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Optimum gain for plasmonic distributed feedback lasers

Although nanolasers typically have low Q-factors and high lasing thresholds, they have been successfully implemented with various gain media. Intuitively, it seems that an increase in the gain coefficient would improve the characteristics of nanolasers. For a plasmonic nanolaser, in particular, a distributed-feed-back (DFB) laser, we propose a self-consistent model that takes into account both spontaneous emission and the multimode character of laser generation to show that for a given pumping strength, the gain coefficient has an optimal value at which the radiation intensity is at a maximum and the radiation linewidth is at a minimum.


INTRODUCTION
Laser physics has been extensively developed since the discovery of the laser more than half a century ago, [1][2][3] and now the principles of its operation have been well established. A laser consists of a resonator and a gain medium, in which a population inversion is created by external pumping. 4 Spontaneous emission with population inversion results in an increase of the number of photons inside the resonator. The resonator provides a positive feedback stimulating the gain medium to radiate into the resonator modes. The gain medium, as an amplifier, together with the feedback makes a coherent light generator.
propagation through a unit length of the amplifying medium. In traditional lasers the gain coefficient varies in the range of 0.1-1 cm -1 . 8 Lasing begins when the pump power exceeds a threshold at which the energy supplied compensates the loss. Energy is lost in the resonator walls and to radiation of light including laser output. There are several ways of increasing the laser power. One can either increase the pump power or decrease losses. Decreasing losses results in increasing the Q-factor of the resonator and thereby decreasing the lasing threshold. The threshold decrease may also be achieved by using new materials with enhanced gain. This often leads to emergence of lasers with a new design. In traditional macroscopic lasers, however, lasing can be achieved at fairly low values of the gain coefficient and a reasonable pump power. Therefore, there is no need for special efforts to increase the gain coefficient.
In the last decade, a new class of lasers, nanolasers, has become a subject of ever-growing interest because of the prospect of creating a subwavelength source of coherent radiation. [9][10][11][12][13][14][15][16][17][18][19] Owing to its small size, the nanolaser allows for ultrafast field dynamics. 13,15 This leads to the possibility of ultrafast transceivers that are important for developing optoelectronic technologies that require sources of coherent radiation that are capable of rapid transformations of electrical to optical signals. However, the side effect of the subwavelength size of plasmonic nanolasers is a wide angular distribution of the emitted light. To narrow the directional distribution, at least one of the laser dimensions should be of the wavelength size. For this purpose, one can use conically shaped nanoantennas, 20 Yagi-Uda antennas, 21,22 or periodic chains 23 and lattices of nanoparticles. The latter forms a plasmonic DFB nanolaser, [17][18][19][24][25][26][27][28][29][30][31][32][33][34] which is the subject of our paper.
Even though substantial progress has been made in creating plasmonic DFB lasers, 18,[24][25][26][27][28][29][30][31][32][33][34] a number of problems remain unsolved. The main problem with a DFB laser is that a plasmonic system has high losses and, thereby, a low Q-factor and a high value of the first lasing threshold. Consequently, in these systems, the contribution of spontaneous emission is high. Therefore, plasmonic DFB lasers may be overheated at levels of pump power required to achieve coherent lasing. [35][36][37] High-gain materials have been used to address this challenge. 18,28 These are either semiconductor quantum wells 18,24,28,32 or organic dyes 25-27, 29-31, 33, 34 for which gain coefficients are in the range of 100-1,000 cm -1 . These materials, however, are either expensive or require cryogenics for laser operations. Therefore, the search for low-loss plasmonic materials and active materials with high gain continues. [38][39][40][41] In typical high-Q lasers, multimode excitations may result in various effects. These include mode competition, which may suppress lasing from all modes except from the strongest one, mode locking, and mode beating, which are well studied in the literature. 12,42,43 In plasmonic DFB lasers, mode competition is suppressed by low Q-factors. Therefore, these systems are essentially multimode. In this paper, we consider a synergetic cooperation of spontaneous emission and a multimode regime.
We show that an increase in the gain coefficient would not necessarily improve nanolaser output. There is an optimal value of the gain that results from the nonlinear interaction of modes via the gain medium. For this value, a maximum intensity of radiation for a given pump power is realized. If the gain exceeds the optimal value, the radiation intensity decreases and the laser linewidth increases. We study the effect of spontaneous emission on multimode generation in a DFB nanolaser and do not consider effects of nonuniform distributions of a gain medium 42, 44 and a pump power in the resonator, [45][46][47][48][49] as well as the non-monotonic dependence of the laser generation threshold on parameters of the resonator. [50][51][52]

THE SYSTEM
In experiments, to create an inverted population of an amplifying medium, an optical pulse pumping is usually used. [25][26][27] The duration of the pulse varies from 40 fs to 200 ns. 25,26 A pulse must not overheat and destroy an active medium, which limits the pulse duration. A characteristic time of the longitudinal relaxation of active medium atoms is less than 10 ns. If a pump pulse has a duration of 100 ns or more, then in a laser stationary oscillations are established. Such a pulse, therefore, can be approximately considered a CW regime. This is the regime that we consider below.
To demonstrate the concept, we consider a DFB laser in which plasmonic nanoparticles are periodically positioned in a layer of a gain medium (see Fig. 1). A similar system was realized experimentally in Ref. 27.
Eigenmodes of such a laser are the Bloch modes, whose wavenumbers B k lie in the plane parallel to the gain layer and satisfy the phase condition of the laser generation 50,55 Re arg , where L is the length of the laser side, n is an integer, and r is the coefficient of reflection from the laser boundary. This coefficient is determined by both the laser structure and properties of the surrounding medium. We assume that arg 0 r = . For calculation of eigenmodes of the plasmonic DFB laser, we apply Bloch boundary conditions to the elementary cell of the periodic lattice. By going through the Bloch wavenumbers satisfying Eq. (1), we find the corresponding complex eigenfrequencies, i ω γ + shown in Fig. 2. Fig. 1. Schematics of the DFB laser. This system consists of a 300-nm layer of a mixture of polymer SU8 and dye R101 that is applied on a quartz substrate. Inside the layer, cylindrical silver nanoparticles are placed. The diameter and the height of a nanoparticle are 100 nm and 30 nm, respectively. The nanoparticles are positioned in periodic square-lattice sites with the period of 390 l = nm, the size of the system is L = 10l. The transition frequency of dye molecules is Note that the radiation from the surface of the DFB laser arises if the spatial spectrum of the generated Bloch wave contains wavenumbers which tangential components are smaller than the wavenumber of the surrounding space. 18,24,27 Below we consider the case when the radiation is in the direction perpendicular to this plane. Despite the small thickness of the active layer the width of the radiation pattern of the emitter may be narrow because it depends on the size of the radiation aperture which, in our case, is the transverse size of the laser.

MATHEMATICAL MODEL
A consistent study of plasmonic DFB lasers should take into account the multimode character of laser generation and spontaneous radiation of "atoms" of the gain medium. There are two equivalent approaches to the description of spontaneous radiation in multimode lasers. The first one is based on the solution of the operator version of the Maxwell-Bloch equations -the Heisenberg-Langevin equations. These equations include noise terms, thanks to which they take into account spontaneous emission processes. 56,57 However, operators describing noise in the Heisenberg-Langevin equations, are non-commutative. Therefore, working with them is difficult. For a large number of atoms, operator equations can be replaced by equations for c-numbers to obtain Maxwell-Bloch equations. 58 Such an approach is commonly used for the consideration of laser generation [59][60][61][62] including studies of multimode lasers. 63 In our paper, we use another approach, which is based on equations for expectation values for operators of the number of photons, the population inversion of atoms of the gain medium, and operators of coupling between these atoms and the electromagnetic field. 16,[64][65][66][67] These equations can be obtained by using the Lindblad form 68, 69 of the master equation for the density matrix (see Appendix A). Such a system of equations does not contain a noise term because the master equation for the density matrix is deterministic. The main problem in such an approach is an infinite number of equations. However, in many problems it is possible to uncouple some operator correlators to obtain a finite system. For a laser system, a common approach is uncoupling of the correlator of the number of photon, ˆâ a + , and the population inversion, D , namely,  68 which justifies the applicability of such an approximation for DFB lasers. This approach is suitable for both numerical simulations and analytical evaluations, and it is used in our paper.
To derive the laser equations that take into account both spontaneous radiation and multimode character of the laser generation, we use the Jaynes-Cummings Hamiltonian: 57,68 ( ) where ˆi a + and ˆi a are creation and annihilation operators of a quantum in the i-th resonator mode, i ω is the frequency of this mode, ˆm σ + and ˆm σ are raising and lowering operators for the transition of the m-th two-level atom, σ ω is the frequency of this transition, and im Ω is the coupling constant between the field in the i-th mode and the m-th atom.
Using Hamiltonian (2) (A6)-(A9)]: , where i γ is the relaxation rate of the photons in the i-th resonator mode and d γ and σ γ are the rates of the longitudinal and transverse relaxations in an atom. The term 0 d m D γ describes processes of incoherent pumping of the two-level active medium. 70 The system of equations (3)-(6) describes interactions between each resonator mode and each atom of the amplifying medium. If we assume that the constants of interactions of the i-mode with each atom are the same, im i Ω = Ω , then, in Eqs. (3)-(6), we can sum up over all atoms of the amplifying medium. With this assumption, the Maxwell-Bloch equations predict a single-mode regime for any gain. 64 As a result, we obtain: where D is the average population inversion of atoms of the amplifying medium, i I is the total energy flux from all atoms of the amplifying medium into the i-th resonator mode, i V is the energy of the interaction between the i-th resonator mode and all atoms of the amplifying medium, and N is the number of atoms. Note, that energy loss in the resonator is due to both losses in the material and radiation. Therefore, for each mode, the relaxation rate i γ can be represented as a sum of rates due to non-radiation, NRi γ , and radiation losses, Ri γ .
The Jaynes-Cummings Hamiltonian that was used for deriving Eqs. (7)-(10) describes the interaction of the field with two-level atoms. In reality, amplifying media have complicated multilevel structures. Often, a multi-level system can be approximately considered as a three-or fourlevel system which, in turn, can be reduced to an effective two-level system. 70 Below we assume that the concentration of atoms of the active medium is 17 3 , which allows us to neglect nonradiation interactions among the atoms of an amplifying medium (e.g., the Förster transitions).
Note, that in high Q-factor lasers, i σ γ γ >> . Therefore, variables i I and i V can be adiabatically excluded from Eqs. (7)- (10). As a result, these equations become identical to the rate equations for the number of photons and the population inversion. In low-quality resonators, such as plasmonic resonators that are considered here, the condition i σ γ γ >> is not fulfilled for some modes. In this case, Eqs. (7)-(10) cannot be reduced to the rate equations, and one has to solve the system (7)- (10).
The coupling constant between the i-th resonator mode and the amplifying medium, 2 , where ( ) r ε is the coordinate dependent dielectric permittivity of the DFB laser in which all media are considered as nonmagnetic. The dispersion of the dielectric permittivity must be taken into account for the correct calculation of the field in the structures with negative permittivities such as plasmonic structures.
As shown in Appendix B, the gain coefficient can be expressed as: where d is the dipole moment of an atom, G ε is the dielectric permittivity of the active medium, and c is the speed of light in vacuum.

SEMICLASSICAL MODELS OF LASING
By solving Eqs. (7)- (10) numerically we show that the number of photons in system modes is always non-zero. This number increases monotonically with an increase in the pump rate as shown in Fig. 3. In the stationary state, the number of photons in the i-th eigenmode is: the population inversion is In Eqs. (9) and (14), spontaneous transitions are described by the term ( ) This value coincides with the first generation threshold obtained from both the Maxwell-Bloch equations 57 and the rate equations when spontaneous radiation is not taken into account. 70 However, away from this threshold, the results of our theory are different. Note that in the semiclassical Maxwell-Bloch theory due to the mode competition, photons exist in one mode only. 64  Moreover, spontaneous emission goes into all modes. Consequently, owing to spontaneous emission, there are photons in all modes, the stimulated emission occurs into all modes as well.
Below the generation threshold, the Maxwell-Bloch equations predict the absence of photons in the modes. According to Eqs. (7)-(10) for / 1 i i i G D α η γ << , the number of photons in each mode is proportional to i α and is approximately the same for each mode (see Fig. 3).In this case, the DFB laser works in the multimode regime.
With an increase in the pump rate, the fastest growth of the number of photons is in the mode with the largest i α , i.e. in the mode with the lowest generation threshold. For / 1 , the number of photons in this mode is much greater than in all other modes. This is similar to the single-mode regime except for the dependence on G.

NON-MONOTONIC BEHAVIOR OF THE PHOTON NUMBER
In a real system, the characteristic gain parameter is GD. The population inversion D depends on time and the pump rate. Assuming that the pump rate 0 D is fixed, we analyze the dependence of numbers of photons in resonator modes on the gain coefficient G of the active medium. From Eqs. (7)-(10), one can find that when the gain coefficient exceeds a threshold value th G , the number of photons in any mode starts sharply increasing (Fig. 4). th G is defined by the condition (14):  Our computer simulation based on the solution of Eqs. (7)- (10) shows that a further increase in the gain coefficient G results in suppression of the lasing rate even in the mode with the lowest generation threshold. At a certain value of the gain coefficient, opt G , the number of photons in this mode has a maximum (see Fig. 4).
In the stationary generation regime, the total rate of stimulated and spontaneous transitions into all modes cannot be greater than the pump rate. However until 0 D > , an increase in the gain coefficient G results in an unlimited increase in the rate of spontaneous emission (9) and (11)]. To stop the increase in the number of spontaneously emitted photons, the depletion of the population inversion should be sufficiently high to yield 1 D = − . Note that, for 0 D < , the photon generation does not occur because stimulated transitions lead to photon absorption.
In a single-mode laser, for 0 D < , the lasing stops but the total number of photons does not decrease due to the growing number of spontaneously emitted photons. In a multi-mode laser, in the mode with the lowest generation threshold, the generation breakdown causes a decrease in both the number of coherent photons and the total photon number, because of the increase in spontaneous emission into the other modes.
Let us consider the change in the rate of spontaneous and stimulated transitions with a variation of the gain coefficient G . The rate of induced transitions is proportional to the number of photons in the mode, The rate of the spontaneous transitions does not depend on i n , As it follows from Eqs. (20) and (21)

A. The generation linewidth
When the amplification is close to the optimum, the number of photons in a mode is large, see Fig. 4. Then, the generation linewidth is determined by the Schawlow-Townes formula: 4, 57, 70 The dependence of the linewidth on gain, G, is shown in Fig. 6. As follows from Eq.

B. The maximum value of the photon number in the mode with the lowest generation threshold
Above its generation threshold, for the mode with the lowest threshold, the value of D is approximately the same as the threshold value 0 / 0 We make an estimate Γ Γ has its maximum as well.

C. The transition from a single-mode to a multimode regime
When G differs from SE G , the ratio of the rates of induced and spontaneous transitions decreases. This leads to the transition from a single-mode to a multimode regime. In the former regime, in the mode with the lowest generation threshold, the number of photons is many times greater than in other modes. This number decreases significantly when the system transitions to the multimode regime.
Let us introduce the parameter i i j j n n χ = ∑ (27) showing the ratio of the number of photons in the i-th mode to the total number of photons in all modes. For the mode with the lowest generation threshold, this parameter depends on the gain coefficient non-monotonically (Fig. 7). As shown in Fig. 7, it reaches the maximum value when To summarize, when the gain coefficient reaches its optimum value, SE G G = , the generation linewidth has a minimum, and the ratio of the number of photons in the mode with the lowest generation threshold to the total number of photons in all modes reaches its maximum. Thus, the system transitions into a single-mode regime. In addition, in the mode with the lowest generation threshold, the number of photons is the greatest when for SE G G ≈ .

CONCLUSION
In this paper, we have studied generation regimes of a plasmonic multimode DFB laser for various parameters of an active medium using a self-consistent model that takes into account spontaneous transitions and the multimode character of laser generation. We have shown that even when the pump power is substantially higher than threshold, the laser is multimode in a low-Q plasmonic structure. In turn, the intensity and the width of the laser line depends nonmonotonically on the gain coefficient of the active medium due to the nonlinear interaction of a large number of modes within the gain medium. We have demonstrated that there is an optimal value of the gain coefficient, , at which the ratio of the rates of induced and spontaneous transitions has a maximum and the generation linewidth is at minimum.
For the optimal value of the gain coefficient, a plasmonic DFB laser supports a single-mode in which the generation occurs in the mode with the lowest generation threshold. When G deviates from SE G , the ratio of the rates of induced and spontaneous transitions decreases, and the laser works in a multimode regime. This greatly reduces the generated intensity.
Note that modes of DFB lasers are leaky waves radiating at various angles to the slab of an active medium. 24,28,32 Usually, the structure of a DFB laser is optimized in a way that the mode with lowest threshold radiates perpendicular to the laser plane. For The optimal value of the gain coefficient for the plasmonic DFB laser considered in our paper is . This value corresponds approximately to the gain coefficient of organic dye R101. 53,54 We have used parameters of this dye in our model. Thus, one can use active media with gain coefficients of the order of a few hundred inverse centimeters in DFB lasers similar to the ones considered in our paper. These could be organic dyes or colloids of quantum dots with optical pumping or bulk semiconductors with current injection. 71 The value of the optimal gain coefficient depends on the detuning between the eigenmode with the smallest generation threshold and the transition frequency of the gain medium. This value is inversely proportional to the ratio of the energy of the electric field of this mode in the volume of the amplifying medium to the total energy of this mode. An optimal laser can, therefore, be realized either by choosing the best active material for a given laser structure or by changing the geometry of the plasmonic structure. If the gain coefficient of the active medium is smaller than optimal, this energy ratio can be increased and vice versa.

APPENDIX A: DERIVATION OF LASER EQUATIONS
In the Markovian approximation, the dynamics of two-level atoms interacting with an external electromagnetic fields may be described with a master equation in the Lindblad form 68 where ρ is density matrix, Ĥ is the Jaynes-Cummings Hamiltonian 57, 68 , Eq. (2). The term Equations (A6)-(A9) allow for describing the spontaneous emission with the accuracy of 1/N, where N is a number of atoms. 64,68

APPENDIX B: EXPRESSING CONSTANTS OF LASER EQUATIONS VIA PARAMETERS OF AN AMPLIFYING MEDIUM
The interaction constant between a field and an atom of an amplifying medium can be expressed as where ε is the dielectric permittivity of the medium which is considered as non-magnetic.

Equation (B2) gives ( )
i m E r when the field distribution in a resonant mode is known.
If the field distribution is sufficiently uniform, the electric field in the position of each atom can be considered as approximately equal to the average field in the amplifying medium. In this case, in Eqs. (3)-(6) of the paper, one can perform the summation over the number of atoms to obtain system (7)-(10).
To calculate the coupling constant 2 i Ω between the field and an amplifying medium, we introduce the parameter i η , which is the ratio of the energy of the electric field of the i-th eigenmode in the volume of the amplifying medium and the total energy of this mode defined by Eq. (B2): where G V is the volume of the gain medium. Again, assuming that

APPENDIX C: THE DEPENDENCE OF THE RATE OF INDUCED TRANSITIONS ON THE GAIN COEFFICIENT
The ratio of the rates of induced and spontaneous transitions is given by Eq. (22): Using Eqs. (15) and (18)  As follows from Eqs. (7) and (8) The minimum of the denominator in Eq. (14) is achieved when Since the number of modes in the laser is much smaller than the number of atoms in the gain medium, the term with 2 2 / K N can be neglected. Then, we obtain Note, that in deriving Eq. (D13) we assume that the number of atoms N remains constant when the amplification coefficient changes.