N. F. Borrelli, “Room-temperature gain at 1.3 |

13. | P. Ramvall, Y. Aoyagi, A. Kuramata, P. Hacke, K. Domen, and K. Horino, “Doping-dependent optical gain in GaN,” Appl. Phys. Lett. |

14. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B |

15. | L.A. Coldren and S. W. Corzine, |

16. | N. A. Pikhtin, S. O. Sliptchenko, Z. N. Sokolova, and I. S. Tarasov, “Analysis of threshold current density and optical gain in InGaAsP quantum well lasers,” Semiconductors |

17. | S. Y. Hu, D. B. Young, S. W. Corzine, A.C. Gossard, and L. A. Coldren, “High-efficiency and low-threshold InGaAs/AlGaAs quantum-well lasers,” J. of Appl. Physics |

18. | N. Kirstaedter, O. G. Schmidt, N. N. Ledentsov, D. Bimberg, V. M. Ustinov, A. Y. Egorov, A.E. Zhukov, M. V. Maximov, P.S. Kopev, and Z. I. Alferov, “Gain and differential gain of single layer InAs/GaAs quantum dot injection lasers,” Appl. Phys. Lett. |

19. | J. R. Sambles, “Grain-boundary scattering and surface-plasmon attenuation in noble-metal films,” Solid State Comm. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

(250.3140) Optoelectronics : Integrated optoelectronic circuits

(260.3910) Physical optics : Metal optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 18, 2004

Revised Manuscript: August 13, 2004

Published: August 23, 2004

**Citation**

Maziar Nezhad, Kevin Tetz, and Yeshaiahu Fainman, "Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides," Opt. Express **12**, 4072-4079 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-4072

Sort: Journal | Reset

### References

- C. Vassalo, Optical waveguide concepts (Elsevier, 1991).
- F. A. Fernández and Y. Lu, Microwave and optical waveguide analysis by the finite element method, (Wiley, 1996).
- W. L. Barnes, A. Dereux and T. W. Ebbesen, �??Surface plasmon subwavelength optics,�?? Nature 424, 824�?? 830 (2003). [CrossRef] [PubMed]
- G. A. Plotz, H. J. Simon, J. M. Tucciarone, �??Enhanced total reflection with surface plasmons,�?? JOSA 69, 419-421 (1979). [CrossRef]
- B. Ya Kogan, V. M. Volkov and S. A. Lebedev, �??Superluminescence and generation of stimulated radiation under internal-reflection conditions,�?? JETP Lett. 16, 100 (1972).
- A. N. Sudarkin and P. A. Demkovich, �??Excitation of surface electromagnetic waves on the boundary of a metal with an amplifying medium,�?? Sov. Phys.Tech. Phys. 34, 764-766 (1989).
- H. Raether, Surface plasmons on smooth and rough surfaces and on gratings (Springer Verlag, 1988).
- C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, �??Longwavelength (λ-�?? 8-11.5 μm) semiconductor lasers with waveguides based on surface plasmons,�?? 23, 1366 (1998). [CrossRef]
- E. D. Palik, Handbook of optical constants of solids vol. I (Academic Press, 1985).
- T. Saitoh and T Mukai, �??1.5 μm GaInAsP traveling-wave semiconductor laser amplifier,�?? IEEE J. Quant. Elec. QE-23, 1010-1020 (1987). [CrossRef]
- N. Hatori, M. Sugawara, K. Mukai, Y. Nakata and H. Ishikawa, �??Room-temperature gain and differential gain characteristics of self-assembled InGaAs/GaAs quantum dots for 1.1-1.3 m semiconductor lasers,�?? Appl. Phys. Lett. 77, 773-775 (2000). [CrossRef]
- K. Wundke, J. Auxier, A. Schulzgen, N. Peyghambarian and N. F. Borrelli, �??Room-temperature gain at 1.3 μm in PbS-doped glasses,�?? Appl. Phys. Lett. 75, 3060-3062 (1999). [CrossRef]
- P. Ramvall, Y. Aoyagi, A. Kuramata, P. Hacke, K. Domen and K. Horino, �??Doping-dependent optical gain in GaN,�?? Appl. Phys. Lett. 76, 2994-2996 (2000). [CrossRef]
- P. Berini, �??Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,�?? Phys. Rev. B 61, 10484 (2000). [CrossRef]
- L.A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (Wiley , 1995).
- N. A. Pikhtin, S. O. Sliptchenko, Z. N. Sokolova and I. S. Tarasov, �??Analysis of threshold current density and optical gain in InGaAsP quantum well lasers,�?? Semiconductors 36, 344-353 (2002). [CrossRef]
- S. Y. Hu, D. B. Young, S. W. Corzine, A.C. Gossard, L. A. Coldren, �??High-efficiency and low-threshold InGaAs/AlGaAs quantum-well lasers,�?? J. of Appl. Physics 76 , 3932-3934 (1994). [CrossRef]
- N. Kirstaedter, O. G. Schmidt, N. N. Ledentsov, D. Bimberg, V. M. Ustinov, A. Y. Egorov, A.E. Zhukov, M. V. Maximov, P.S. Kopev and Z. I. Alferov, �??Gain and differential gain of single layer InAs/GaAs quantum dot injection lasers,�?? Appl. Phys. Lett. 69, 1226-1228 (1996). [CrossRef]
- J. R. Sambles, �??Grain-boundary scattering and surface-plasmon attenuation in noble-metal films,�?? Solid State Comm. 49, 343-345 (1984). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.