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

doi:10.1364/OE.20.000462

« Show journal navigation

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

Kwang-Hyon Kim, Anton Husakou, and Joachim Herrmann »View Author Affiliations

Optics Express, Vol. 20, Issue 1, pp. 462-473 (2012)
http://dx.doi.org/10.1364/OE.20.000462

View Full Text Article

Acrobat PDF (1102 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Full Text
Article Info
References (49)
Cited By
Figures (5)
Metrics

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.

1. Introduction

Surface plasmon polaritons (SPPs) allow the localization of light below the diffraction limit and promise progress in miniaturization of sophisticated compact optical devices with new functionalities. One of the main limitation for applications of SPPs is its short propagation length which is typically in the range of 30 ∼ 200 μm 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

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]

] and amplification of spontaneous emission [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]

] of SPPs. The next step in this development was the realization of lasers on the nanoscale by appropriate feedback [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]

]. The smallest laser reported to date has been achieved [10

] by the realization of a concept developed by Bergman and Stockman [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]

] by using a single gold core and dye doped silica shell structure. Additional theoretical studies of spasers [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]

] and SPP amplification [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]

] have been published. Especially, the possibility of ultrafast plasmon amplification has been proposed in Ref. [20

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

The main purpose of the present paper is to investigate the possibility of ultrashort surface plasmon pulse generation which may find applications for ultrafast plasmonic applications including ultrafast surface spectroscopy, ultrafast surface nonlinear optics, highly integrated plasmonic information processors and others. We theoretically study, for the first time to our knowledge, passive mode-locking in SPP lasers by slow saturable absorbers and predict the possibility of femtosecond plasmon pulse generation in such lasers.

We consider a configuration (Fig. 1) with a dielectric layer c composed of a gain sublayer g and a sublayer with a saturable absorber sa adjacent to the metal film b. At the interface of the flat continuous metal film b SPPs can be excited which are confined to the proximity of the metal-dielectric interface and decay exponentially in both media. Optical pumping from below the dielectric layer c leads to a population inversion in the gain medium. Feedback in this scheme is realized by using a SPP grating at both end sides of the metal layer [25

A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Light. Tech. 24(2), 912–918 (2006). [CrossRef]

]. Such a grating with an appropriate period coherently couples the forward-propagating plasmon mode to the backward-propagating mode providing efficient reflection as well as laser outcoupling to the traveling waves. Passive mode-locking can be achieved by the combined action of slow saturable gain and slow saturable absorption in one round trip. Such a mode-locking mechanism is realized in standard dye lasers [26

J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light Pulses (North-Holland, Amsterdam, 1987).

–28

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

] where both media exhibit different saturation intensities and different focusing. In a SPP laser considered here, the saturation in the absorber layer sa is stronger than in the gain sublayer g because of the stronger field localization of the SPP modes as shown in Fig. 1. Everywhere in the paper below, we denote the quantities related to gain by subindex g and for saturable absorber by subindex q.

Fig. 1 (Color online) Configuration of the SPP laser. The layer c is a dielectric layer with gain (green). For mode locking, we add a saturable absorber layer (gray-colored) into the layer c adjacent to the metal film layer b. The absorber and gain media are doped in the same host material with permittivity ɛ_c. For the feedback we take a Bragg reflectors of SPPs as the resonator mirrors.

2. Surface plasmon polariton laser equation

In this section, we study the evolution of longitudinal SPP modes. First, we consider a passive SPP resonator. The dispersion relation for propagating SPPs is given by (see Ref. [29

M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, Chichester-New York-Brisbane-Toronto, 1981).

–31

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

])

(ɛ_{a} α_{b} + ɛ_{b} α_{a}) (ɛ_{b} α_{c} + ɛ_{c} α_{b}) + (ɛ_{a} α_{b} - ɛ_{b} α_{a}) (ɛ_{b} α_{c} - ɛ_{c} α_{b}) e^{- 2 α_{b} d} = 0,

(1)

where d is the sub-wavelength thickness of the metal film, ɛ_a (ω), ɛ_b (ω) and ɛ_c (ω) are the frequency-depending dielectric functions of layers a, b and c taken from Ref. [32

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985).

]. The constants α_a, α_b and α_c are directly related with the wavenumber K of propagating SPP by

α_{j}^{2} = K^{2} - ɛ_{j} ω^{2} / c^{2}

, where j = a, b and c, ω is the angular frequency of light and c is the light velocity in vacuum. For the passive SPP resonator in Fig. 1, the counterpropagating fields interfere resulting in intracavity modes described as E_n = U_ne^{−(γ_n+iΩ_n)t}, where indices n indicate longitudinal mode numbers and the mode function U_m (x, z) is given by

\begin{array}{l} U_{n} = A_{a n} q_{a n}^{-} e^{- α_{a n} x} & (\infty > x \geq 0) \\ U_{n} = A_{b n}^{+} q_{b n}^{+} e^{α_{b n} x} + A_{b n}^{-} q_{b n}^{-} e^{- α_{b n} x} & (0 > x \geq - d) \\ U_{n} = A_{c n} q_{c n}^{+} e^{α_{c n} (x + d)} & (- d > x \geq - \infty) \end{array},

(2)

where

q_{j n}^{\pm} = (\hat{x} sin K_{n} z \pm \hat{z} (α_{j n} / K_{n}) cos K_{n} z)

, A_j (j = a, b and c) are field amplitudes for each layers, x̂ and ẑ are unit vectors along the x and z axes, respectively, K_n = n_eff,_nΩ_n/c, n_eff,_n is the linear mode index and Ω_n is the angular frequency of the n-th longitudinal mode of the passive SPP resonator. From the field equations, we can see that SPP polarization direction is altered from normal to parallel at the interface in a half spatial period of the longitudinal interferometric fringe of each mode. Nevertheless, normal components are dominant in layers a and c because |α_jn/K_n| ≪ 1, j = a, and c. The relations between the field amplitudes are defined by boundary condition for the electromagnetic field and are given by

A_{a n} = \frac{ɛ_{c} α_{b n} + ɛ_{b} α_{c n}}{ɛ_{a} α_{b n} - ɛ_{b} α_{a n}} e^{α_{b n} d} A_{c n},

(3)

A_{b n}^{\pm} = \frac{(ɛ_{c} α_{b n} \pm ɛ_{b} α_{c n})}{2 ɛ_{b} α_{b n}} e^{\pm α_{b n} d} A_{c n} .

(4)

In the expressions for the mode fields the wavenumber K_n, the field attenuation factors in x-direction α_jn and the resonator loss γ_n obey the following relations:

K_{n}^{2} = α_{j n}^{2} - \frac{ɛ_{j} (Ω_{n})}{c^{2}} {(γ_{n} + i Ω_{n})}^{2},

(5)

where j = a, b and c. In fact, additional loss in the resonator arises due to the imperfect coupling and reflectance of the resonator mirrors which is included into the parameter γ_n. The additional modification for modal structure and loss is related to the finite width of the waveguide in y direction. For a waveguide thickness much larger than d but smaller than a few μm there exist only one mode in this direction and its properties can be calculated using the effective index approach.

Next, we consider the evolution of modes of an active SPP resonator with real wavenumbers and complex frequencies. The fields in layers 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

E_{c} = \frac{1}{2} \sum_{n} A_{n} (t) {\hat{x} sin (K_{n} z) + \hat{z} (α_{c n} / K_{n}) cos (K_{n} z)} e^{α_{c n} (x + d) - i ω_{n} t} + c . c .,

(6)

where ω_n is the angular frequency of n-th mode in the active resonator and we assume that ω_n ≈ Ω_n. The induced polarization is described by the same expression replacing the field amplitude A_n (t) with P_n (t).

Substituting the above equation into Maxwell’s equations for TM waves and using the slowly varying envelope approximation (SVEA) and the rotating-wave approximation for the induced polarization [33

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

], we obtain the master equation for the SPP laser

{\dot{A}}_{n} + γ_{n} A_{n} = \frac{σ_{g} \sqrt{ɛ_{c}} 𝒟_{n} Γ_{g} A_{n}}{M_{n}} \int_{0}^{L} \int_{- D_{g} - d_{g}}^{- d_{g}} {| U_{n} |}^{2} N_{g} d z d x,

(7)

where ρ_g is electric dipole moment of the gain, 𝒟_n = [Γ_g + i (ω_n – ω_L)]⁻¹,

T_{2 g} = Γ_{g}^{- 1}

is the dephasing time,

σ_{g} = ρ_{g}^{2} ω_{n} / (2 ɛ_{0} ɛ_{c} \bar{h} Γ_{g})

is the gain cross-section, N_g = N_g (x, z,t) is a space-time dependent population inversion,

M_{n} = \int_{0}^{L} \int_{- \infty}^{\infty} d z d x {| U_{n} (x, z) |}^{2} Re [n (x)]

is a normalization factor, n(x,ω_n) is the refractive index different at each layer, L is the length of the SPP resonator, ɛ₀ 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.

We describe the right hand of Eq. (7) as g_nA_n, where g_n is the transient nonlinear mode gain. Using Eq. (2) we obtain

g_{n} = β_{n} σ_{g} \int_{- D_{g} - d_{g}}^{- d_{g}} {\bar{N}}_{g} (x, t) e^{κ (x + d)} d x,

(8)

where

β_{n} = Γ_{g} 𝒟_{n} (1 + {| \frac{α_{c n}}{K_{n}} |}^{2}) \sqrt{ɛ_{c}} / (2 M_{n})

, κ= 2|Re[α_c(ω_L)]|, ω_L is the lasing frequency,

{\bar{N}}_{g} (x, t) = \int_{0}^{L} N_{g} (x, z, t) d z / L

. The analogous procedure can be applied for the slow saturable absorber which yields similar expressions as above. For the study of mode-locking, below we consider the modes as a continuum by replacing ω_n, K_n, γ_n and n_eff,n with ω, K, γ, and n_eff, respectively.

In difference to bulk mode-locked lasers, in the case of SPP lasers the mode field is confined in the vicinity of the metallic layer. Hence, the strength of gain and absorption saturation also depend on the position 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

J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena , 2nd ed. (Academic Press, San Diego, 2006).

, 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

The sum in Eq. (6) is replaced by the forward propagating field A(T,τ) = ∫A(t,k)e^{−iδωt+ikz}dk, where k = K – K_L = (n_eff/c)(ω – ω_L), δω = ω – ω_L, and 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_g, where 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

T_{R} \frac{\partial A}{\partial T} = [g (τ) - q (τ) - γ_{0}] A + δ_{1} \frac{\partial A}{\partial τ} + δ_{2} \frac{\partial^{2} A}{\partial τ^{2}},

(9)

where g(τ), q(τ) and γ₀ are the total gain, the absorber loss and the resonator loss for a round trip, γ₀ = γ|_k₌₀,

δ_{1} = {- i v_{g}^{- 1} (\partial γ / \partial k) |}_{k = 0} + g_{i} Γ_{g}^{- 1} - q_{i} Γ_{q}^{- 1}

and

δ_{2} = g_{i} Γ_{g}^{- 2} - q_{i} Γ_{q}^{- 2} + {(2 v_{g}^{2})}^{- 1} {(\partial^{2} [i ω (k) (T_{R} + g_{i} Γ_{g}^{- 1} - q_{i} Γ_{q}^{- 1}) + γ (k)] / \partial k^{2}) |}_{k = 0}

, respectively. The detailed derivation can be found in the Appendix. Here the dispersion of the plasmonic modes is explicitly taken into account by the function ω (k) defined by the inverse function of k = (n_eff/c)(ω – ω_L). The dependence of δ₁ and δ₁ 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.

For stable mode-locked operation, g_i must be partly recovered during one round trip. The linear parts of gain and loss rates are represented by

g_{i} = T_{R} β \int_{- D_{g} - D_{q} - d}^{- D_{q} - d} g_{l} e^{κ (x + d)} d x

and

q_{i} = T_{R} β \int_{- D_{q} - d}^{- d} q_{l} e^{κ (x + d)} d x

, respectively, and β is defined below Eq. (8) with the substitution ω_n → ω_L, g_l and q_l are the nonlinear local gain and loss rates dependent on the spatial and temporal variables, 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

J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena , 2nd ed. (Academic Press, San Diego, 2006).

]

\frac{\partial g_{l}}{\partial τ} = - \frac{g_{l} - g_{0} (x)}{τ_{0 g}} - g_{l} \frac{{| E_{c} (x, τ) |}^{2}}{A_{s g}^{2} τ_{0 g}},

(10)

\frac{\partial q_{l}}{\partial τ} = - \frac{q_{l} - q_{0} (x)}{τ_{0 q}} - q_{l} \frac{{| E_{c} (x, τ) |}^{2}}{A_{s q}^{2} τ_{0 q}},

(11)

where τ_0g and τ_0q are the upper-level lifetimes, A_sg and A_sg are the saturation fields for gain and absorber dyes, respectively.

The SPP field intensity distribution is nonuniform and the saturation at the position nearer to metal surface is stronger [20

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

]. We do not use a power expansion for the gain with respect to the intensity [27

J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena , 2nd ed. (Academic Press, San Diego, 2006).

] but self-consistently solve the combined Eqs. (9–11). For slow passive mode-locking the gain and absorber dyes exhibit a longer relaxation time compared with the pulse duration: τ_0q, τ_0g < τ₀. Besides, stable mode-locking operation is possible only if the conditions 0.1 ≤ T_R/τ_0g ≤ 10 and τ_0q < T_R are fulfilled [26

J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light Pulses (North-Holland, Amsterdam, 1987).

–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]

]. The master equation for mode-locking of SPP lasers Eq. (9) can not be analytically solved because both the local gain and nonlinear absorption Eq. (10) depend on time and space, therefore we apply the split-step-Fourier method [35

G. P. Agrawal, Nonlinear Fiber Optics , 4th ed. (Elsevier, Amsterdam, 2007).

] to solve Eq. (9).

4. Design of long range SPP lasers

In this section, we discuss parameters for appropriate gain and absorber medium and determine the main structural parameters supporting lasing of SPPs.

The dielectric layers 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_L) = 2.20 (PMMA) and ɛ_b (λ_L) = −35.99+ 2.20i (silver), respectively [32

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985).

]. We restrict ourselves to the symmetric SPP mode [36

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

] which has the smallest propagation loss.

We assume that the gain and the saturable absorber sublayers are doped with dyes Styryl-9 (Ref. [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]

]) and IR-26 (Refs. [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]

]) in a PMMA polymer host. Pumping and lasing wavelengths are taken to be 532 nm and 900 nm, respectively. The main parameters of the dyes are as follows: For the gain medium using Styryl-9, cross-sections for stimulated emission at 900 nm and absorption at 532 nm are σ_s = 1.8 × 10⁻¹⁶ cm² and σ_ag = 1.2 × 10⁻¹⁶ cm², respectively, the upper-state relaxation time (or longitudinal relaxation time) is τ_0g = 400 ps and the dephasing rate is Γ_g = 3.3 × 10¹³ Hz (see Ref. [37

]). For saturable absorber molecules IR-26, absorption cross-section σ_aq = 1.5 × 10⁻¹⁶ cm² (Ref. [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]

]), upper-state relaxation time (or absorber recovery time) is τ_0q = 22 ps (Ref. [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]

]) and dephasing rate is Γ_q = 2.5 × 10¹³ 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 × 10¹⁸ cm⁻³ and 1 × 10¹⁷ cm⁻³, respectively.

Under the condition of CW pump operation, a severe problem in the use of dyes is the long-lived transient triplet-triplet absorption that gradually reduce the net gain and ultimately terminates the lasing process. In liquid dye lasers [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]

], this problem is solved by free-flying dye jets [43

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

] and in solid dye lasers by a rotating disc [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]

]. Here we consider as an alternative a solid-state dye gain medium that is optically pumped with pulsed light sources with ns duration.

We choose a length of the SPP laser resonator of 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 μm for a thickness 30 nm of the Ag layer b [29

M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, Chichester-New York-Brisbane-Toronto, 1981).

–31

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

]. Here we take it to be 1.8 μm and obtain for the effective index and propagation loss n_eff = 1.4988 and γ₀ = 44.78, respectively. For the structure described above, we find δ₁ = (2.9 – 3.0i) × 10⁻¹² s and δ₂ = (8.2 – 1.4i) × 10⁻²⁶ s².

Below we determine the appropriate range of thicknesses for the saturable absorber sublayer and metal layer 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/cm². 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/cm² for D_q = 400 nm and D_g = 5 μm, 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.

Fig. 2 (Color online) Long range SPP intensity profile normalized by its maximum (a) and the dependence of attenuation γ₀ and linear gain g₀ on the metal film thickness d (b). Incident pump intensities 5, 10 and 15 MW/cm² were considered. Resonator length is 1 cm, Styryl-9 concentration is N_g = 2.5 × 10¹⁸ cm⁻³, lasing wavelength is λ_L = 900 nm, the thicknesses of gain and absorber sublayers are D_q = 400 nm and D_g = 5 μm. The permittivities of the layers are ɛ_a (λ_L) = ɛ_c (λ_L) = 2.20 (PMMA) and ɛ_b (λ_L) = −35.99 + 2.20i (silver), respectively.

In the final part of the section, we consider the influence of the inhomogeneity of the pump. The intensity of the pump beam becomes inhomogeneous [see Fig. 3(a)] due to the absorption by the gain molecules and the formation of standing waves by the reflection from the metal film surface. The spatial modulation of the pump beam is calculated by using the matrix formulation [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]

,47

P. Yeh, Optical Waves in Layered Media (John Wiley, New York, 1988).

] and is taken into account in all simulations for the mode-locking behavior. The pump absorption in the saturable absorber layer, for this case, is negligible (see [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]

]).

Fig. 3 Pump intensity distribution I_p (x) in the gain sublayer (a) and normalized lifetime of the upper state for both gain and saturable absorber sublayers. In (a), the intensity of incident pump beam is I_p₀ = 15 MW/cm². In (b) doted line shows the interface between gain (right) and absorber sublayers; the thickness of the Ag layer b and the saturable absorber sublayer are d = 30 nm and D_q = 400 nm, respectively.

In addition the shortening of τ_0g due to fluorescence quenching of the gain molecules by the dipole energy transfer to the metal layer [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]

], is calculated by using the method represented in Refs. [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]

] [see Fig. 3(b)]. This effect is very weak because the gain sublayer is located 400 nm far from the metal film and the quantum yield of Styryl-9 molecules is relatively small (0.05, see Ref. [37

]). As for the saturable absorber, the decrease of the upper level lifetime responsible for fluorescence increases the ground state recovery time and makes the absorption to be recovered faster. However, the calculation shows that this effect is valid only in a narrow region [below ∼ 10 nm from the metal surface, see Fig. 3(b)] because the fluorescence quantum yield of IR-26 dye is as small as ∼ 10⁻³ (Refs. [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]

]). Taking into account the very weak lifetime shortening, we neglected this effect for both gain and absorber.

5. Numerical results and discussion

In Fig. 4 we show an example of our numerical study of the pulse evolution in mode-locked long range SPP lasers. Figure 4 shows the evolutions of gain and total loss (a), and the generated pulse profile at the position adjacent to the metal film in the dielectric layers (b). For self-starting stable mode-locking, three conditions have to be satisfied [26

J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light Pulses (North-Holland, Amsterdam, 1987).

]: absorption saturation has to be stronger than gain saturation, the unsaturated net gain has to take a positive value and the net gain has to be negative before and after the pulse. As mentioned above, for design parameters as discussed in section IV the absorber is stronger saturated than the gain because of the high confinement of the plasmon field. The unsaturated net gain per round trip is positive with a value of 59.6 (the contribution of the linear loss in the saturable absorber to this value is 15.6). However, in the steady state of pulse formation, net gain before and after the pulse is negative and its value just before the pulse is about −2.02 [see Fig. 4(a)]. The Figure 4(b) illustrates the dynamics of ultrashort pulse formation in the considered SPP laser: even though the response time of the gain and absorber medium is longer than the pulse duration in this regime a stable fs pulse regime is established because noise at the wings of the pulse is suppressed due to the negative net gain in this region. In the initial stage of pulse formation the evolving pulse is shortened due to the different saturation behavior of the gain and the absorber because the absorber recovery is faster than that of the gain. This general scenario of passive mode-locking is analogous as in bulk femtosecond dye lasers [26

J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light Pulses (North-Holland, Amsterdam, 1987).

, 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]

]. In the example given in Fig. 4 for a SPP laser with a pump intensity of 10.26 MW/cm², a stable pulse train is formed with ∼ 20 mJ/cm² 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/cm².

Fig. 4 (Color online) Evolution of gain g_i (dashed blue line) and total loss q_i + l₀ (dash-dot green line) (a) and their behavior near the pulse maximum (b). The pulse shape is presented at the position adjacent to the Ag film in the dielectric layers a and c. The concentrations of gain and absorber molecules are N_g = 2.5 × 10¹⁸ cm⁻³ and N_q = 1 × 10¹⁷ cm⁻³, respectively. The thicknesses of Ag film, gain and absorber sublayers are d = 30 nm, D_q = 400 nm and D_g = 5 μm, respectively. The intensity of the incident pump beam is I_p₀ = 10.26 MW/cm².

Note that from Fig. 4(b) we can see that for the considered SPP laser parameters both the gain and the loss of the absorber are high and both are strongly saturated. This means that a simplified approach based on the assumption of small net gain and weak saturation [27

J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena , 2nd ed. (Academic Press, San Diego, 2006).

] can not be applied for this case.

In Fig. 5 we show the dependence of the output pulse fluence and the pulse duration on the pump intensity. Figure 5 shows that the pulse fluence linearly increases and the pulse duration decreases in a nonlinear way with increasing pump intensity. The limits of shortest pulse duration and largest output fluence is set by damage threshold at roughly 1 J/cm² as well as by available pump sources.

Fig. 5 (Color online) Dependence of maximum pulse fluence F_max (at the position adjacent to the Ag film) and pulse duration τ on the pump intensity I_p. The other parameters are the same as in Fig. 4.

6. Conclusion

In this paper we studied the possibility of femtosecond plasmon pulse generation by mode-locking of long range SPP lasers. We developed a theory of passive mode-locking of SPP lasers with a Bragg resonator consisting of a silver film, a saturable absorber layer adjacent to the metal film, and a gain medium. For the specific example of solid dyes acting both as slow saturable gain medium and a slow saturable absorber the characteristic laser pulse parameters are calculated numerically. The results show that SPP femtosecond pulses with maximum peak intensity in the range of ∼ 140 GW/cm² 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.

Appendices

7. Appendix: Derivation of Eq. (9)

The SPP field in laser resonator given by Eq. (6) can be written by a sum of forward and backward waves:

E_{c} = E_{c}^{+} + E_{c}^{-},

where

E_{c}^{+} = \frac{1}{4} \sum_{n} A_{n} (t) e^{- i ω_{n} t + K_{n} z} (- i \hat{x} + \frac{α_{c n}}{K_{n}} \hat{z}) e^{α_{c n} (x + d)} + c . c .,

(A-1)

E_{c}^{-} = \frac{1}{4} \sum_{n} A_{n} (t) e^{- i ω_{n} t - K_{n} z} (i \hat{x} + \frac{α_{c n}}{K_{n}} \hat{z}) e^{α_{c n} (x + d)} + c . c .

(A-2)

Here we consider only forward propagating wave, because the corresponding counterpropagating waves behave analogously. Equation (A-1) can be approximately written by

E_{c}^{+} (x, z, t) = \frac{1}{4} (- i \hat{x} + \frac{α_{c}}{K_{L}} \hat{z}) e^{α_{c} (x + d)} \sum_{n} A_{n} (t) e^{- i ω_{n} t + K_{n} z} + c . c .

(A-3)

We introduce the slowly varying envelope

A (t, z) = \sum_{n} A_{n} (t) e^{- i (ω_{n} - ω_{L}) t + (K_{n} - K_{L}) z} .

(A-4)

where K_L is the wavenumber for the central lasing frequency ω_L. We neglect the discrete spectral structure and consider the SPP field as a continuum:

A (t, z) = \int A (t, k) e^{- i δ ω t + i k z} d k,

(A-5)

where

k = K - K_{L} = \frac{n_{eff}}{c} (ω - ω_{L}),

(A-6)

δω = ω – ω_L, and A (t, k) is the continuous form of the mode fields A_n (t) (slowly varying envelope).

From Eq. (A-5),

\frac{\partial A (t, z)}{\partial t} = \int [\frac{\partial A (t, k)}{\partial t} - i δ ω A (t, k)] e^{- δ ω t + i k z} d k .

(A-7)

δω can be expanded as follows:

δ ω = v_{g} k + \frac{1}{2} {\frac{\partial^{2} ω}{\partial k^{2}} |}_{k = 0} k^{2} + \dots

(A-8)

On the other hand, from Eq. (7, 8) we have

\frac{\partial A (t, k)}{\partial t} = [- γ (k) + g (t, k)] A (t, k),

(A-9)

where γ(k) and g(t,k) are the continuous form of γ_n and g_n (t). The passive resonator loss γ(k) can be expanded as follows:

γ (k) = γ_{0} + {\frac{\partial γ}{\partial k} |}_{k = 0} k + \frac{1}{2} {\frac{\partial^{2} γ}{\partial k^{2}} |}_{k = 0} k^{2} + \dots

(A-10)

The frequency dependent gain is

g (t, k) = g (t) 𝒟_{n} Γ_{g},

(A-11)

where

g (t) = β σ_{g} \int_{- D_{g} - d_{g}}^{- d_{g}} {\bar{N}}_{g} (x, t) e^{κ (x + d)} d z d x,

(A-12)

and β is a value of β_n given at ω_L. From Eq. (A-11),

g (t, k) = g (t) [1 + \frac{i δ ω}{Γ_{g}} - {(\frac{δ ω}{Γ_{g}})}^{2} + \dots] .

(A-13)

By using Eq. (A-8), we can rewrite the above equation:

g (t, k) = g (t) [1 + \frac{i v_{g}}{Γ_{g}} k - (\frac{i}{2 Γ_{g}} {\frac{\partial^{2} ω}{\partial k^{2}} |}_{k = 0} - \frac{v_{g}^{2}}{Γ_{g}^{2}}) k^{2} + \dots] .

(A-14)

Summarizing Eqs. (A-7, A-9, A-10, and A-14),

\begin{matrix} \frac{\partial A (t, z)}{\partial t} = \int {(g - γ_{0}) - i (\frac{i}{v_{g}} {\frac{\partial γ}{\partial k} |}_{k = 0} - \frac{g}{Γ_{g}} + 1) (v_{g} k) \\ + [\frac{1}{2 v_{g}^{2}} {\frac{\partial γ}{\partial k} |}_{k = 0} - g (\frac{i}{2 Γ_{g} v_{g}^{2}} {\frac{\partial^{2} ω}{\partial k^{2}} |}_{k = 0} - \frac{1}{Γ_{g}^{2}}) - \frac{i}{2 v_{g}^{2}} {\frac{\partial^{2} ω}{\partial k^{2}} |}_{k = 0}] {(v_{g} k)}^{2}} \\ \times A (t, k) e^{- i δ ω t + i k z} d k . \end{matrix}

(A-15)

We add the terms from the saturated absorber into Eq. (A-15) and multiplying round trip time T_R on the both sides:

\begin{matrix} T_{R} \frac{\partial A (t, z)}{\partial t} = \int {(g^{'} - q^{'} - γ_{0}^{'}) - i (\frac{i}{v_{g}} {\frac{\partial γ^{'}}{\partial k} |}_{k = 0} - \frac{g^{'}}{Γ_{g}} + \frac{q^{'}}{Γ_{q}} + T_{R}) (v_{g} k) \\ + (\frac{g^{'}}{Γ_{g}^{2}} - \frac{q^{'}}{Γ_{q}^{2}} + \frac{1}{2 v_{g}^{2}} {[i (T_{R} + \frac{g^{'}}{Γ_{g}} - \frac{q^{'}}{Γ_{q}}) \frac{\partial^{2} ω}{\partial k^{2}} + \frac{\partial^{2} γ^{'}}{\partial k^{2}}] |}_{k = 0}) {(v_{g} k)}^{2}} \\ \times A (t, k) e^{- i δ ω t + i k z} d k, \end{matrix}

(A-16)

where g′ = gT_R, q′ = qT_R, and γ′ = γT_R. 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 T = t,

τ = t - v_{g}^{- 1} z

, where 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_gk can be changed by the derivative for local time τ and we obtain Eq. (9).

References and links

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]
2.	M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett. 101, 226806 (2008). [CrossRef] [PubMed]
3.	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. 8, 3998–4001 (2008). [CrossRef] [PubMed]
4.	I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4, 382–387 (2010). [CrossRef]
5.	M. C. Gather, K. Meerholz, N. Danz, and K. Lesson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nature Photon. 4, 457–461 (2010). [CrossRef]
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]
9.	M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Noetzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (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]
14.	I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71, 063812 (2005). [CrossRef]
15.	J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]
16.	Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Resonance amplification of left-handed transmission at optical frequencies by stimulated emission of radiation in active metamaterials,” Opt. Express 16, 20974–20980 (2008). [CrossRef] [PubMed]
17.	M. Wegener, J. L. Garcia-Pomar, N. M. C. M. Soukoulis, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16, 19785–19798 (2008). [CrossRef] [PubMed]
18.	S.-W. Chang, C.-Y. A. Ni, and S.-L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 024301 (2008). [CrossRef]
19.	A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009). [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]
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]
24.	D. Yu. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011). [CrossRef] [PubMed]
25.	A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Light. Tech. 24(2), 912–918 (2006). [CrossRef]
26.	J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light Pulses (North-Holland, Amsterdam, 1987).
27.	J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena , 2nd ed. (Academic Press, San Diego, 2006).
28.	H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975). [CrossRef]
29.	M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, Chichester-New York-Brisbane-Toronto, 1981).
30.	L. Wendler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59, 3289–3291 (1986). [CrossRef]
31.	K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (John Wiley and Sons, New York, 2001). [CrossRef]
32.	E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985).
33.	P. Meystre and M. Sargent III, Elements of Quantum Optics , 4th ed. (Springer Verlag, Berlin, 2007). [CrossRef]
34.	J. Herrmann and F. Weidner, “Theory of passively mode-locked cw dye lasers,” Appl. Phys. B 27, 105–113 (1982). [CrossRef]
35.	G. P. Agrawal, Nonlinear Fiber Optics , 4th ed. (Elsevier, Amsterdam, 2007).
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]
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]
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]
45.	I. G. Kytina, V. G. Kitin, and K. Lips, “High power polymer dye laser with improved stability,” Appl. Phys. Lett. 84, 4092–4904 (2004). [CrossRef]
46.	R. Bornemann, U. Lemmer, and E. Thiel, “Continuous-wave solid-state dye laser,” Opt. Lett. 31, 1669–1671 (2006). [CrossRef] [PubMed]
47.	P. Yeh, Optical Waves in Layered Media (John Wiley, New York, 1988).
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]

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

Sort: Author | Year | Journal | Reset

References

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]
M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett.101, 226806 (2008). [CrossRef] [PubMed]
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.8, 3998–4001 (2008). [CrossRef] [PubMed]
I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon.4, 382–387 (2010). [CrossRef]
M. C. Gather, K. Meerholz, N. Danz, and K. Lesson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nature Photon.4, 457–461 (2010). [CrossRef]
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]
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]
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,” Nature461, 629–632 (2009). [CrossRef] [PubMed]
M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Noetzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express17, 11107–11112 (2009). [CrossRef] [PubMed]
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,” Nature460, 1110–1112 (2009). [CrossRef] [PubMed]
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.90027402 (2003). [CrossRef] [PubMed]
M. I. Stockman, “Spasers explained,” Nature Photon.2, 327–329 (2008). [CrossRef]
K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71, 115409 (2005). [CrossRef]
I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71, 063812 (2005). [CrossRef]
J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express15, 2622–2653 (2007). [CrossRef] [PubMed]
Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Resonance amplification of left-handed transmission at optical frequencies by stimulated emission of radiation in active metamaterials,” Opt. Express16, 20974–20980 (2008). [CrossRef] [PubMed]
M. Wegener, J. L. Garcia-Pomar, N. M. C. M. Soukoulis, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express16, 19785–19798 (2008). [CrossRef] [PubMed]
S.-W. Chang, C.-Y. A. Ni, and S.-L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express16, 024301 (2008). [CrossRef]
A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B79, 241104 (2009). [CrossRef]
M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt.12, 024004 (2010). [CrossRef]
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]
I. D. Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B78, 161401 (2008). [CrossRef]
I. D. Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express17, 20191–20202 (2009). [CrossRef] [PubMed]
D. Yu. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express19, 12524–12531 (2011). [CrossRef] [PubMed]
A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Light. Tech.24(2), 912–918 (2006). [CrossRef]
J. Herrmann and B. Wilhelmi, Lasers for Ultrashort Light Pulses (North-Holland, Amsterdam, 1987).
J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, San Diego, 2006).
H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron.QE-11, 736–746 (1975). [CrossRef]
M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, Chichester-New York-Brisbane-Toronto, 1981).
L. Wendler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys.59, 3289–3291 (1986). [CrossRef]
K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (John Wiley and Sons, New York, 2001). [CrossRef]
E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985).
P. Meystre and M. Sargent, Elements of Quantum Optics, 4th ed. (Springer Verlag, Berlin, 2007). [CrossRef]
J. Herrmann and F. Weidner, “Theory of passively mode-locked cw dye lasers,” Appl. Phys. B27, 105–113 (1982). [CrossRef]
G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier, Amsterdam, 2007).
P. Berini, “Long-range surface plasmon polaritons,” Advances in Optics and Photonics1, 484–588 (2009). [CrossRef]
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]
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]
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. B29, 15–18 (1982). [CrossRef]
A. A. Ishchenko, “Laser media based on polymethine dyes,” Quantum Electron.24, 87–172 (1994). [CrossRef]
B. H. Soffer and B. B. McFarland, “Continuously tuable, narrow band organic dye lasers,” Appl. Phys. Lett.10, 266–267 (1967). [CrossRef]
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]
P. Runge and R. Rosenberg, “Unconfined flowing-dye films for CW dye lasers,” IEEE J. Quantum Electron.8, 910–911 (1972). [CrossRef]
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]
I. G. Kytina, V. G. Kitin, and K. Lips, “High power polymer dye laser with improved stability,” Appl. Phys. Lett.84, 4092–4904 (2004). [CrossRef]
R. Bornemann, U. Lemmer, and E. Thiel, “Continuous-wave solid-state dye laser,” Opt. Lett.31, 1669–1671 (2006). [CrossRef] [PubMed]
P. Yeh, Optical Waves in Layered Media (John Wiley, New York, 1988).
G. Ford and W. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep.113, 195–287 (1984). [CrossRef]
W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt.45, 661–699 (1998). [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.

Click here to see a list of articles that cite this paper