## Theory of plasmonic femtosecond pulse generation by mode-locking of long-range surface plasmon polariton lasers |

Optics Express, Vol. 20, Issue 1, pp. 462-473 (2012)

http://dx.doi.org/10.1364/OE.20.000462

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### 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

*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

1. J. Seidel, S. Grafstroem, 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]

6. A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. **11**, 2231–2235 (2011). [CrossRef] [PubMed]

7. 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. **35**, 1197–1199 (2010). [CrossRef] [PubMed]

8. 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]

10. 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**, 1110–1112 (2009). [CrossRef] [PubMed]

10. 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**, 1110–1112 (2009). [CrossRef] [PubMed]

11. 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]

12. M. I. Stockman, “Spasers explained,” Nature Photon. **2**, 327–329 (2008). [CrossRef]

13. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B **71**, 115409 (2005). [CrossRef]

20. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. **12**, 024004 (2010). [CrossRef]

21. G. Winter, S. Wedge, and W. L. Barnes, “Can lasing at visible wavelengths be achieved using the low-loss long-range surface plasmon-polariton mode?” New J. Phys. **8**, 211102 (2006). [CrossRef]

24. D. Yu. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express **19**, 12524–12531 (2011). [CrossRef] [PubMed]

20. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. **12**, 024004 (2010). [CrossRef]

## 2. Surface plasmon polariton laser equation

*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 where

*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*).

33. P. Meystre and M. Sargent III, *Elements of Quantum Optics*, 4th ed. (Springer Verlag, Berlin, 2007). [CrossRef]

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

_{g}*= [Γ*

_{n}*+*

_{g}*i*(

*ω*–

_{n}*ω*)]

_{L}^{−1},

*N*=

_{g}*N*(

_{g}*x, z,t*) is a space-time dependent population inversion,

*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.

*g*, where

_{n}A_{n}*g*is the transient nonlinear mode gain. Using Eq. (2) we obtain where

_{n}*κ*= 2|Re[

*α*(

_{c}*ω*)]|,

_{L}*ω*is the lasing frequency,

_{L}*ω*,

_{n}*K*,

_{n}*γ*and

_{n}*n*

_{eff,n}with

*ω*,

*K*,

*γ*, and

*n*

_{eff}, respectively.

*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

28. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. **QE-11**, 736–746 (1975). [CrossRef]

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

*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 where

*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},

*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.

*g*must be partly recovered during one round trip. The linear parts of gain and loss rates are represented by

_{i}*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] where

*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}20. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. **12**, 024004 (2010). [CrossRef]

*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

_{R}28. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. **QE-11**, 736–746 (1975). [CrossRef]

34. J. Herrmann and F. Weidner, “Theory of passively mode-locked cw dye lasers,” Appl. Phys. B **27**, 105–113 (1982). [CrossRef]

## 4. Design of long range SPP lasers

*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

36. P. Berini, “Long-range surface plasmon polaritons,” Advances in Optics and Photonics **1**, 484–588 (2009). [CrossRef]

37. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. **20**, 395–431 (1988). [CrossRef]

38. D. P. Benfey, D. C. Brown, S. J. Davis, L. G. Piper, and R. F. Foutter, “Diode-pumped dye laser analysis and design,” Appl. Opt. **31**(33), 7034–7041 (1992). [CrossRef] [PubMed]

40. A. A. Ishchenko, “Laser media based on polymethine dyes,” Quantum Electron. **24**, 87–172 (1994). [CrossRef]

*σ*= 1.8 × 10

_{s}^{−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

37. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. **20**, 395–431 (1988). [CrossRef]

*σ*= 1.5 × 10

_{aq}^{−16}cm

^{2}(Ref. [38

38. D. P. Benfey, D. C. Brown, S. J. Davis, L. G. Piper, and R. F. Foutter, “Diode-pumped dye laser analysis and design,” Appl. Opt. **31**(33), 7034–7041 (1992). [CrossRef] [PubMed]

*τ*

_{0q}= 22 ps (Ref. [39

39. B. Kopainsky, P. Qiu, W. Kaiser, B. Sens, and K. H. Drexhage, “Lifetime, photostability, and chemical structure of IR heptamethine cyanine dyes absorbing beyond 1 mm,” Appl. Phys. B **29**, 15–18 (1982). [CrossRef]

40. A. A. Ishchenko, “Laser media based on polymethine dyes,” Quantum Electron. **24**, 87–172 (1994). [CrossRef]

*= 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.

41. B. H. Soffer and B. B. McFarland, “Continuously tuable, narrow band organic dye lasers,” Appl. Phys. Lett. **10**, 266–267 (1967). [CrossRef]

42. A. Costela, I. Garcia-Moreno, and C. Gomez, “Efficient and stable dye laser action from modified dipyrromethene BF2 complexes,” Appl. Phys. Lett. **79**, 305–307 (2001). [CrossRef]

43. P. Runge and R. Rosenberg, “Unconfined flowing-dye films for CW dye lasers,” IEEE J. Quantum Electron. **8**, 910–911 (1972). [CrossRef]

44. A. Costela, I. Garcia-Moreno, R. Sastre, D. W. Coutts, and C. E. Webb, “High repetition- rate polymeric solid-state dye lasers pumped by a copper-vapor laser,” Appl. Phys. Lett. **79**, 452–454 (2001). [CrossRef]

46. R. Bornemann, U. Lemmer, and E. Thiel, “Continuous-wave solid-state dye laser,” Opt. Lett. **31**, 1669–1671 (2006). [CrossRef] [PubMed]

*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

31. K. Kawano and T. Kitoh, *Introduction to Optical Waveguide Analysis* (John Wiley and Sons, New York, 2001). [CrossRef]

*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}.

*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*= 400 nm and

_{q}*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.

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

23. I. D. Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express **17**, 20191–20202 (2009). [CrossRef] [PubMed]

38. D. P. Benfey, D. C. Brown, S. J. Davis, L. G. Piper, and R. F. Foutter, “Diode-pumped dye laser analysis and design,” Appl. Opt. **31**(33), 7034–7041 (1992). [CrossRef] [PubMed]

*τ*

_{0g}due to fluorescence quenching of the gain molecules by the dipole energy transfer to the metal layer [48

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

49. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. **45**, 661–699 (1998). [CrossRef]

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

23. I. D. Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express **17**, 20191–20202 (2009). [CrossRef] [PubMed]

49. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. **45**, 661–699 (1998). [CrossRef]

37. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. **20**, 395–431 (1988). [CrossRef]

^{−3}(Refs. [38

**31**(33), 7034–7041 (1992). [CrossRef] [PubMed]

40. A. A. Ishchenko, “Laser media based on polymethine dyes,” Quantum Electron. **24**, 87–172 (1994). [CrossRef]

## 5. Numerical results and discussion

28. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. **QE-11**, 736–746 (1975). [CrossRef]

34. J. Herrmann and F. Weidner, “Theory of passively mode-locked cw dye lasers,” Appl. Phys. B **27**, 105–113 (1982). [CrossRef]

^{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}.

^{2}as well as by available pump sources.

## 6. Conclusion

^{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)

*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).

*δω*can be expanded as follows: On the other hand, from Eq. (7, 8) we have where

*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: The frequency dependent gain is where and

*β*is a value of

*β*given at

_{n}*ω*. From Eq. (A-11), By using Eq. (A-8), we can rewrite the above equation: Summarizing Eqs. (A-7, A-9, A-10, and A-14),

_{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*,

*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).

## References and links

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**OCIS Codes**

(140.4050) Lasers and laser optics : Mode-locked lasers

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: September 30, 2011

Revised Manuscript: October 31, 2011

Manuscript Accepted: October 31, 2011

Published: December 21, 2011

**Citation**

Kwang-Hyon Kim, Anton Husakou, and Joachim Herrmann, "Theory of plasmonic femtosecond pulse generation by mode-locking of long-range surface plasmon polariton lasers," Opt. Express **20**, 462-473 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-462

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