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Abstract
Distributed feedback plasmonic laser based on the periodic array of holes in the silver half-space and covered with gain medium is considered. Square, hexagonal and several rectangular lattices are studied. It is demonstrated that the bound states in the continuum provide substantially lower threshold than radiating modes. Moreover, it is shown that while the hole size increases the lasing threshold of some modes decreases. Among the studied types of lattices, lasing in the hexagonal lattice requires the lowest material gain of only 18 cm−1. Our results pave the way to the development of the efficient low-threshold distributed feedback plasmonic lasers.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the seminal paper by Bergman and Stockman [1] where the concept of spaser (surface plasmon amplification by stimulated emission of radiation) was theoretically proposed, it has been demonstrated experimentally multiple times [2–14]. Ultimately, all spasers (also often referred to as plasmonic lasers) can be divided into several groups based on their geometry: spasers based on the localized surface plasmon resonances of a single particles [2–4], distributed feedback (DFB) lasers on the arrays of the plasmonic particles [5–7], DFB lasers based on the arrays of holes in metal films [8–10] and other Fabry-Perot-like configurations [11–13]. One of the main motivations behind the development of plasmonic lasers is design of the optical computer components [14–17]. It is also believed that such devices can be used in active biosensing platforms [18, 19], high density data storage systems [20–22], intra-cell dynamics visualizations [23, 24] and DNA sequencing [25]. To be able to employ the plasmonic laser in these applications one should have a fine control over its radiation properties which is a challenging task. Moreover, in case of a single-particle spasers or DFB lasers on the arrays of holes in metal films, achieving lasing can itself pose a problem because of the high material gain required. The core-shell particle geometry of a laser was already a subject of research to minimize required gain [26]. It was found that minimal threshold gain attainable in such geometry is around 103 cm−1 in the near IR band. In the present work we have focused on a distributed feedback plasmonic laser formed by the dielectric-metal interface with nanoholes in the metal and investigated the influence of laser geometry and the choice of modes on the lasing threshold with the goal of finding the geometry that provides the lowest threshold.
Distributed feedback plasmonic lasers usually consist of the metal film perforated with the periodic array of holes and covered with the gain medium (see Fig. 1(a)). There are experimental realizations with semiconductors [8] and dye molecules [9,10] serving as a gain medium. The idea behind the distributed feedback mechanism is that while traveling along the metal surface, plasmons are scattered by the holes and thus couple to the plasmons propagating in the opposite and in the perpendicular directions. One of the features of the infinite lattices of scatterers (whether it has metal parts or no) is the existence of the radiationless modes. Such modes, called bound states in the continuum (BIC), also often referred to as “dark” modes, may exist in the infinite periodic systems with the C2 symmetry [27]. Such modes are Bloch waves, that decay exponentially in the half-spaces surrounding the periodic structure (see Fig. 1(b) and 1(c)) in contrast to radiating modes, that are non-zero even at the distance from the structure (Fig. 1(d)). Since radiation losses are absent for BIC, we can expect that their lasing threshold will be lower than the threshold of radiating modes and our results confirm this intuition. BIC modes do not radiate and are in essence Bloch surface plasmon modes in our case, so stimulated emission in such modes can be called “spasing”, however we will use the usual term “lasing”. In the finite, but sufficiently large lattices, similar modes exist that do not radiate at the angles normal to the plane of periodicity. Their lasing properties have been studied experimentally [28,29].
Fig. 1 (a) Schematic view of several lattice periods of the distributed feedback plasmonic laser. It consists of a periodic array of holes in the silver half-space covered by the dielectric host medium with active molecules. (b–d) Electric field norm distribution of the eigenmodes of the plasmonic lattice above the metal surface: non-radiating or BIC modes (b, c) and one of the radiating modes (d). Non-radiating modes decay exponentially away from the metal surface in contrast to radiating mode.
In this paper we focus on comparing the lasing thresholds of the radiative and BIC states present in the DFB laser. We also investigate the influence of the geometrical parameters of the laser on the threshold. The paper is organized as follows. In Section 2 we discuss the gain model that we use and the methods of calculating lasing thresholds. In Section 3 the design of the distributed feedback laser is presented. In Section 4 we present main results and discuss them. Section 5 contains the final conclusions of our work.
2. Mathematical model of the threshold calculation
2.1. Main equations
In the present work we treat the gain medium as a material with complex permittivity. This model can be derived from the semiclassical Maxwell-Bloch system of equations that is often used to describe the laser dynamics. For the gain medium consisting of a two-level molecules embedded in a dielectric matrix this system of equations takes the following form [30,31]:
where E and P are a position- and time-dependent electric field and macroscopic polarization of the active medium respectively, C(r) is the concentration of the active molecules, n is the population inversion of the active molecules, ε̂(r, t) is an operator that describes the response of the dielectric and metal in the absence of gain molecules, ω21 is the transition frequency between the two levels of the active molecules, μ21 is the off-diagonal matrix element of the dipole momentum of the molecules, τp and τn are relaxation times of the polarization and population inversion and n0 is the population inversion determined by the external pump intensity. In the equations above E is usually the electric field generated by the active medium. However, these equations can be adapted to describe external light amplification, if we add the corresponding source term to the right part of the first equation. In this case E is the field excited by this external source.If we consider the harmonic time dependence of the electric field and polarization and stationary population inversion, after some transformations we will arrive at the nonlinear Helmholtz equation:
where now E is the position- and frequency-dependent electric field, ε(r, ω) is the spatial distribution of a complex permittivity and is the complex permittivity that describes the nonlinear material response of the gain medium to a field with a frequency ω, α = |μ21|2τpn0/3ħε0, β = |μ21|2τpτn/3ħ2.When the pump is off or low, population inversion n0 is below zero and permittivity (3) describes absorbing medium. With the sufficient pump intensity inversion n0 is above zero and the medium becomes amplifying, with the β|E|2 term in the denominator putting the limit on how big the field can grow. When the pump is off or not high enough there is only a trivial solution to the nonlinear Helmholtz Eq. (2), if we constrain the frequency to be purely real. Above some threshold value of Imεg, solving Eq. (2) might yield a non-zero electric fields, which will correspond to the lasing regime. However, when there are several solutions to this equation, the applicability of the time harmonic approximation fails. Such regime corresponds to the mode competition and potentially multimode laser operation. Both effects are not accounted for in the present model. In literature one can find an approach that includes mode interaction and allows one to find the solution in the multimode regime [32]. Even when the approximations used to derive Eq. (2) are valid, it is difficult to find non-trivial solution of this equation due to the nonlinear dependence of the permittivity εg on the electric field intensity and frequency. Only some simple configurations of the laser allow one to find the exact analytic solution [33,34]. However, just above the threshold, when the condition β|E|2 ≪ 1 holds true, we can neglect the dependence of the permittivity on the electric field, making it much easier to find a solution. We can also get rid of the frequency nonlinearity, if we further assume that the laser frequency is equal to the central emission wavelength of the active molecules, εg(r, ω) = εg(r, ω21) = iε″g(r). Then we can rewrite the Eq. (2) in the following way:
which we now have to solve with respect to the electric field E, spatial distribution of ε and magnitude and spatial distribution of ε″g. In another words, permittivity is a function of geometrical parameters (such as lattice pitch, size of the scatterers, thickness of layers and various others depending on the base geometry) expressed as a single vector ξ, ε = ε(r, ω21, ξ). We can also divide the gain permittivity into a constant magnitude and spatially varying part, ε″g(r) = ε″gψ(r, ξ), where ψ is a known piecewise function proportional to the active molecule concentration C(r, ξ) (here we restrict the range of it’s possible values to 0 and 1). Solving Eq. (4) with respect to ξ and ε″g is still cumbersome. Instead we will use two workaround approaches to find non-trivial solution as described below.2.1.1. Approach 1
Instead of solving Eq. (4) with respect to ξ and ε″g, we define a range of values for the geometry ξ and permittivity ε″g and solve the eigenfrequency problem
for each ξn and ε″g,m in the ranges, with respect to the unknown eigenfrequency Ωnm. In order to find the non-trivial solution to this kind of problem we can employ one of the various well-developed numerical methods such as FDTD, FEM or other. Then we can construct the interpolating function Ω(ξ, ε″g), so that Ω(ξn, ε″g,m) = Ωnm, and find ξ and ε″g for which Ω = ω21. We need to stress that not only real part of the eigenfrequency should be equal to ω21, but also its imaginary part should be equal to zero to satisfy conditions for lasing. Found values of the ξ and ε″g will be the non-trivial solution to the Eq. (4). Once we find ε″g, we can go back to the parameters of the dye molecules of our choice and infer the required population inversion and pump from ε″g by equating it to the imaginary part of (3).2.1.2. Approach 2
Non-trivial solution to the Eq. (4) can be also found in a different way. We can add the source term to the right side of the equation and then use the similar procedure with solving the equation for a range of values of ξ and ε″g as in the previous method. However, since now we have the source on the right side of the equation we need to solve scattering problem instead of the eigenfrequency problem. Within this approach solution of the Eq. (4) manifests itself as a singularity in the transmission, reflection and absorption coefficients, since it is related to the excitation of the eigenmodes. Therefore in the second approach we find ξ and ε″g employing the following procedure. First, we solve the equation
where J source is current density of the source of the plane waves, that can excite the modes of the system, for a range of values of ξn and ε″g,m. Then we find the sum of the absorption and reflection coefficients Φnm(ξn, ε″g,m) (there is no transmission coefficient in our geometry) and construct the interpolating function Φ(ξ, ε″g). Values of ξ and ε″g at which Φ diverges are then the solutions to the Eq. (4). This method requires much finer mesh in ξn and ε″g,m than the previous one so its use is generally discouraged. However, as will be discussed in the next section, in certain situations finding non-trivial solution of the eigenfrequency problem according to the first approach proves to be hardly possible and the second approach such as discussed is required.In the rest of the paper we will not consider the specifics of the gain medium and will only specify the emission wavelength. We will tune our laser geometry to have resonances at the 850 nm, which roughly corresponds to the central emission wavelength of the IR-140 dye. To characterize the medium we will use a bulk value of gain, which is related to the complex permittivity as follows:
where εh is the permittivity of the host medium in which active molecules are embedded. Even though (3) suggests that the attainable amplification of the gain medium is proportionate to the active molecules concentration, there is a limit to the maximum possible gain imposed by the effects of self-absorption and quenching at high concentrations. Specifically, the maximum value of gain for the IR-140 dye found in literature is 68 cm−1 [35].2.2. Numerical modeling
We use COMSOL Multiphysics with Wave Optics module to solve homogeneous Helmholtz Eq. (5) for the unknown electric field and complex eigenfrequency. The simulation domain consists of one unit cell of the periodic lattice of the holes in the metal half-space. On the boundaries that limit the simulation domain in the directions of periodicity we generally employ periodic boundary conditions. However, certain lattices have symmetries that allow one to shrink the simulation domain and use different combinations of perfect electric conductor and perfect magnetic conductor boundary conditions to truncate the domain. Apart from the computational and memory simplification this provides the insight into the symmetries of the eigenmodes. In the dielectric above the metal the simulation domain is truncated with the perfectly matched layer (PML). Introduction of the PML can lead to the appearance of the spurious eigenmodes due to the reflection from the PML at grazing angles. These modes are essentially the modes of the artificial Fabry-Perot resonator, formed by the metal surface on one side and by the PML on another. These spurious modes can hybridize with the physical modes and change their spectral position and lasing threshold. However, in the majority of our calculations we do not encounter problems with the spurious modes, since either they are far away from the physical ones or do not hybridize with them. They do influence the physical modes, when the latter ones are leaking or radiating at the grazing angles to the PML. In this case, we employ the second approach from the Section 2.1 by solving Eq. (6), where we use external electric field source to excite modes of the system. We use port boundary condition to excite the incident plane waves impinging on the structure and also to truncate the simulation domain instead of PML.
In order to find non-trivial solution to the Eq. (4) we need to find the eigenmode whose eigenfrequency has real part equal to ω21 and imaginary part equal to zero. In the previous section we discussed general approach to solving this problem. Here we further specify the details. While solving the Eq. (5), at first we keep ε″g = 0 and perform parametric sweep across different geometrical parameters of the laser to find configurations in which wavelengths of the eigenmodes are equal to 850 nm (emission wavelength of the dye molecules). Imaginary part of the eigenfrequency is unconstrained during this process. Then for such configurations we sweep across different values of ε″g and interpolate obtained complex eigenfrequencies to find the imaginary part of the permittivity of the gain medium for which imaginary part of the eigenfrequency is equal to zero. Usually the real part of the eigenfrequency does not change substantially during the ε″g sweep, however, if it does deviate from the emission wavelength of the molecules substantially we tune the laser geometry accordingly. In cases when spurious modes related to the reflection from the PML are present and we have to use the second approach with port boundary conditions as an external source of the electromagnetic field, we scan geometrical parameters in the same way. In the absence of the spurious modes both methods give similar results, however, the major downside of the second approach is that it requires much smaller steps in ε″g and geometrical parameters so it should be avoided when possible.
3. Geometry of the laser under study
Lasing threshold depends on the materials used, lattice shape, lattice pitch, size and the shape of the scattering elements, thicknesses of the metal and the gain. In this work we limit ourselves to the case of the cylindrical holes as scattering elements (there are several experimental works that employ this kind of scatterers [8–10] and essentially they are the easiest to fabricate). Furthermore, we consider the metal half-space instead of the metal film, because it significantly simplifies the numerical calculations. It is a reasonable approximation because we do not have to consider the non-lasing surface plasmon modes on the opposite side of the metal film. We use silver as a plasmonic material, since it has low losses in the visible and near IR. Possible degradation of the optical properties of silver due to oxidation can be eliminated by covering it with a thin layer of SiO2 as was demonstrated in [9]. As a results, such structures will possess both low losses and high stability. Refractive index of silver is taken from [36]. We will optimize our structure so that the lasing wavelength is 850 nm, which roughly corresponds to the emission wavelength of the IR-140 dye. Lasing with this dye has already been demonstrated in the DFB lasers with the waveguide modes and plasmonic scatterers [5,6] and in the spaser with localized plasmons [3], where the gain coefficient of IR-140 was theoretically estimated to be 2630 cm−1. However, the maximum gain measured experimentally is about 68 cm−1 [35]. Refractive index of the host medium is 1.4692, which corresponds to the PVA matrix. We consider two cases: infinite PVA half-space and finite layer of PVA (see Fig. 2). In case of the infinite PVA (Fig. 2(a)) the gain molecules are embedded only in the 1.2 μm layer adjacent to the metal surface. This is a realistic assumption, since there could be no infinite gain medium in practice and refractive indices of the host matrix for the dye molecules and covering medium can be closely matched. In case of the finite layer of PVA (Fig. 2(b)) with active molecules its thickness is equal to 400 nm.
Fig. 2 Geometries of the distributed feedback plasmon lasers, xz cross-section through the center of the holes. Infinite dielectric with finite gain layer thickness (a), finite dielectric host with gain (b).
We will also consider different types of lattices — square, hexagonal and three rectangular ones (see Fig. 3). In order to solve Eq. (4) and satisfy condition ω = ω21 we have to restrict laser geometry. We will change the lattice pitch T and hole radius r and, therefore, plot position of the eigenmodes in the (r, T) coordinates.
Fig. 3 Lattice shapes under study: (a) square, (b) hexagonal, (c) rectangular with Tx −Ty = Δ, where Tx,y are lattice pitches along the x and y directions, respectively; Δ = 10, 15 and 30 nm. Hole diameter is d.
4. Results and discussion
First, we consider the eigenmodes of the plasmonic lattice in the empty lattice approximation in the conventional (kx, ω) coordinates, where ω is free space wavelength. In this approximation the scatterers are considered infinitesimally small. In this case the position of the eigenmodes is given by [37]:
where keig is the wavevector of the eigenmode, which lays in the (kx, ky) plane (in case of surface plasmon on the metal-dielectric interface keig = ksp = k0 [εmεd/(εm + εd)]1/2, k0 is the free-space wavenumber, εm,d are the permittivity of metal and dielectric, respectively), k̃eig is the wavevector of the eigenmode in the Brillouin zone, G1,2 are the primitive vectors of the reciprocal lattice, n and m are integer numbers. In case of the square lattice |G1| = |G2| = G = 2π/T, where T is lattice pitch.Figure 4 shows the surface plasmon dispersion on the silver-PVA interface according to the Eq. (8) in square lattice. The (n, m) = (±1, 0) eigenmodes are obtained by translating the keig(ω) by the reciprocal vector along the kx axis, which can be easily seen from the figure. The (n, m) = (0, ±1) modes are not so intuitive since they are obtained by translation along the ky axis, perpendicular to the figure plane. It can be seen from the figure that several eigenmodes lay within the light cone, which means that they in principle can radiate plane waves or be excited by the plane waves. Such plane waves will have the projection of their wavevector on the (kx, ky) plane equal to the k̃eig. In particular, when k̃eig = 0 the eigenmodes can emit light and can be excited at the angle normal to the metal surface. At this specific point eigenmodes form the standing wave. For the finite hole size the bandgap will form around this point due to anticrossing, with zero group velocity at the band edges [37]. This band edges are suitable for the lasing, so our task is to determine which of those modes has the lowest threshold and at which radius of the hole. It is important to emphasize that there are four modes at the k̃eig = 0 point in Fig. 4. Two of them with (n, m) = (±1, 0) can be seen in the figure as the red lines and the other two (0, ±1) are degenerate and can be seen as the yellow line. All four of them are degenerate at the point k̃eig = 0. This degeneracy originates from the four possible directions (±x and ±y) in which surface plasmon can propagate. Specifically, at this point there can be a standing wave in either x or y direction with either node or antinode above the infinitesimally small hole. The degeneracy is lifted when we consider finite size holes and different polarizations of the emitted radiation. The extension of the analytical approach as it is expressed in Eq. (8) to include the non-zero hole size and polarization is out of the scope of this paper and has already been done in [38]. We will use the numerical modelling as was discussed in Section 3 to account for these effect.
Fig. 4 Surface plasmon dispersion on the silver-PVA interface in the empty lattice approximation in the square lattice with T = 561 nm. Different colors show different (n, m) combinations. Black dashed lines show light cone in the air.
Here it is important for us that even in the case of non-zero hole sizes two out of four surface plasmon modes at the k̃eig = 0 point remain “dark” in a sense that they neither radiate nor can be excited by a plane wave. Such modes are usually referred to as bound in the continuum or BIC [39]. The reason for those modes to remain “dark” is the destructive interference of the radiation emitted by the eigenmodes that propagate in the opposite directions. It follows from the Fourier series expansion of the magnetic field that modes, that are odd under the 180-degree rotation around the z-axis, cannot radiate at the angle normal to the surface of the plasmonic crystal. Detailed discussion of this effect can be found in [27]. We can deduce that the eigenmodes, that we are studying, are in fact BIC states from their symmetry and the absence of emitted radiation. From the symmetry of the BIC states it follows that they must satisfy perfect magnetic conductor boundary conditions (for surface plasmons, for other types of modes it can also be perfect electric conductor) on all of the boundaries truncating the simulation domain in the x and y directions [39]. Eigenmodes that satisfy such boundary conditions cannot radiate at the angle normal to the lattice surface. Furthermore, if the lattice pitch is less than the wavelength in the topmost halfspace, there are no non-zero diffraction orders and the eigenmodes cannot radiate at all. This allows us to make the conclusion that such modes are BIC states.
Since BIC states have no radiation losses, we expect such modes to have lower lasing thresholds compared to the radiative modes. The absence of the radiation losses can be beneficial when, for example, coupling surface plasmon emitting laser to a surface plasmon waveguide. However, often lasing radiation is required. In case of the BIC states this can be achieved by introducing asymmetries in the geometry, for example by using triangular or ellipsoidal holes rotated slightly around the z-axis. The amount of radiation emitted by the lasing eigenmode in this case can be smoothly controlled by the degree of deviation from the perfectly symmetrical geometry.
Next we are going to consider eigenmodes in three types of plasmonic lattices (square, hexagonal and rectangular, see Fig. 3) for the geometry depicted in Fig. 2(a). We limit ourselves to the case of k̃eig = 0, since only at this point non-radiating modes, whose lasing threshold is minimal, exist in our structure. The dispersion relation in the case of non-zero hole sizes for the similar structure can be found in [38].
In Fig. 5 eigenmode’s positions in the (r, T) coordinate space are presented for different lattices (for the rectangular lattices T stands for Tx). In order to plot this figure we have solved Eq. (5), while for simplicity keeping ε″g = 0 and finding T = T(r) from Re [Ω(r, T)] = ω21 and keeping Im[Ω(r, T)] unconstrained. As was discussed earlier, position of the eigenmodes does not change substantially while we change ε″g, therefore position of the eigenmodes in Fig. 5 coincides with their position at the threshold. From these figures one can see that for the square and hexagonal lattices as the hole radius tends to zero, all mode’s pitches tend to a single value that can be found in the empty lattice approximation. For the rectangular lattices the difference between Tx and Ty lifts the degeneracy of the plasmons traveling in x and y directions, hence there are two different lattice pitches at r = 0. Some of the modes in Fig. 5 are degenerate even at r ≠ 0, those are radiating modes in square and hexagonal lattices (blue lines), and the lower one of the BIC modes in hexagonal lattice.
Fig. 5 Position of the eigenmodes in the (r, T) coordinates at fixed wavelength in the square (a), hexagonal (b), rectangular with Δ = 10 nm (c), 15 nm (d), 30 nm (e) lattices. The lattice is covered with infinite dielectric (see Fig. 2(a)), ε″g = 0. Blue line correspond to radiating modes (annotated by r), red line — to the non-radiative modes (nr). Horizontal black lines show the position of the Wood anomaly. Eigenmode above the Wood anomaly is shown as dotted line. Color on the background shows the logarithm of the absorptance (blue — low absorption, yellow — high absorption).
For the major part of the modes their lattice pitch decreases while the radius of the holes increases. However, there is one BIC state for each lattice shape that has a maximum lattice pitch for the non-zero hole size. In rectangular lattices surface plasmon can even cross the Wood anomaly, which corresponds to the lattice pitch at which the first diffraction order arises. In case of Δ = 30 nm one of the radiating modes and the BIC mode lay above the Wood anomaly even at r = 0 (see Fig. 5(e)). The position of the Wood anomalies is governed by the condition k = |nG1 + mG2|, where k = ndk0 is the wavenumber in the dielectric half-space with refractive index nd. In case of the Fig. 5(c)–5((e) the surface plasmon mode (0, ±1) crosses the (±1, 0) anomaly. It means that the plasmons that are traveling in the ±y direction scatter into the plane waves with kx = ±2π/Tx and ky = 0 above the anomaly. The schematics of this effect are depicted in Fig. 6. In contrast to the odd Hx and Ey components that interfere destructively, the Hy component of the eigenmode that appears due to the scattering on the hole is even and interfere constructively in the diffraction orders. Hence, such modes above the Wood anomaly cannot be referred to as BIC modes, since they have radiation losses. However, for the sake of simplicity we will still call them BIC or “dark” modes.
Fig. 6 Scheme that shows the eigenmodes and their radiation channels in the rectangular lattice with Tx − Ty = Δ. Red arrows show the standing waves in the ±x direction (left) and ±y direction (right). The lattice pitch Tx on the right happens to be above the Wood anomaly so there are the diffraction orders that the eigenmode is scattered into (shown as green arrows).
The modes above the Wood anomaly in rectangular lattices radiate into the diffraction orders. Because of the proximity to the Wood anomaly the angle of these diffraction orders is close to 90° to the normal and the substantial portion of radiation is reflected by the PML layer truncating the simulation domain — the issue that was discussed earlier. This is the reason we need to employ the second approach to find the mode position above the Wood anomaly. In Fig. 5(c)–5(e) the background color shows the logarithm of the absorptance. As can be seen from the Fig. 5(c) and 5(d), close to the Wood anomaly high absorptance and position of the eigenmode coincide. However, farther from the Wood anomaly the discrepancy grows. From the Fig. 5(e), which corresponds to the lattice with Δ = 30 nm, we can see that above the Wood anomaly, around r = 200 nm the eigenmode substantially deviates from the position of maximum absorption. This is due to the hybridization with the spurious Fabry-Perot mode and anticrossing behavior. Below the Wood anomaly, where the mode position can be calculated correctly, eigenmode position coincides with the maximum absorption.
The rapid change of the lattice pitch as the hole radius tends to 200 nm is due to the hybridization of the surface plasmon mode with the waveguide mode of the cylindrical hole.
The lasing thresholds for the configurations same as in Fig. 5 are shown in Fig. 7. In order to find lasing thresholds shown in this figure, we have employed the procedure discussed in sections 2.1–2.2. Specifically, for the modes below the Wood anomaly we have solved Eq. (5) for the range of values of ε″g and geometry configurations from Fig. 5 and found the ε″g for which ImΩ = 0. For the modes above the Wood anomaly we have similarly searched for the value of ε″g for which Φ diverges (see Eq. (6)).
Fig. 7 Gain required to reach the lasing threshold for eigenmodes in the square (a), hexagonal (b), rectangular with Δ = 10 nm (c), 15 nm (d), 30 nm (e) lattices. Lattice pitches can be found in Fig. 5. The lattice is covered with the infinite dielectric (see Fig. 2(a)). Blue lines correspond to the radiating modes (annotated by r), red lines — to the non-radiative modes (nr). Solid lines correspond to the threshold found by solving equation for the eigenmodes and complex eigenfrequencies (r and nr), dashed lines () correspond to the thresholds found through the singularity in absorption coefficient for the incident plane waves (above the Wood anomaly).
As one can see from the figure, for the majority of the modes the lasing threshold grows with the increase of the hole size. However, for each lattice shape there might exist one eigenmode, for which lasing threshold decreases up to a certain hole size. This mode is the eigenmode with the highest pitch from the Fig. 5. Moreover, one can notice that the minimum in the lasing threshold corresponds to the maximum in the lattice pitch for the square and hexagonal lattices. The reason for this is that the penetration depth of the surface plasmon mode into the dielectric increases as the lattice pitch comes closer to the Wood anomaly. As the penetration depth increases, the fraction of the mode’s energy concentrated in the dielectric grows and the fraction of the energy concentrated in the metal diminishes. Hence, the Joule losses also become lower and the threshold drops. We can illustrate this effect if we plot lasing threshold as a function of the penetration depth for the square and hexagonal lattices. In case of infinite homogeneous dielectric covering the metal surface (as in Fig. 2(a) for ε″g = 0), the penetration depth can be determined from the z-component of the eigenmode’s wavevector:
where L is the penetration depth of the eigenmode in the dielectric half-space and k is the wavenumber in the dielectric half-space with refractive index nd, k = ndω21/c. While the eigenmode pitch is below the Wood anomaly, is purely imaginary. The closer lattice pitch comes to the Wood anomaly, the smaller becomes and the larger L becomes. The threshold dependencies of the eigenmodes in square and hexagonal lattices are shown in Fig. 8. The threshold almost perfectly coincides for both lattices up to the point, where the threshold in the square lattice reaches its minimum value (the same holds for other BIC modes in both lattices, even though they are not shown in the figure). We need to stress out that hole size changes at different rates along the curves in the figure for different lattices. This means that up to the turning point (around L = 2000 nm for the square lattice and L = 7000 nm for the hexagonal lattice) the lasing threshold is completely determined by the penetration depth. As for the rectangular lattices, the effect of increased penetration depth on the lasing threshold also works for them but up to a moment, where the mode crosses the Wood anomaly in the perpendicular direction. Above the Wood anomaly, as was discussed earlier, these modes acquire radiation losses, which lead to the increase in their lasing threshold.Fig. 8 Gain required to reach lasing threshold for the lowest threshold eigenmodes in square (blue line) and hexagonal (red line) lattices vs. penetration depth of the eigenmodes into the dielectric half-space for a range of hole radii from 0 nm (indicated by the black dot) to 200 nm (arrows show the direction of increasing radius). The lattice is covered with the infinite dielectric (see Fig. 2(a)).
The thresholds of the topmost eigenmodes from Fig. 5 are shown in Fig. 9(a). As can be seen from the figure, hexagonal lattice has the lowest threshold with g = 18 cm−1 at its minimum. It is about 8 times less than the threshold in the absence of the holes, which is 146 cm−1. The square lattice has its minimal threshold of 63 cm−1 or about 2.3 times less than in the absence of the holes. The threshold for the rectangular lattices with Δ = 10 and 15 nm has two minimums — around the radii where eigenmode crosses the Wood anomaly. The precise gain required at the minimums is hard to calculate, since both approaches that we use become less accurate when the eigenmode pitch comes very close to the Wood anomaly. For the rectangular lattice with Δ = 30 nm the threshold gain starts to grow right from the r = 0 nm, since even at the infinitely small hole sizes its position is above the Wood anomaly.
Fig. 9 (a) Threshold gain for the topmost modes from Fig. 5 for each lattice type (with infinite dielectric covering the metal, see Fig. 2(a)). (b) Threshold gain for the eigenmodes that have the lowest threshold for each lattice type (thickness of the gain medium covering the metal is 400 nm, see Fig. 2(b)). Figure parts (a) and (b) share the same legend.
In the laser configuration with the finite dielectric/gain layer above the metal surface (see Fig. 2(b)) there is no Wood anomaly issue, since its position in the upper half-space is substantially shifted in the higher pitch region. We considered the configuration with the gain thickness of 400 nm. In this case for each lattice configuration there is one mode whose pitch substantially increases with the hole radius. Same as before this surface plasmon mode starts to hybridize with the waveguide mode of the hole around the r = 180 nm and the pitch becomes smaller. The threshold gain for each of these modes is shown in Fig. 9(b). The threshold at zero hole size is larger than in the case of infinite dielectric, because the thickness of the gain layer is smaller (1200 nm vs. 400 nm). The relative decrease in threshold is smaller for the same reason. The hexagonal lattice provides the lowest threshold with g = 124 cm−1 at r ≈ 165 nm. That is 35% less than in the absence of the holes. Square lattice and rectangular lattices with Δ = 10 and 15 nm have almost the same thresholds, around 145–150 cm−1 or 23% less than in the absence of the holes. Rectangular lattice with Δ = 30 nm has the worst performance of all studied here, with the threshold equal to 160 cm−1. These threshold values are substantially higher than in the case of infinite dielectric covering the metal (see Fig. 2(a)). In case of dye molecules in the dielectric host (like water, polymer matrix or others with refractive index around 1.5) increase in the dielectric thickness leads to the decrease in the gain required for the lasing. However, as was shown in [40], this is not the case for the semiconductor gain with permittivity around 7 and higher, since it might have optimal finite thickness that provides the smallest possible threshold.
We can now compare the threshold gain of the proposed laser configuration to the other configurations found in literature. For the core-shell geometry of the spaser the best theoretical estimate of the threshold gain is higher than 2000 cm−1 for the gain-silver-gain configuration [26]. We can also estimate the threshold pump intensity for our configuration and compare it to the experimental results. In order to make such estimates, one needs first to specify the parameters of the active medium. For instance, for the IR-140 dye molecules serving as the gain medium we can refer to the experimental results from [35]. With this external data we obtain the required average pump intensity in the laser pulse of 77 kW/cm2 to provide 18 cm−1 gain. In practice this value will depend on the pump conditions, such as angle of incidence, however we do not expect for this number to deviate more than one order of magnitude. Now we can compare our estimate to the threshold pump intensity of the other surface plasmon lasers. The most similar configuration can be found in [10], where the average pulse intensity is equal to 9.5 MW/cm2 at the threshold, which is two orders of magnitude higher than in our case. The DFB plasmonic laser on the array of metal particles studied in the [7] has the average threshold pulse intensity equal to 3 MW/cm2. From this we can conclude, that our results can lead to the design of new, advanced lasers with substantially lower thresholds and pumping rates and therefore better characteristics.
5. Conclusion
We have shown that the generation threshold in the distributed feedback plasmonic laser substantially depends on the geometrical parameters (such as hole radius, lattice pitch and lattice shape) as well as on the choice of the lasing modes. It was shown that bound states in the continuum require the lowest material gain to reach generation threshold. Decrease in generation threshold in each lattice can be attributed to the increase in the penetration depth into the dielectric and, hence, to the decrease of the Joule losses in the metal parts. Out of all lattice shapes considered in this study, the hexagonal lattice has the lowest threshold. In case of the infinite dielectric covering the silver half-space with the array of holes, this mode can have lasing threshold as low as 18 cm−1, which is attainable with the lasing dye such as IR-140. These results provide valuable insights into the ways to substantially lower lasing thresholds and pumping rates of the distributed feedback surface plasmon lasers.
Funding
Advanced Research Foundation (7/004/2013-2018); Russian Foundation for Basic Research (15-52-52006 and 18-02-00315); MEPhI Academic Excellence Project (02.a03.21.0005).
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