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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 1 — Jan. 2, 2012
  • pp: 462–473
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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


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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

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.

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 ɛc. For the feedback we take a Bragg reflectors of SPPs as the resonator mirrors.

2. Surface plasmon polariton laser equation

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
Ec=12nAn(t){x^sin(Knz)+z^(αcn/Kn)cos(Knz)}eαcn(x+d)iωnt+c.c.,
(6)
where ωn is the angular frequency of n-th mode in the active resonator and we assume that ωn ≈ Ωn. The induced polarization is described by the same expression replacing the field amplitude An (t) with Pn (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

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
A˙n+γnAn=σgɛc𝒟nΓgAnMn0LDgdgdg|Un|2Ngdzdx,
(7)
where ρg is electric dipole moment of the gain, 𝒟n = [Γg + i (ωnωL)]−1, T2g=Γg1 is the dephasing time, σg=ρg2ωn/(2ɛ0ɛch¯Γg) is the gain cross-section, Ng = Ng (x, z,t) is a space-time dependent population inversion, Mn=0Ldzdx|Un(x,z)|2Re[n(x)] is a normalization factor, n(x,ωn) is the refractive index different at each layer, L is the length of the SPP resonator, ɛ0 is the vacuum permittivity, and 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 gnAn, where gn is the transient nonlinear mode gain. Using Eq. (2) we obtain
gn=βnσgDgdgdgN¯g(x,t)eκ(x+d)dx,
(8)
where βn=Γg𝒟n(1+|αcnKn|2)ɛc/(2Mn), κ= 2|Re[αc(ωL)]|, ωL is the lasing frequency, N¯g(x,t)=0LNg(x,z,t)dz/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 ωn, Kn, γn and neff,n with ω, K, γ, and neff, 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

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

, 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

The sum in Eq. (6) is replaced by the forward propagating field A(T,τ) = ∫A(t,k)eiδωt+ikzdk, where k = KKL = (neff/c)(ωωL), δω = ωωL, and A (t, k) is the continuous form of the mode fields An (t). We apply a coordinate transformation T = t, τ = tz/vg, where T is the laboratory time, τ is the local time, γ(ω) is the frequency-dependent loss, TR is the round trip time, and vg 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
TRAT=[g(τ)q(τ)γ0]A+δ1Aτ+δ22Aτ2,
(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, δ1=ivg1(γ/k)|k=0+giΓg1qiΓq1 and δ2=giΓg2qiΓq2+(2vg2)1(2[iω(k)(TR+giΓg1qiΓq1)+γ(k)]/k2)|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 = (neff/c)(ωωL). 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, gi must be partly recovered during one round trip. The linear parts of gain and loss rates are represented by gi=TRβDgDqdDqdgleκ(x+d)dx and qi=TRβDqddqleκ(x+d)dx, respectively, and β is defined below Eq. (8) with the substitution ωnωL, gl and ql 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 gl (x,τ) and ql (x,τ) are given by the equations [27

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

]
glτ=glg0(x)τ0ggl|Ec(x,τ)|2Asg2τ0g,
(10)
qlτ=qlq0(x)τ0qql|Ec(x,τ)|2Asq2τ0q,
(11)
where τ0g and τ0q are the upper-level lifetimes, Asg and Asg 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

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

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. (911). For slow passive mode-locking the gain and absorber dyes exhibit a longer relaxation time compared with the pulse duration: τ0q, τ0g < τ0. Besides, stable mode-locking operation is possible only if the conditions 0.1 ≤ TR/τ0g ≤ 10 and τ0q < TR are fulfilled [26

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

28

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

, 34

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

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 ɛa (λL) = ɛc (λL) = 2.20 (PMMA) and ɛb (λL) = −35.99+ 2.20i (silver), respectively [32

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

]. 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]

] which has the smallest propagation loss.

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

]) and IR-26 (Refs. [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]

40

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 σs = 1.8 × 10−16 cm2 and σag = 1.2 × 10−16 cm2, respectively, the upper-state relaxation time (or longitudinal relaxation time) is τ0g = 400 ps and the dephasing rate is Γg = 3.3 × 1013 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]

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

]), upper-state relaxation time (or absorber recovery time) is τ0q = 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

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

]) and dephasing rate is Γq = 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

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

, 42

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

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

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

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

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

31

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 neff = 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 Dq = 400 nm and Dg = 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 Ng = 2.5 × 1018 cm−3, lasing wavelength is λL = 900 nm, the thicknesses of gain and absorber sublayers are Dq = 400 nm and Dg = 5 μm. The permittivities of the layers are ɛa (λL) = ɛc (λL) = 2.20 (PMMA) and ɛb (λL) = −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

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

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

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

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 Ip (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 Ip0 = 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 Dq = 400 nm, respectively.

In addition the shortening of τ0g 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

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

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

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

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

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]

]). 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

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

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

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

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

, 28

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

, 34

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 Imax = 143 GW/cm2.

Fig. 4 (Color online) Evolution of gain gi (dashed blue line) and total loss qi + 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 Ng = 2.5 × 1018 cm−3 and Nq = 1 × 1017 cm−3, respectively. The thicknesses of Ag film, gain and absorber sublayers are d = 30 nm, Dq = 400 nm and Dg = 5 μm, respectively. The intensity of the incident pump beam is Ip0 = 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

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 Fmax (at the position adjacent to the Ag film) and pulse duration τ on the pump intensity Ip. 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.

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:
Ec=Ec++Ec,
where
Ec+=14nAn(t)eiωnt+Knz(ix^+αcnKnz^)eαcn(x+d)+c.c.,
(A-1)
Ec=14nAn(t)eiωntKnz(ix^+αcnKnz^)eαcn(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
Ec+(x,z,t)=14(ix^+αcKLz^)eαc(x+d)nAn(t)eiωnt+Knz+c.c.
(A-3)
We introduce the slowly varying envelope
A(t,z)=nAn(t)ei(ωnωL)t+(KnKL)z.
(A-4)
where KL is the wavenumber for the central lasing frequency ωL. We neglect the discrete spectral structure and consider the SPP field as a continuum:
A(t,z)=A(t,k)eiδωt+ikzdk,
(A-5)
where
k=KKL=neffc(ωωL),
(A-6)
δω = ωωL, and A (t, k) is the continuous form of the mode fields An (t) (slowly varying envelope).

From Eq. (A-5),
A(t,z)t=[A(t,k)tiδωA(t,k)]eδωt+ikzdk.
(A-7)
δω can be expanded as follows:
δω=vgk+122ωk2|k=0k2+
(A-8)
On the other hand, from Eq. (7, 8) we have
A(t,k)t=[γ(k)+g(t,k)]A(t,k),
(A-9)
where γ(k) and g(t,k) are the continuous form of γn and gn (t). The passive resonator loss γ(k) can be expanded as follows:
γ(k)=γ0+γk|k=0k+122γk2|k=0k2+
(A-10)
The frequency dependent gain is
g(t,k)=g(t)𝒟nΓg,
(A-11)
where
g(t)=βσgDgdgdgN¯g(x,t)eκ(x+d)dzdx,
(A-12)
and β is a value of βn given at ωL. From Eq. (A-11),
g(t,k)=g(t)[1+iδωΓg(δωΓg)2+].
(A-13)
By using Eq. (A-8), we can rewrite the above equation:
g(t,k)=g(t)[1+ivgΓgk(i2Γg2ωk2|k=0vg2Γg2)k2+].
(A-14)
Summarizing Eqs. (A-7, A-9, A-10, and A-14),
A(t,z)t={(gγ0)i(ivgγk|k=0gΓg+1)(vgk)+[12vg2γk|k=0g(i2Γgvg22ωk2|k=01Γg2)i2vg22ωk2|k=0](vgk)2}×A(t,k)eiδωt+ikzdk.
(A-15)
We add the terms from the saturated absorber into Eq. (A-15) and multiplying round trip time TR on the both sides:
TRA(t,z)t={(gqγ0)i(ivgγk|k=0gΓg+qΓq+TR)(vgk)+(gΓg2qΓq2+12vg2[i(TR+gΓgqΓq)2ωk2+2γk2]|k=0)(vgk)2}×A(t,k)eiδωt+ikzdk,
(A-16)
where g′ = gTR, q′ = qTR, and γ′ = γTR. 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=tvg1z, 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 vgk 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. 90027402 (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


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References

  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,” Nature461, 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. Express17, 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,” Nature460, 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.90027402 (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. B71, 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. A71, 063812 (2005). [CrossRef]
  15. 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]
  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. Express16, 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. Express16, 19785–19798 (2008). [CrossRef] [PubMed]
  18. S.-W. Chang, C.-Y. A. Ni, and S.-L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express16, 024301 (2008). [CrossRef]
  19. A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B79, 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. B78, 161401 (2008). [CrossRef]
  23. I. D. Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express17, 20191–20202 (2009). [CrossRef] [PubMed]
  24. D. Yu. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express19, 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, 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. B27, 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 Photonics1, 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. B29, 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]

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