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Coupling of surface plasmon (SP) waves between two metal-dielectric interfaces of a dielectric-metal-dielectric (DMD) waveguide, in which one of the dielectric layers is incorporated with optical gain, is proposed to realize plasmonic lasing. The propagation characteristics of the SP waves supported inside the DMD waveguides are studied by finite-difference time-domain method. It is found that there are optimized thicknesses for the metal film and gain region to obtain minimum propagation loss. Furthermore, a coupled-mode model is developed to analyze the lasing characteristics of the DMD waveguides with coherent optical feedback. The conditions to achieve single-longitudinal-mode lasing from the DMD waveguides are also investigated.
©2008 Optical Society of America
1. Introduction
Surface-plasmon (SP) waves travelling at metal-dielectric interface have shown potential applications in the realization of compact integrated optical devices and circuits. This is because the phenomenon of surface plasmon polariton concentrates light in dimensions much smaller than the diffraction limit [1]. However, the propagation distance of SP waves, which is much shorter than that of the conventional dielectric optical waveguides, is limited by the high absorption loss of the metal film. Therefore, the use of optical gain is suggested to amplify the localized SP waves in nanostructures to achieve strong light localization effects [2]. In fact, the enhancement of localized SP waves at the interface of metallic thin films has been demonstrated experimentally by using dye solution as an optical gain medium [3], [4]. On the other hand, the confinement characteristics of long-range SP waves on metallic waveguide adjacent to a gain medium were analyzed. It was estimated that the presence of optical gain with magnitude less than 400 cm-1 could realize lossless metallic waveguides [5]. However, the possibilities to realize coherent optical feedback to sustain lasing from the metallic waveguides are yet to be investigated.
In this paper, the coupling behavior of propagating long-range SP waves between the two metal-dielectric interfaces of a dielectric-metal-dielectric (DMD) waveguide is analyzed by finite-difference time-domain (FDTD) technique. It can be shown that the SP waves are coupled together via the formation of supermodes (i.e., equivalent to directional coupling in the conventional waveguide directional couplers) within the DMD waveguides. Hence, the incorporation of gain medium onto one side of the dielectric layers is suggested to amplify the coupled SP waves. It is found that the absorption loss of the DMD waveguide is dependent on both the thickness of the metal film and gain region; however, the corresponding coupling coefficient is mainly affected by the thickness of metal film. Furthermore, a coupled-mode model is developed to study the lasing characteristics of DMD waveguides with coherent optical feedback. It is also proposed to realize grating along the cavity length on the dielectric layer opposite to the gain region so that single-mode plasmonic lasing can be achieved from the DMD waveguides.
2. Propagation characteristics of SP waves - FDTD analysis
The propagation characteristics of SP waves along the metal-dielectric interfaces of a DMD waveguide can be investigated by FDTD method. In the calculation, it is assumed that the thin film, which has thickness of 20 nm embedded inside a dielectric medium, is extended infinitely along the x and z directions where x-coordinate is the direction of propagation. In addition, the refractive indices of the dielectric medium and metal are assumed equal to be 1.54 and 0.0537+3.948i respectively. A free plane-wave with wavelength of ~594 nm is end-fired in the x-direction at the top dielectric layer at few tens of nanometers above the surface of the metal film (i.e., at y>15 nm) and coupled to the SP waves on the top of the metal film.
Fig. 1. Instant distribution of |Hz|2 intensity for the coupling SP waves propagating along the two metal interfaces of the DMD waveguide (a) without and (b) with gain at the bottom of the dielectric medium (at y<0). The corresponding schematic diagrams of the DMD waveguides under end-fire coupling to the SP waves are also given.
Figure 1(a) shows the longitudinal propagation of SP waves with wavelength of ~594 nm inside a DMD waveguide. It is observed that the SP wave on the top dielectric layer coupled to the bottom dielectric layer and vice verse when the SP waves are moving along the x direction. It must be noted that the SP waves are excited only on one side of the metal-dielectric interface. Hence, this asymmetric excitation of SP waves along the y direction causes the coupling of SP waves between the two interfaces of the DMD waveguides. Furthermore, the asymmetric excitation causes the modal profile of SP waves differently to that of the eigenmodes excited symmetrically inside the DMD multilayer [6]. The field intensity distribution given in Fig. 1 is in fact the combination of symmetric and asymmetric SP waves. It is also observed that the SP waves completely disappeared at x>12 µm. On the other hand, for the case given in Fig. 1(b) where the bottom dielectric layer has an optical gain of ~600 cm-1, the SP waves can propagate much longer than that given in Fig. 1(a). It must be noted that the real part of refractive indices of the top dielectric layer and bottom gain region are assumed to be the same. This indicated that the introduction of optical gain to the bottom dielectric layer can help to sustain the propagation of SP waves. This is because the extra optical gain compensates the propagation loss of the SP waves inside the dielectric layers.
It is noted that the propagation characteristics of SP waves in DMD waveguide are similar to the exchange of power between guided modes in a lossy directional coupler [7]. Hence, it is necessary to study the equivalent propagation loss and coupling coefficient of the DMD waveguides. In addition, the lasing characteristics of SP waves can be studied by using coupled-mode theory if the corresponding propagation loss and coupling coefficient are known. From Fig. 1, it is observed that the SP waves couple together through the presence of metal thin film. This implies that the corresponding propagation loss and coupling coefficient are dependent on the physical properties of the metal film. Furthermore, the thickness of gain medium determines the amplification efficiency of the SP waves. Hence, the dependence of propagation loss and coupling coefficient on the thickness of metal film and gain medium are required to be investigated.
Fig. 2. Variation of (a) propagation loss, α, and (b) coupling coefficient, M, of the DMD waveguide with different thickness of metal film,
Figure 2 shows the calculated values of propagation loss, α, and coupling coefficient, M, of the DMD waveguide versus the thickness of metal film, tm. M is defined as the inverse of the distance between the two adjacent field intensity peaks along the propagation direction x, see also Fig 1(b). The parameter M reviews the efficiency of energy transfer between the two metal-dielectric interfaces when the SP waves propagating along the x direction. It is observed that the values of α and M increase with the reduction of tm. This is because the electric field of asymmetric SP waves is strongly confined (i.e., more portion of electric field being interacted with the metal layer) inside metal layer with thinner thickness so that the value of α increases with the decrease of tm. Furthermore, it is found that the propagation coefficient of the SP waves, β, remains unchanged at 16 µm-1 for various values of tm.
Fig. 3. Variation of (a) propagation loss, α, and (b) coupling coefficient, M, of the DMD waveguide with different thickness of gain region, tg.
Figure 3 shows the calculated values of α and M of the DMD waveguide versus the thickness of gain region, tg. In the calculation, different values of tm (40 to 60 nm) and imaginary refractive index of the gain region, nimag, (-0.005 to -0.01 which correspond to the optical gain of ~260 to ~510 cm-1) were used. It is observed that the values of α reduced with tg and saturated for tg greater than 400 nm for most of the cases. On the other hand, the values of M are less dependent on the variation of tg. It is noted that there is a minimum value of tg to maximize the confinement of SP waves within the gain region. However, the values of M is less dependent on tg as M is mainly determine by the thickness of the metal thin film as well as the refractive indices of the dielectric materials surrounding the metal thin film. From the above calculations, it is found that the optimized values of tm and tg are ~60 and ~400 nm respectively. This is because these values of parameters can obtain a small value of α for the SP waves as well as the dimensions of the dielectric waveguide can be minimized to subwavelength dimensions.
Fig. 4. Plot of (a) propagation coefficient, α, and (b) coupling coefficient, M, of a waveguide with the variation of real part of refractive index of the gain region, nreal.
The real part of refractive index, n real, of the host matrix (i.e., the dielectric layer with gain to be introduced) may vary due to the introduction of dopants and this may lead to the excitation of conventional waveguide modes. Hence, it is interesting to study the influence of small variation of n real on the propagation characteristics of DMD waveguides. Figure 4 shows the variation of α and M with the small change of nreal. It is observed that a slight decrease of n real reduces the propagation loss of the waveguides. This is because the reduction of refractive index reduces the confinement of SP waves at the dielectric-metal interface so that the SP waves experience less absorption loss from the metal. Hence, for the design of DMD waveguides, it is preferred that the gain medium has slightly less refractive index when compared to that of the cladding layer in order to achieve relatively low loss waveguide. However, if the different of n real between gain region and cladding layer is larger than 0.01, this may lead to the excitation of conventional waveguide modes. As a result, the coupling of SP waves between the two dielectric-metal interfaces may not be supported.
3. Coupled-mode model for the DMD waveguides
The physics of plasmonic lasing in DMD waveguide is investigated by assuming that the device consists of cladding layer, a thin layer of metal film, and an active layer on a bulk substrate. Coherent optical feedback is obtained from the facets with high reflection coating. This is possible as the SP waves are propagating in the direction perpendicular to the facets so that there is no interference of plasmonic effects with the optical reflection. The laser is optically pumped from the substrate by a high power light source. In addition, the cladding layer, metal layer, gain layer and substrate are made by SiO2, Ag, dye doped polymer, and bulk SiO2 substrate respectively.
In order to match the refractive index of SiO2 as well as to obtain high optical gain under optical excitation, Rhodamine 6G (R6G) doped PVC active medium is selected to amplify ~594 nm stimulated emission under 532 nm light excitation [4], [8]. Figure 5 shows a four-level model used to describe the recombination characteristics of R6G dye. The energy states on the left-hand-side of the figure represent a four-level system along with the triplet states on the right [9]. The ground state is split into two vibronic levels with population distribution governed by Boltzmann distribution. In fact, it can be shown that the four-level model can be simplified to an equivalent two-level model [10]. This is possible as the non-radiative decay rate of populations at the highest energy level is larger than that of that the other energy levels and pumping rate. The simplified two-level model for R6G dye can be expressed as:
where N 2 and N 1 are the populations at the upper and lower energy levels respectively of the two-level system. N dye is the total dye concentration and τR2 is the carrier lifetime. The polarization dependent pump absorption rates (R 13) and signal absorption (Wa) and emission rates (We) can be written by
where h is Planck’s constant, νp(s) is the frequency of pumped (signal) light, Aeff is the effective area, σ 13 P is the absorption cross section of the pump light and, σaS and σeS is the absorption and emission cross section of the signal light respectively. P pump and P 0 S(λs) is the pump and signal power respectively. λp (λs) is the pumped (signal) wavelength and S is the normalized signal power.
As shown in Fig. 1, directional coupling of SP waves between the top and bottom dielectric layers of the DMD waveguides is observed. Therefore, the propagation characteristics of the SP waves, which have similarity to the guided modes in the directional coupler, can be studied by using coupled-mode theory. The normalized SP propagation waves, S + C(A) and S - C(A), which propagation characteristics are shown in Fig. 6, can be described by a modified coupled-mode equations [7]
where δβ is the detuning from the propagation coefficient of the DMD waveguides, Γ is the confinement factor related to the confinement of SP waves inside the gain medium, and the signal optical gain, gS, can be expressed as
The parameters α and M have been defined in the above section. The normalized signal power in active layer, SA, given in (2) is related to S + A and S - A by
It is also defined the normalized signal power in cladding layer as
The boundary conditions for the lasers are given by
where r L and rR are the facets reflectivity at x=0 and x=L respectively.
Fig. 6. Schematic diagram of a SP laser with cladding layer, gain medium and substrate of same refractive index.
In (3), χ ± A is the spontaneous emission noise coupled to the signal fields. It is assumed that i) the distribution of spontaneous emission noise has a Gaussian profile and ii) the spontaneous emission fields coupled into the forward and reverse fields have the same magnitude. Hence, the spontaneous emission noise can be generated from a Gaussian distributed random number generator that satisfies the following correlation functions [11]
where Pspont is the magnitude of the spontaneous emission power per area and is defined as
where βs (=10-5) is the spontaneous coupling factor. Hence, by solving the rate equations of populations and coupled-mode equations of the SP waves self-consistently [11], the lasing characteristics of the DMD waveguides can be reviewed. This can be done by dividing the waveguide into discrete spatial sections of length Δx with temporal time steps taken as Δt=Δx/νg, so that the optical field is sampled at the rate fs=l/Δt.
4. DMD waveguide with coated facets
In this section, the steady state and transient response of the proposed dye doped DMD waveguides with coated facets are analyzed. In the calculation, it is assumed that the length of the waveguide, L is 300 µm and the facet coatings, r L and rR, both have equal reflectivity of 0.92 (i.e., Ag coated). In order to model the propagation characteristics of SP waves by using coupled mode equations, it is assumed that α=500 cm-1, M=0.2 cm-1 and Γ~1 as deduced from section 2 for the optimized dimensions (i.e., optimized values of tm and tg) of the DMD waveguides. The material parameters for the dye doped polymer [4] are assumed to be Aeff=3×10-11 m2, N dye=1.3×1025 m-3, σ 13 P=4.3×10-20 m2, σaS=2.1×10-20 m2, σeS=2.0×10-20 m2, and τR2=3.0×10-9 s. The refractive indices of SiO2 and doped PVC are assumed equal to 1.54. In the calculation, the laser is divided into 600 (=L/Δx) equal length sections and the laser is switched on by the pump power with a step function
where Pext is the excitation power.
To study the modal behavior of DMD waveguides with facet reflection, the finite gain bandwidth of the dye solution should be taken into consideration. This can be accomplished by digitally filtering the SP waves [12]. Mode competition insures that only gain curvature around the central frequency is important. The frequency response of the filter, H(ω), used to model the optical gain profile can be expressed as
where b∈(0,1) represents the filter width, ωp is the frequency of the gain peak and Δt is the time step. The discrete implementation of this filter for the forward mode (and similarly for the reverse mode) is
where B=bexp(jω pΔt). A digital filter with b=0.009 and Δt=20 fs is used to approximate the optical gain of dye solution with bandwidth of ~3 THz.
Fig. 7. Transient response of (a) SA and SC and (b) N 2 for DMD waveguide with coated facets r L=r R=0.92 pumped at Pext=1.03 times threshold. (c) Steady-state lasing spectra of SA for the facet coated DMD laser with Pext equals to P 1 (1.03×threshold), P 1 (2.0×threshold) and P 3 (3×threshold).
Figure 7 shows the transient response of the DMD waveguide under the excitation of Pext equals to 1.03 times its threshold. It is observed that both SA and SC (N 2) exhibit in-phase (out-of-phase) relaxation oscillation. It is noted that mode beating of both S A and SC is due to the excitation of side modes. In addition, the formation of closely spaced Fabry Perot modes is the result of high facet reflectivity and long cavity length of the DMD waveguides.
Fig. 8. (a). Plot of spatial distribution of SA (black) and SC (red). (b) Calculated steady-state lasing spectra without (top) and with (bottom) plasmonic coupling taken into consideration.
The coupling behavior between SA and SC is explained in Fig. 8(a). It is clearly shown that SA and SC are varied out of phase along the laser cavity (i.e., energy transfers from active region to cladding region and vice versa), similar to the intensity distribution in Fig. 1. Figure 8(b) shows the influence of plasmonic coupling on the resultant lasing spectra. It is noted that the mode spacing of Fabry Perot modes is about ~300 GHz without the influence of plasmonic coupling. However, the presence of plasmonic coupling, which is equivalent to the coupling of two Fabry Perot cavities together, induced extra modes in the emission spectrum.
5. DMD waveguides with periodic grating
In order to suppress side modes, grating is introduced along the cladding layer of the DMD waveguides. Figure 9 shows the schematic diagram of the DMD waveguide with grating. It is assumed that the effective refractive index of cladding layer remains unchanged as the corresponding variation of refractive index inside the grating is in order of 10-3.
If first order periodic grating is assumed to be fabricated at the cladding layer, the coupled-mode equations given in (3a) can be modified as
The coupling coefficient, κ, can be deduced from [12]
where Λ is the periodic of the grating, b is the lower index region of the grating, Δn is the refractive index different of the grating, and Γg is the confinement factor of the SP waves inside the grating. If the grating is realized by direct UV imprinting technique with Δn~5×10-3 and b/Λ ~0.75, it can be shown that κ ~80 cm-1 for Γg ~1. It is possible to obtain Γg ~1 as the depth of the grating can match with the penetration depth of the SP waves inside the cladding layer provided that the depth of the grating is more than 200 nm.
Fig. 10. Transient response of (a) SA and SC and (b) N 2 for the AR coating (i.e., r L=rR=0) DMD waveguide with grating κ=80 cm-1 and Pext=1.03 times its threshold.
Figure 10 shows the transient response of the DMD waveguide with κ ~80 cm-1 pumped at 1.03 times threshold. It is assumed that the cleaved facets are anti-reflection (AR) coated to avoid the formation of Fabry Perot modes. The transient response of the DMD waveguide with periodic grating is similar to that of the Fabry Perot cavities. However, mode beating is still observed in its transient response. Figure 11 shows the corresponding steady-state spatial distributions of S A, S C, N 2 as well as the lasing spectra of S A at different Pext. In Fig. 11(a) and 11(b), the spatial distributions of SA and SC at Pext=P 1 are magnified by 5 times whereas that at Pext=P 2 and P 3 are shifted up by 500 mW and 1000 mW respectively so that they can be easily read from the figures. It is observed that the mode beating is due to the excitation of two bandgap modes. This is expected as 1st order uniform grating will support bandgap modes [13].
Fig. 11. Spatial distribution of (a) SA, (b) SC, (c) N 2 and (d) steady-state lasing spectra of SA for the AR coated DMD laser with κ=80 cm-1 and Pext equals to P 1 (1.03×threshold), P1 (2.0.threshold) and P 3 (3×threshold).
From the coupling wave theory, it is noted that the symmetry of the emission spectra can be broken if facet reflection is introduced into the consideration. Figure 12 repeats the investigation of grating coupled DMD waveguides with r L and r R ~0.2. The presence of small facet reflectivity breaks the symmetric of the periodic grating as well as excites bandgap mode. It is also observed that stable single-mode operation can be obtained even at higher pump intensities. Hence, it has shown that it is possibility to achieve single-mode lasing from DMD waveguides.
Fig. 12. Spatial distribution of (a) SA, (b) SC, (c) N 2 and (d) steady-state lasing spectra of SA for the cleaved facets DMD laser with κ=80 cm-1 and Pext equals to P 1 (1.03×threshold), P 1 (2.0×threshold) and P 3 (3×threshold)..
6. Discussions and conclusions
Propagation characteristics of SP waves, with wavelength of ~594 nm, along the DMD waveguides are studied by FDTD technique. It is observed that there is an exchange of power between the SP waves at the two Ag-SiO2 interfaces due to the asymmetric excitation. The propagation characteristics of SP waves in the DMD waveguides are equivalent to that of the guided modes in the conventional directional couplers. It is found that the coupling characteristics of SP waves are dependent on the thickness of Ag film as well as the refractive index of dielectric layers. It is also observed that the DMD waveguides with the Ag thickness equal to ~60 nm can minimize the corresponding propagation loss to less than 500 cm-1. However, if dielectric layers with higher refractive index are used for the design of DMD waveguides, coupling characteristics of SP waves will not be supported unless longer wavelength of SP waves is used. For example, for SP waves with wavelength of ~1550 nm, materials such as InP or GaAs (i.e., refractive index of 3.4) can be used as the dielectric layers to realize coupling of SP waves.
By introducing optical gain on one side of the dielectric layers, it can be shown that the amplification of SP waves can be obtained for the DMD waveguides. In addition, the thickness of the gain medium should be larger than 400 nm in order to confine the SP waves. It is proposed to use R6G doped PVC as the dielectric layer with gain medium under the optical excitation of 532 nm light. This is because the refractive index of R6G doped PVC is matched with the cladding layer of DMD waveguides. In addition, R6G dye can provide optical gain to more than 103 cm-1 at ~600 nm [3],[4]. It must be noted that the refractive indices of the two dielectric layers should be close in order to allow effective coupling of SP waves. However, slightly reduction of refractive index in the gain medium can further enhance the amplification of the SP waves.
As it is not possible to simulate the roundtrip propagation characteristics of the SP waves by FDTD technique due to the requirement of extensive computational time, a coupled-mode model is developed to study the lasing characteristics of DMD waveguides. This is possible as the coupling characteristics of the two adjacent SP waves are similar to that of the guided modes inside the directional couplers. Hence, it is proposed to study the lasing characteristics of DMD waveguides with R6G doped polymer as the gain medium. In addition, the use of periodic grating along the cladding layer is suggested to realize single-mode lasing. It is found that single-mode lasing can be supported in 300 µm long DMD waveguides with cleaved facets provided that the corresponding value of κ is around 80 cm-1 in order to suppress the influence of facet reflection.
Acknowledgment
This project is supported by Singapore MoE grant ARC 02/06
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