## Surface plasmonic lasing via the amplification of coupled surface plasmon waves inside dielectric-metal-dielectric waveguides

Optics Express, Vol. 16, Issue 20, pp. 16113-16123 (2008)

http://dx.doi.org/10.1364/OE.16.016113

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### Abstract

Coupling of surface plasmon (SP) waves between two metal-dielectric interfaces of a dielectric-metal-dielectric (DMD) waveguide, in which one of the dielectric layers is incorporated with optical gain, is proposed to realize plasmonic lasing. The propagation characteristics of the SP waves supported inside the DMD waveguides are studied by finite-difference time-domain method. It is found that there are optimized thicknesses for the metal film and gain region to obtain minimum propagation loss. Furthermore, a coupled-mode model is developed to analyze the lasing characteristics of the DMD waveguides with coherent optical feedback. The conditions to achieve single-longitudinal-mode lasing from the DMD waveguides are also investigated.

© 2008 Optical Society of America

## 1. Introduction

2. 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]

3. 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]

4. M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express **16**, 1385–1392 (2008). [CrossRef] [PubMed]

5. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express **12**, 4072–4079 (2004). [CrossRef] [PubMed]

## 2. Propagation characteristics of SP waves - FDTD analysis

*x*and

*z*directions where

*x*-coordinate is the direction of propagation. In addition, the refractive indices of the dielectric medium and metal are assumed equal to be 1.54 and 0.0537+3.948i respectively. A free plane-wave with wavelength of ~594 nm is end-fired in the

*x*-direction at the top dielectric layer at few tens of nanometers above the surface of the metal film (i.e., at

*y*>15 nm) and coupled to the SP waves on the top of the metal film.

*x*direction. It must be noted that the SP waves are excited only on one side of the metal-dielectric interface. Hence, this asymmetric excitation of SP waves along the

*y*direction causes the coupling of SP waves between the two interfaces of the DMD waveguides. Furthermore, the asymmetric excitation causes the modal profile of SP waves differently to that of the eigenmodes excited symmetrically inside the DMD multilayer [6

6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B. **61**, 10484–10503 (2000). [CrossRef]

*x*>12 µm. On the other hand, for the case given in Fig. 1(b) where the bottom dielectric layer has an optical gain of ~600 cm

^{-1}, the SP waves can propagate much longer than that given in Fig. 1(a). It must be noted that the real part of refractive indices of the top dielectric layer and bottom gain region are assumed to be the same. This indicated that the introduction of optical gain to the bottom dielectric layer can help to sustain the propagation of SP waves. This is because the extra optical gain compensates the propagation loss of the SP waves inside the dielectric layers.

*M*, of the DMD waveguide versus the thickness of metal film,

*t*.

_{m}*M*is defined as the inverse of the distance between the two adjacent field intensity peaks along the propagation direction

*x*, see also Fig 1(b). The parameter

*M*reviews the efficiency of energy transfer between the two metal-dielectric interfaces when the SP waves propagating along the

*x*direction. It is observed that the values of α and

*M*increase with the reduction of

*t*. This is because the electric field of asymmetric SP waves is strongly confined (i.e., more portion of electric field being interacted with the metal layer) inside metal layer with thinner thickness so that the value of α increases with the decrease of

_{m}*t*. Furthermore, it is found that the propagation coefficient of the SP waves, β, remains unchanged at 16 µm

_{m}^{-1}for various values of

*t*.

_{m}*M*of the DMD waveguide versus the thickness of gain region,

*t*. In the calculation, different values of

_{g}*t*(40 to 60 nm) and imaginary refractive index of the gain region,

_{m}*n*, (-0.005 to -0.01 which correspond to the optical gain of ~260 to ~510 cm

_{imag}^{-1}) were used. It is observed that the values of α reduced with

*t*and saturated for

_{g}*t*greater than 400 nm for most of the cases. On the other hand, the values of

_{g}*M*are less dependent on the variation of

*t*. It is noted that there is a minimum value of

_{g}*t*to maximize the confinement of SP waves within the gain region. However, the values of

_{g}*M*is less dependent on

*t*as

_{g}*M*is mainly determine by the thickness of the metal thin film as well as the refractive indices of the dielectric materials surrounding the metal thin film. From the above calculations, it is found that the optimized values of

*t*and

_{m}*t*are ~60 and ~400 nm respectively. This is because these values of parameters can obtain a small value of α for the SP waves as well as the dimensions of the dielectric waveguide can be minimized to subwavelength dimensions.

_{g}*n*

_{real}, of the host matrix (i.e., the dielectric layer with gain to be introduced) may vary due to the introduction of dopants and this may lead to the excitation of conventional waveguide modes. Hence, it is interesting to study the influence of small variation of

*n*

_{real}on the propagation characteristics of DMD waveguides. Figure 4 shows the variation of α and

*M*with the small change of nreal. It is observed that a slight decrease of

*n*

_{real}reduces the propagation loss of the waveguides. This is because the reduction of refractive index reduces the confinement of SP waves at the dielectric-metal interface so that the SP waves experience less absorption loss from the metal. Hence, for the design of DMD waveguides, it is preferred that the gain medium has slightly less refractive index when compared to that of the cladding layer in order to achieve relatively low loss waveguide. However, if the different of

*n*

_{real}between gain region and cladding layer is larger than 0.01, this may lead to the excitation of conventional waveguide modes. As a result, the coupling of SP waves between the two dielectric-metal interfaces may not be supported.

## 3. Coupled-mode model for the DMD waveguides

_{2}, Ag, dye doped polymer, and bulk SiO

_{2}substrate respectively.

_{2}as well as to obtain high optical gain under optical excitation, Rhodamine 6G (R6G) doped PVC active medium is selected to amplify ~594 nm stimulated emission under 532 nm light excitation [4

4. M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express **16**, 1385–1392 (2008). [CrossRef] [PubMed]

8. H. M. A. Moneim, L. Z. Ismail, G. A. Fatah, and Z. A. Zohdy, “Radiative fluorescence lifetime of rhodamin doped in PVC,” Polym. Test. **20**, 135–139 (2001). [CrossRef]

9. B. G. Huth and M. R. Kagan, “Dynamics of Flashlamp-pumped Rhodamine 6G Laser,” IBM J. Res. Dev. **15**, 278 (1971). [CrossRef]

*N*

_{2}and

*N*

_{1}are the populations at the upper and lower energy levels respectively of the two-level system.

*N*

_{dye}is the total dye concentration and

*τ*is the carrier lifetime. The polarization dependent pump absorption rates (

_{R2}*R*

_{13}) and signal absorption (

*W*) and emission rates (

_{a}*W*) can be written by

_{e}*h*is Planck’s constant,

*ν*is the frequency of pumped (signal) light,

_{p(s)}*A*is the effective area,

_{eff}*σ*

^{13}

*is the absorption cross section of the pump light and,*

_{P}*σ*and

^{a}_{S}*σ*is the absorption and emission cross section of the signal light respectively.

^{e}_{S}*P*

_{pump}and

*P*

^{0}

*(*

_{S}*λ*) is the pump and signal power respectively.

_{s}*λ*(

_{p}*λ*) is the pumped (signal) wavelength and

_{s}*S*is the normalized signal power.

*S*

^{+}

*and*

_{C(A)}*S*

^{-}

*, which propagation characteristics are shown in Fig. 6, can be described by a modified coupled-mode equations [7]*

_{C(A)}*r*

_{L}and

*r*are the facets reflectivity at

_{R}*x*=0 and

*x*=

*L*respectively.

*χ*

^{±}

*is the spontaneous emission noise coupled to the signal fields. It is assumed that i) the distribution of spontaneous emission noise has a Gaussian profile and ii) the spontaneous emission fields coupled into the forward and reverse fields have the same magnitude. Hence, the spontaneous emission noise can be generated from a Gaussian distributed random number generator that satisfies the following correlation functions [11*

_{A}11. L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a 2nd-order dfb laser using time-domain large-signal travelingwave model,” IEEE J. Quantum Elect. **30**, 1389–1395 (1994). [CrossRef]

*P*is the magnitude of the spontaneous emission power per area and is defined as

_{spont}*β*(=10

_{s}^{-5}) is the spontaneous coupling factor. Hence, by solving the rate equations of populations and coupled-mode equations of the SP waves self-consistently [11

11. L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a 2nd-order dfb laser using time-domain large-signal travelingwave model,” IEEE J. Quantum Elect. **30**, 1389–1395 (1994). [CrossRef]

*x*with temporal time steps taken as Δ

*t*=Δ

*x*/

*ν*, so that the optical field is sampled at the rate

_{g}*f*=l/Δ

_{s}*t*.

## 4. DMD waveguide with coated facets

*L*is 300 µm and the facet coatings,

*r*

_{L}and

*r*, both have equal reflectivity of 0.92 (i.e., Ag coated). In order to model the propagation characteristics of SP waves by using coupled mode equations, it is assumed that α=500 cm

_{R}^{-1},

*M*=0.2 cm

^{-1}and Γ~1 as deduced from section 2 for the optimized dimensions (i.e., optimized values of

*t*and

_{m}*t*) of the DMD waveguides. The material parameters for the dye doped polymer [4

_{g}4. M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express **16**, 1385–1392 (2008). [CrossRef] [PubMed]

*A*=3×10

_{eff}^{-11}m

^{2},

*N*

_{dye}=1.3×10

^{25}m

^{-3},

*σ*

^{13}

*=4.3×10*

_{P}^{-20}m

^{2},

*σ*=2.1×10

^{a}_{S}^{-20}m

^{2},

*σ*=2.0×10

^{e}_{S}^{-20}m

^{2}, and

*τ*=3.0×10

_{R2}^{-9}s. The refractive indices of SiO

_{2}and doped PVC are assumed equal to 1.54. In the calculation, the laser is divided into 600 (=

*L*/Δ

*x*) equal length sections and the laser is switched on by the pump power with a step function

*P*is the excitation power.

_{ext}12. S. F. Yu, T. I. Yuk, and P. Shum, “Dynamic analysis of erbium-doped optically pumped waveguide lasers using a time-domain travelling wave model,” Opt. Quantum Electron. **29**, 683–696 (1997). [CrossRef]

*H*(

*ω*), used to model the optical gain profile can be expressed as

*b*∈(0,1) represents the filter width,

*ω*is the frequency of the gain peak and Δ

_{p}*t*is the time step. The discrete implementation of this filter for the forward mode (and similarly for the reverse mode) is

*B*=

*b*exp(

*jω*

_{p}Δ

*t*). A digital filter with

*b*=0.009 and Δ

*t*=20 fs is used to approximate the optical gain of dye solution with bandwidth of ~3 THz.

*P*equals to 1.03 times its threshold. It is observed that both

_{ext}*S*and

_{A}*S*(

_{C}*N*

_{2}) exhibit in-phase (out-of-phase) relaxation oscillation. It is noted that mode beating of both

*S*

_{A}and

*S*is due to the excitation of side modes. In addition, the formation of closely spaced Fabry Perot modes is the result of high facet reflectivity and long cavity length of the DMD waveguides.

_{C}*S*and

_{A}*S*is explained in Fig. 8(a). It is clearly shown that

_{C}*S*and

_{A}*S*are varied out of phase along the laser cavity (i.e., energy transfers from active region to cladding region and vice versa), similar to the intensity distribution in Fig. 1. Figure 8(b) shows the influence of plasmonic coupling on the resultant lasing spectra. It is noted that the mode spacing of Fabry Perot modes is about ~300 GHz without the influence of plasmonic coupling. However, the presence of plasmonic coupling, which is equivalent to the coupling of two Fabry Perot cavities together, induced extra modes in the emission spectrum.

_{C}## 5. DMD waveguides with periodic grating

^{-3}.

*κ*, can be deduced from [12

12. S. F. Yu, T. I. Yuk, and P. Shum, “Dynamic analysis of erbium-doped optically pumped waveguide lasers using a time-domain travelling wave model,” Opt. Quantum Electron. **29**, 683–696 (1997). [CrossRef]

*b*is the lower index region of the grating, Δ

*n*is the refractive index different of the grating, and Γ

_{g}is the confinement factor of the SP waves inside the grating. If the grating is realized by direct UV imprinting technique with Δ

*n*~5×10

^{-3}and

*b*/Λ ~0.75, it can be shown that κ ~80 cm

^{-1}for Γ

_{g}~1. It is possible to obtain Γ

_{g}~1 as the depth of the grating can match with the penetration depth of the SP waves inside the cladding layer provided that the depth of the grating is more than 200 nm.

*κ*~80 cm

^{-1}pumped at 1.03 times threshold. It is assumed that the cleaved facets are anti-reflection (AR) coated to avoid the formation of Fabry Perot modes. The transient response of the DMD waveguide with periodic grating is similar to that of the Fabry Perot cavities. However, mode beating is still observed in its transient response. Figure 11 shows the corresponding steady-state spatial distributions of

*S*

_{A},

*S*

_{C},

*N*

_{2}as well as the lasing spectra of

*S*

_{A}at different

*P*. In Fig. 11(a) and 11(b), the spatial distributions of

_{ext}*S*and

_{A}*S*at

_{C}*P*=

_{ext}*P*

_{1}are magnified by 5 times whereas that at

*P*=

_{ext}*P*

_{2}and

*P*

_{3}are shifted up by 500 mW and 1000 mW respectively so that they can be easily read from the figures. It is observed that the mode beating is due to the excitation of two bandgap modes. This is expected as 1

^{st}order uniform grating will support bandgap modes [13

13. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Elect. **21**, 144–150 (1985). [CrossRef]

*r*

_{L}and

*r*

_{R}~0.2. The presence of small facet reflectivity breaks the symmetric of the periodic grating as well as excites bandgap mode. It is also observed that stable single-mode operation can be obtained even at higher pump intensities. Hence, it has shown that it is possibility to achieve single-mode lasing from DMD waveguides.

## 6. Discussions and conclusions

_{2}interfaces due to the asymmetric excitation. The propagation characteristics of SP waves in the DMD waveguides are equivalent to that of the guided modes in the conventional directional couplers. It is found that the coupling characteristics of SP waves are dependent on the thickness of Ag film as well as the refractive index of dielectric layers. It is also observed that the DMD waveguides with the Ag thickness equal to ~60 nm can minimize the corresponding propagation loss to less than 500 cm

^{-1}. However, if dielectric layers with higher refractive index are used for the design of DMD waveguides, coupling characteristics of SP waves will not be supported unless longer wavelength of SP waves is used. For example, for SP waves with wavelength of ~1550 nm, materials such as InP or GaAs (i.e., refractive index of 3.4) can be used as the dielectric layers to realize coupling of SP waves.

^{3}cm

^{-1}at ~600 nm [3

3. 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]

**16**, 1385–1392 (2008). [CrossRef] [PubMed]

^{-1}in order to suppress the influence of facet reflection.

## Acknowledgment

## References and links

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

2. | 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. |

3. | 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. |

4. | M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express |

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

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

7. | S. L. Chuang, |

8. | H. M. A. Moneim, L. Z. Ismail, G. A. Fatah, and Z. A. Zohdy, “Radiative fluorescence lifetime of rhodamin doped in PVC,” Polym. Test. |

9. | B. G. Huth and M. R. Kagan, “Dynamics of Flashlamp-pumped Rhodamine 6G Laser,” IBM J. Res. Dev. |

10. | Ya. I. Khanin, |

11. | L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a 2nd-order dfb laser using time-domain large-signal travelingwave model,” IEEE J. Quantum Elect. |

12. | S. F. Yu, T. I. Yuk, and P. Shum, “Dynamic analysis of erbium-doped optically pumped waveguide lasers using a time-domain travelling wave model,” Opt. Quantum Electron. |

13. | R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Elect. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(140.3460) Lasers and laser optics : Lasers

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 31, 2008

Revised Manuscript: June 7, 2008

Manuscript Accepted: July 21, 2008

Published: September 26, 2008

**Citation**

Ashwani Kumar, S. F. Yu, X. F. Li, and S. P. Lau, "Surface plasmonic lasing via the amplification of coupled surface plasmon waves inside dielectric-metal-dielectric waveguides," Opt. Express **16**, 16113-16123 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16113

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
- 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]
- 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]
- M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, "Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium," Opt. Express 16, 1385-1392 (2008). [CrossRef] [PubMed]
- M. P. Nezhad, K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides," Opt. Express 12, 4072-4079 (2004). [CrossRef] [PubMed]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B. 61, 10484-10503 (2000). [CrossRef]
- S. L. Chuang, Physics of Optoelectronic Devices (John Wiley and Sons, 1995).
- H. M. A. Moneim, L. Z. Ismail, G. A. Fatah, and Z. A. Zohdy, "Radiative fluorescence lifetime of rhodamin doped in PVC," Polym. Test. 20, 135-139 (2001). [CrossRef]
- B. G. Huth and M. R. Kagan, "Dynamics of Flashlamp-pumped Rhodamine 6G Laser," IBM J. Res. Dev. 15, 278 (1971). [CrossRef]
- Ya. I. Khanin, Fundamentals of Laser Dynamics (Cambridge International Science Publishing, 2005).
- L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, "Dynamic analysis of radiation and side-mode suppression in a 2nd-order dfb laser using time-domain large-signal traveling-wave model," IEEE J. Quantum Elect. 30, 1389-1395 (1994). [CrossRef]
- S. F. Yu, T. I. Yuk, and P. Shum, "Dynamic analysis of erbium-doped optically pumped waveguide lasers using a time-domain travelling wave model," Opt. Quantum Electron. 29, 683-696 (1997). [CrossRef]
- R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Elect. 21, 144-150 (1985). [CrossRef]

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