Mode cooperation in two-dimensional plasmonic distributed-feedback

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Mode cooperation in two-dimensional plasmonic distributed-feedback laser Nikita Nefedkin, Alexander Andreevitch Zyablovsky, Evgeny S. Andrianov, A.A. Pukhov, and Alexey P. Vinogradov ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00265 • Publication Date (Web): 05 Jul 2018 Downloaded from http://pubs.acs.org on July 5, 2018

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Mode cooperation in two-dimensional plasmonic distributed-feedback laser Nikita E. Nefedkin,∗,†,‡,¶ Alexander A. Zyablovsky,∗,†,‡ Evgeny S. Andrianov,†,‡ Alexander A. Pukhov,†,‡,¶ and Alexey P. Vinogradov†,‡,¶ †Dukhov Research Institute of Automatics (VNIIA), 22 Sushchevskaya, Moscow 127055, Russia ‡Moscow Institute of Physics and Technology, Moscow 141700, Russia ¶Institute for Theoretical and Applied Electromagnetics, 13 Izhorskaya, Moscow 125412, Russia E-mail: [email protected]; [email protected]

Abstract Plasmonic distributed-feedback lasers based on a two-dimensional periodic array of metallic nanostructures are the main candidate for nanoscale sources of coherent electromagnetic field. In this work we show that, when the pumping spot is smaller than the size of the laser, the nonlinear interaction between the modes of a periodic plasmonic structure via the active medium leads to a new effect, namely, mode cooperation. Mode cooperation is manifested as the lasing of the bright modes in the allowed band with a high generation threshold instead of the dark modes at the edge of the band gap with a low generation threshold. This effect is inherent for two-dimensional distributedfeedback lasers and does not appear in one-dimensional lasers. Mode cooperation arises due to non-orthogonality of the laser modes in the pumped area where the bright modes become synchronized by the phase and their amplitudes interfere constructively. We

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demonstrate that mode cooperation results in broadening of the radiation pattern above the generation threshold, which has been observed in recent experiments. The obtained results provide new possibilities for effective control and manipulation of the radiation pattern of nanoscale systems.

The creation of nanosized sources of coherent radiation is a challenging problem in modern laser science. 1,2 The size of ordinary lasers is constrained by the diffraction limit. To overcome this, it has been proposed to use metallic nanostructures as laser cavities 3,4 with resonance plasmonic excitations as cavity modes. In the last decade, substantial progress has been achieved in creating and investigating lasers based on plasmonic nanostructures, such as metallic nanoparticles, 5–8 waveguides, 9–13 etc. To enhance feedback and obtain beam directionality and a narrow radiation pattern it has been suggested to use periodic metallic structures. The periodicity leads to the formation of allowed and forbidden bands for the electromagnetic field. The electromagnetic field in periodic structures is presented as a superposition of Bloch modes with frequencies belonging to an allowed band and wavenumbers from 0 to 2π/a, where a is the system’s period. 14,15 These Bloch modes are distributed throughout the resonator. The interaction of the Bloch modes with the active medium creates positive distributed feedback. 16–28 Such plasmonic distributed-feedback (DFB) lasers have been realized in a series of experiments, where periodic plasmonic structures, such as metallic films perforated by holes 17–21,29,30 or two-dimensional arrays of plasmonic nanoparticles 16,23–26,28,31–34 play the role of the resonator. Plasmonic DFB lasers are very promising for a wide range of applications, from sensing and bioimaging to optoelectronics. 2 For spectroscopic and optoelectronic applications, it is necessary that a laser has a narrow tunable radiation pattern. The radiation pattern of a plasmonic DFB laser is created by the radiation pattern of the Bloch modes which propagate along the plane of the surface of the DFB laser. Standard qualitative arguments predict that a DFB laser has to radiate in a narrow cone close to the normal direction. Indeed, the radiation angle of these modes as well as their radiation losses depend on the ratio of the 2


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Bloch wavenumber to the free space wavenumber. 19 For eigenmodes at the edge of a band gap, this ratio tends to zero, and such modes radiate close to the normal direction of the DFB laser surface while modes in the allowed band radiate at a nonzero angle. In the vicinity of the edge of the band gap, there are dark modes for which the radiation losses are the smallest. 35,36 These modes have the lowest generation threshold and have to be generated with the highest amplitude. This should lead to radiation in a narrow cone close to the normal direction, and prevent the tunability of the radiation pattern. Such qualitative arguments are confirmed by the well-known theory of DFB lasers. 37 However, recent experiments 17–20 show much more complex behavior of the radiation pattern, which cannot be explained by these qualitative arguments. Namely, the radiation pattern is much wider, than in the case, when the dark modes at the edge of the band gap generate. 37 This circumstance indicates that the lasing can be observed in the bright modes lying in the allowed band. 20 In the present paper, we show that in the two-dimensional plasmonic DFB laser, where the pumping area is smaller than the laser’s size, nonlinear interaction between modes of the plasmonic lattice via the active medium is accompanied by a new effect which we call mode cooperation. This effect occurs due to non-orthogonality of the laser modes in the pumped area and is the opposite of mode competition in usual lasers. Mode cooperation appears as a growth of stimulated emission in the modes in the allowed band with a high threshold instead of the modes localized near the band gap with a low threshold. Suppression of the lasing of the modes near the band gap results in the widening of the radiation pattern above the generation threshold. We show that mode cooperation depends strongly on the density of states of the system and, therefore, mode cooperation takes place for two-dimensional DFB lasers and does not appear in one-dimensional ones. Mode cooperation gives the opportunity for effective control and manipulation of the radiation pattern of nanolaser systems, which can be widely applied in spectroscopy and optoelectronics.

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Figure 1: A sketch of 2D plasmonic DFB laser consisting of two layers and the radiation pattern created by the laser. The first layer is a gold film perforated by an array of nanoholes and the second layer is an active medium located beneath the gold film.

The model for describing the dynamics of a plasmonic DFB laser The description of the dynamics of the electromagnetic field of a periodic plasmonic lattice and the atoms of the active medium (see Fig. 1) is based on the approach which has been proposed in 38 for considering ultrafast phenomena in dispersive dissipative media. The basis of this approach is the description of the collective dynamics of the photon number in each 4


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mode, the population inversion of each atom, and the energy flows between the different modes through the active medium. For the system under consideration, this approach leads to the following equations (see Note S1, S4 in Supp. Inf.) X |Ωjm |2 γσj dnjj (2njj Dm + Dm + 1) + = −2γj njj + 2 dt |∆ | j m X Ωjm Ω∗ njk km + + c.c. Dm ∆ j m,k,k6=j

(1)

dnjl = − (γj + γl ) njl + i (ωj − ωl ) njl + dt X Ωlm Ω∗jm (Dm + 1) (∆j + ∆∗l ) + + ∗ 2∆ ∆ j l m X Ωlm Ω∗ njk Ωkm Ω∗jm nlk km + + Dm ∗ ∆ ∆ j l m,k,k6=j

(2)

dDm = −γD (1 + Dm ) + γpump (1 − Dm ) − dt X |Ωjm |2 γσj −2 (2njj Dm + Dm + 1) − |∆j |2 j X Ωjm Ω∗ njk km −2 + c.c. Dm ∆ j j,k,k6=j

(3)

Here, njj is the photon number in the jth mode, and njl is an interference term responsible for the flow of photons from the lth cavity mode to the jth cavity mode when j 6= l. Dm is the mean value of the population inversion of the mth atom. The parameters ωi , γi correspond to the real and imaginary parts of the modes’ eigenfrequencies (see Note S2 in Supp. Inf.); ωσ , γph , γD and γpump are the atom transition frequency and atom rates of energy relaxation, phase relaxation, and incoherent pumping, respectively. ∆j = γσj − i (ωj − ωσ ), where γσj = γph + γj + (γD + γpump ) /2. Ωjm is the Rabi constant of interaction between the jth mode and the mth atom (See Note S3 in Supp. Inf.). These equations generalize the rate equations widely used in laser theory. In addition to the dynamics of the photon number in each mode and the population inversion of each atom, they describe additional energy flows between the active medium and the atoms. Such energy flows arise from the 5


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stimulated emission generated by interference terms, njl . The energy flows are described by the last terms in Eqs. (1)–(3) and strongly depend on the spatial overlap of the plasmonic lattice modes in the active medium region. Indeed, the last terms in Eqs. (1)–(3) are proportional to P Ωjm Ω∗km njk ∆j

m,k,k6=j

=

P k,k6=j

∼

P k,k6=j

njk ∆j

P

m ¯ njk D ∆j

+ c.c. Dm =

(Ωjm Ω∗km Dm + c.c.) ∼ R

(4)

(Ek∗ (r) Ej (r) + c.c.) dV

Vgain

¯ is the characteristic value of where the range of integration coincides with the pump area, D the population inversion in the active medium, and the njk play role of the interference terms of the electromagnetic field. 38 When the active medium is pumped uniformly over the whole volume, these energy flows are equal to zero due to the orthogonality of the modes. In such a case, the best lasing condition is for the mode with the smallest dissipation and the highest constant of interaction with the active medium. 39 However, when only part of the active medium is pumped, the modes inside this pumped area are no longer orthogonal. Therefore, the interference terms are nonzero and energy flows become essential. Such energy flows take their maximum values for the modes which are maximally overlapped in the region where the active medium is located. As we show below, these terms are crucial for two-dimensional DFB plasmonic lasers with inhomogeneous pumping.

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1600

a

4

800

3 2

1550 1500 1450 -100

0 kxH2ΠL-1L

50

2 1

400

3

200

1 -50

b

600 Q

1650

Λ HnmL

100

0 1450

1500

1550 Λ HnmL

4 1600

1650

Figure 2: (a) The dispersion curve of the 2D plasmonic DFB laser. The surface area of Au film is L × L = 190 × 190 µm2 . The step of quantization of Bloch wavenumber δkB = π/L. (b) The quality factor of eigenmodes. The green line is an atomic emission line. 1.0 0.8 R00 Ha.u.L

Output HkWcm2L

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0.6 0.4 0.2 0.0 0

5

10 15 Intput HkWcm2L

20

25

Figure 3: The dependence of output power (green line) and the ratio of the photon number in the mode at the edge P of a band gap to the total photon number in all modes R00 = n (kBx = 0, kBy = 0) / nii (blue line) on the input power. i

Results and discussion Lasing curve and the radiation pattern of two-dimensional plasmonic DFB laser We now consider the 2D plasmonic DFB laser with cavity based on Au film perforated by a periodic array of nanosized holes. The active medium is located beneath the surface of the metallic film, see Fig. 1. This structure is typical for experiments. 19,20 The eigenimodes of the plasmonic structure without holes are the TM-modes that are similar to surface plasmons

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at the metal-dielectric interface. Such modes have in-plane component of the magnetic field, Hk , that is perpendicular to the direction of mode’s propagation and both out-of-plane, E⊥ , and in-plane, Ek , components of the electric field. The in-plane electric field, Ek , is in the direction of the mode’s propagation. For a surface plasmon the out-of-plane component of the electric field, E⊥ , is much greater than in-plane component, Ek . To find the eigenmodes of the plasmonic periodic nanostructure, we used the coupledmode theory with the experimental values of the scattering rates (see 19 and Note S2 in Supp. Inf.). We consider that the transition frequency of the active medium is close to the frequency at which the wavelength of the surface plasmon is equal to the period of the plasmonic lattice. Dispersion curves of this system are shown in Fig. 2a. The modes corresponding to different dispersion curves have various radiation losses and field distributions inside the unit cell of the plasmonic crystal (see Note S2 in Supp. Inf.). The coupling of the surface plasmons with the far-field mainly occurs due to the scattering of Hk by the holes. 19 For this reason, the modes that have a minimum value of Hk at the holes are dark, whereas the modes having the maximum value of Hk at the holes are bright. The dark modes lie in the 1st and 2nd dispersion curves at the edge of the band gap (see 19 and Note S2 in Supp. Inf.). Such modes have the highest quality factors. The modes lying in the 1st and 2nd allowed bands (kB 6= 0) radiate at the angle sin θ = kB /(2π/λ), where λ = 2πc/ω and, therefore, have lower quality factors than the dark modes at the edge of the band gap. In the 3rd and 4th dispersion curves, there are the bright modes having maximum of Hk at the holes (see 19 and Note S2 in Supp. Inf.). These modes have even lower quality factors than the modes from the 1st and 2nd allowed bands, Fig. 2b. In our simulations we assume that the width of a transition line of the active medium covers the whole investigated frequency range, Fig. 2b. This condition is satisfied in the experiments. 19,20 We consider the case when the pump beam is focused to a finite spot on the plasmonic laser surface. 16–28 In the pump area, the population inversion is positive, whereas beyond the pump area it remains negative. 16 The electromagnetic field, in turn, is amplified inside 8


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Figure 4: The mode distribution of the photon number below (a) and above (b) the pump threshold. the pump area until its amplitude reaches the stationary value. However, beyond the pump area amplitudes of the electromagnetic waves exponentially decrease at a distance from active region. This structure is similar to the one investigated in the papers. 17,19–21 Mode competition in a multimode laser usually leads to suppression of lasing in modes with a high pump threshold. 10,40,41 For this reason, we can expect that the lasing could be observed in dark modes. The photon number in these modes would be much higher than in other modes. In this case, the radiation of the DFB laser is rather small. The main results of numerical simulations of Eqs. (1)–(3) are shown in Figs. 3–4. The lasing occurs at the modes lying in the 2nd dispersion curve. The dependence of the total output electromagnetic field intensity on the input pump power is shown in Fig. 3. It is seen that the output intensity demonstrates threshold behavior, see a green line in Fig. 3. The distribution of the photon number below and above the pumping threshold over the modes with different values of the Bloch wavevector kB = (kx , ky ) is shown in Figs. 4a and 4b, respectively. On the one hand, below the threshold the photon number is maximal for the dark mode at the edge of a band gap with a low dissipation rate, see Fig. 4a. The photon number monotonically decreases when the mode frequency moves from the edge of the band gap to the allowed band. In other words, the excitation is maximal for the dark modes with 9


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kB = (0, 0). This result is expected from the qualitative considerations. On the other hand, the situation dramatically changes above the threshold. The maximum of the photon number is observed for the modes which lie in the allowed band, see Fig. 4b. Such behavior contradicts the intuitive expectations. Usually, the modes with the lowest threshold have a maximal field amplitude. However, in investigated two-dimensional plasmonic DFB laser, the modes, which lie inside the allowed band and have higher dissipation rates due to radiation, have maximal amplitudes. Thus, stimulated emission is primarily in bright modes with kB 6= (0, 0). In Fig. 3, we also show the ratio of the photon number in the mode at the edge of the band gap to the total photon number in all modes (a blue line in Fig. 3). It is seen that this ratio decreases from maximal value below the threshold to a constant value above the threshold. This corresponds to the fact that above the threshold an essential part of the total output intensity is produced by the bright modes. Transition from spontaneous emission in dark modes to lasing in bright ones leads to both a sharp increase of the photon number and the radiation power. Thus, below the generation threshold, when the main contribution in laser output is produced by spontaneous transitions of atoms of the active medium, the photon number is maximal in the dark mode at the edge of a band gap. At the same time, above the threshold, when the stimulated emission dominates, the maximum photon number is in the bright modes lying inside the allowed band.

Toy model for mode cooperation The described situation is unusual for lasers. Conventionally the modes with the lowest generation threshold has a maximum photon number. In the previous section, we have seen that this behavior takes place only under the generation threshold. The reason can be revealed through a simple system which consists of three modes with eigenfrequencies ω1 , ω2 , ω3 , where ω2 = ω3 , |ω1 − ω2 | γ2 , and dissipation rates γ1 < γ2 = γ3 , which interact 10


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with the active medium. The second and third modes differ in the field distribution. For simplicity we suppose that γph γj , γD , γpump , |ω1 − ω2 | and the population inversions of all atoms are the same. In such a case, Eqs. (1)–(3) can be simplified and take the form Γ2 2Γ2 dn11 = −2γ1 n11 + (D + 1) + n11 D dt γph γph Γ2 dn22 = −2γ2 n22 + (D + 1) + dt γph 2Γ2 2Γ2 + n22 D + 23 n23 D γph γph dn23 Γ2 = − (γ2 + γ3 ) n23 + 23 (D + 1) + dt γph 2 2 2Γ 2Γ + n23 D + 23 n22 D γph γph 6Γ2 dD = −γD (1 + D) + γpump (1 − D) − (D + 1) − dt Nat γph 2D 2Γ2 2D 2Γ223 n23 − (n11 + n22 + n33 ) − + c.c. Nat γph Nat γph where Γ2 =

N at P

(5) (6)

(7)

(8)

Ω∗jm Ωjm is a net interaction constant of the active medium with field of these

m=1

three modes and Γ223 =

N at P

Ω∗2m Ω3m is an overlap integral of the second and third modes, and

m=1

Nat is the number of atoms in the system. Here we neglect the interference terms between the first and other modes (i.e. the terms n12 , n13 ) due to |ω1 − ω2 | γ2 . In the steady-state, the photon numbers in the first and second modes are given by nst 11 =

1 Γ2 Dst + 1 2 γ1 γph 1 − γ Γγ2 Dst

(9)

1 ph

st nst 22 = n33

1 Γ2ef f Dst + 1 = 2 γ2 γph 1 − Γ2ef f Dst

(10)

γ2 γph

where Γ2ef f = Γ2 +

Γ423 Dst γ2 γph 1 − γ Γγ2 Dst 2 ph 11


(11)

is an effective coupling-constant between the second mode and the active medium.

Output Ha.u.L

25 20 15 10 5 0 0

20

40 60 Input Ha.u.L

80

100

Figure 5: The dependence of the photon number in the first (blue solid line) and second (red dashed line) modes on pump intensity calculating from Eqs. (5) – (8). Here γ1 = 10−4 ωσ ; γ2,3 = 1.5 × 10−4 ωσ ; γD = 10−6 ωσ ; Γ2 /γph = 10−3 ωσ ; Γ223 /γph = 7 × 10−4 ωσ and Nat = 104 , where ωσ is a transition frequency of atoms of the active medium. An increase of the pumping rate γpump leads to the rise of the population inversion D, see Eq. (8). This results in a growth of the effective coupling-constant, Γef f . From Eqs. (9)–(10) it is seen that when Γ2ef f /γ2 < Γ2 /γ1 , the photon number in the first mode is larger than the photon number in the second and third modes, see Fig. 5. However, when the pump power exceeds the critical value such that Γ2ef f /γ2 > Γ2 /γ1 , the photon number in the first mode is less than one in the second and third modes, see Fig. 5. This behavior is caused by the constructive interference of the second and third modes, whose electric field oscillates in phase (n23 > 0). In Eqs. (5)–(8) it reveals as existence of constant Γ23 which describes mode overlap in the active region. Indeed, when Γ23 = 0 the effective coupling-constant Γef f = Γ and the photon number in each mode is determined by the mode dissipation rate, see denominator in Eqs. (9)–(10). However, when Γ23 6= 0 the effective coupling constant Γef f is increased by the growth of the pumping rate. Thus, the constructive interference leads to the following consequences. The interaction between the active medium and the modes with the high threshold (the second and third modes in above example) is stronger than interaction between the active medium and the mode with the low threshold (the first mode in above example). As a result, the photon num12

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ber is the greatest in the modes with high threshold. Enhancement of the electromagnetic field amplitude due to the constructive interference between modes we call mode cooperation.

Mode cooperation of a two-dimensional DFB laser Let us consider how this mode cooperation manifests in a real two-dimensional DFB laser. To estimate the influence of interference interaction on lasing in this case we compare the P 2 direct stimulated emission rate into the jth mode, |Ωjm | njj Dm , and the total stimulated m P emission rate into the jth mode, |Ωjm |2 njl Dm , above the generation threshold. The first m,l value corresponds to photon amplification in the jth mode only due to stimulated emission induced by photons in this mode. The second one, in addition, takes into account photon amplification in the jth mode by stimulated emission induced by the photons in other modes. P P P 2 2 We show the value |Ωjm | njl Dm / |Ωjm | njj Dm in Fig. 6. It is seen that the m

l

m

influence of interference interaction is maximal for modes shifted from the edge of a band gap. The energy flow from atoms of the active medium to the these modes is increased by interference interaction between modes. Such increase leads to the shift of the lasing peak away from the edge of the band gap. Thus, the cavity modes help one another to extract energy from the active medium. This effect is opposite to mode competition in conventional lasers, and therefore, justifies its name by mode cooperation. Let us estimate at which conditions the mode cooperation can take place in the twodimensional plasmonic DFB laser. As it was discussed above, the reason for the mode cooperation is interference interaction between modes. In the Eqs. (1)–(3) the term which is responsible for this interference is a spatial overlap of modes in the region of the pumped active medium. If the population inversion is equal for each atom, then the spatial overlap of the modes in the region of the active medium can be expressed as ! X Γ2jk njk ¯ + c.c. D γ σ k,k6=j 13


(12)

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Figure 6: The ratio of the atoms to the jth energy flow from all mode with and without the P P P |Ωjm |2 njl Dm / |Ωjm |2 njj Dm . interference terms, i.e. m m l

where Γ2jk =

P m

Ωjm Ω∗km . In the case, when the pump area occupies all surface of the DFB

laser, Γjk = 0 because the cavity modes are orthogonal in the active area. The interference flows between modes are mutually cancelled. In a two-dimensional plasmonic DFB laser, usually the pump area is less than the surface of DFB laser and the modes are nonorthogonal in the volume of pumping. In this case, the interference interaction between modes is not zero and leads to creating an additional flow from the atoms to the modes. Let us determine which modes give the main contribution to (12). In order to do this, we note that the inteference interaction between the jth and kth modes, Eq. (12), is comparable 2

¯ Γjj njj , when |Γjk | ∼ Γjj , |njk | ∼ njj , with the direct energy flow from the active medium, D γσ and Re(njk ) > 0. The first condition is fulfiled when the jth and kth modes lie in the same allowed band and their Bloch wavevectors satisfy the following inequality (see the integrand in Eq. (4)): ~ ~ kBj − kBk l ≤ 1

(13)

where l is the size of the pump area. The absolute value of the interference term between the jth and kth modes is proportional to |njk | ∼ |(γj + γl ) − i (ωj − ωl )|−1 while |njj | ∼ 2(γj )−1 , 14


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see Eqs. (1)–(3). Thus, the second condition is fulfiled when

|ωj − ωk | ≤ γj + γk

(14)

The last condition is fulfiled when the electric fields of the modes oscillate in phase. Our calculations show (see Fig. 6) that, above the generation threshold, interference leads to an increase in the energy flux from the active medium to the modes. Consequently, above the generation threshold, the electric fields of different modes begin to oscillate in phase.

Figure 7: The radiation pattern of a 2D plasmonic DFB laser below (a) and above (b) the generation threshold. The dashed lines correspond to the radiation pattern of a onedimensional plasmonic DFB laser with the same parameters.

Comparison of one- and two-dimensional systems The condition (13) is satisfied for all modes in considered range of wavelength. At the same time, the condition (14) enables to distinguish the main feature of the radiation of the oneand two-dimensional plasmonic DFB laser. Indeed, the number of modes which lie in the interval (ωj − (γj + γl ) , ωj + (γj + γl )) may be evaluated as ρ (ωj ) (γj + γl ), where ρ(ω) is d2 k ∼ the density of states. In the two-dimensional case, the density of states ρ (ωj ) = dω 2 ω=ωj kBj dk . At the edge of a band gap, dω/dk ∼ kBj and, as a consequence, ρ (ωj ) ∼ const (π/L)2 dω ω=ωj 15


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. 42 At a distance from the edge of a band gap, the ratio dk/dω slightly varies and the density of states is proportional to kBj , see Fig. 2a. Thus, the density of states tends to a constant at the edge of a band gap and increases in an allowed band. As a result, the number of modes in the allowed band which satisfies the condition (14) is larger than one at the edge of the band gap. So the interference interaction between modes is stronger in the allowed band. Taking into account that the dissipation rate of the modes increases in the allowed band, we come to the conclusion that there is the value of frequency range in the allowed band for which the generation condition is optimal. Numerical simulations confirm this qualitative conclusion, see Fig. 4. In the one-dimensional case, ρ (ωj ) ∼ (L/π) dk/dω and has a maximum at the edge of the band gap, where dω/dk tends to zero . 42 As a result, the dark modes at the edge of the band gap have the optimal condition for generation. Numerical simulations of Eqs. (1) – (3) for the one-dimensional plasmonic DFB laser confirm above conclusion. The maximum of the photon number is observed in the dark mode at the edge of the band gap both below and above the generation threshold (see Note S5, Fig. 1S in Supp. Inf.) Thus, in the one-dimensional DFB lasers the dark modes at the edge of the band gap are generated while in the two-dimensional DFB lasers there are bright modes in the allowed band.

Manifestation of mode cooperation in the two-dimensional plasmonic DFB laser The existence of mode cooperation can be experimentally verified by measuring the radiation pattern, polarization of the far-field and the phase profile of the in-plane component of the intracavity electric field. To compare experimental and theoretical results, in our calculations, we use the same parameters of the system as in Ref. 20 We demonstrate that our theory correctly describes experimental data obtained in Ref. 20 Firstly, our simulations shows that the two-dimensional plasmonic DFB laser above 16


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threshold emits a donut-shaped beam with the opening angle ' 4 · 10−2 rad (see Fig. 7b). Such shape of the beam and opening angle coincide with the experimental measurements, see Figs. 2f and 3g in Ref.

20

As it has been noted in the Introduction, the direction of

the radiation depends on the ratio of the Bloch wavevector to the wavevector in free space. Namely, the angle θx,y at which the mode of the plasmonic DFB laser radiates is determined by the equations kBx,By = (2π/λ) sin θx,y . 19 If the modes at the edge of the band gap with zero Bloch wavevector are generated, then the radiation pattern represents a narrow cone around the normal direction. This takes place in the one-dimensional case, when the generation takes place at the edge of the band gap, see the green dashed curves in Fig. 7b. If the modes in an allowed band with nonzero Bloch vector are generated, then the radiation pattern has a maximum in a direction which is determined by mode cooperation. Such a situation arises above the generation threshold in the two-dimensional case, see Fig. 7b. Secondly, we reproduce the amplitude and phase profiles of the in-plane electric field (in our model phase mismatch between the jth and kth modes is determined according to √ exp (i (ϕk − ϕj )) = nkj / nkk njj , see Ref. 43 and Note S1 in Supp. Inf.). The amplitude and phase profiles of x-polarization of the in-plane electric field are shown in Fig. 8. It is seen that as well as in the experiment, 20 the in-plane electric field is radially polarized. The polarization of the in-plane electric field is determined by mode cooperation. To show this, first remember that the two-dimensional plasmonic DFB laser operates in the TM-modes of the periodic plasmonic structure (see Note S2 in Supp. Inf.). These modes have both out-of-plane, E⊥ , and in-plane, Ek , components of the electric field, and |E⊥ | Ek . Ek determines the radiation of the plasmonic DFB laser, whereas E⊥ , |E⊥ | Ek , determines the interaction of the modes with the active medium. We have demonstrated that the generation threshold is smaller when the electric field of the modes with the same frequencies oscillates in phase (see the section "Mode cooperation..."). In a two-dimensional plasmonic DFB laser, there is a frequency degeneracy of the modes having Bloch wavevectors with the same absolute values but with different directions. 17


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At the laser generation, the out-of-plane electric fields of such modes oscillate in phase. The complex amplitudes of in-plane electric field of each mode also oscillate in phase and the in-plane electric field is co-directed with Bloch wavevector of this mode. Thus, the in-plane electric field is radially polarized. Since the angle θx,y , at which the mode of the plasmonic DFB laser radiates, is determined by the Bloch wavevector kBx,By and the polarization of the in-plane electric field of this mode is in the direction of the Bloch wavevector, then the far field of the two-dimensional DFB plasmonic laser is also radially polarized. Note that the conclusion about the beam polarization is only valid when the laser operates in TM-modes. Thirdly, from the Fig. 8b, it is seen that phase of the in-plane electric field experienses a π jump in the center of the surface of the plasmonic DFB laser and monotonically increases to the boundaries of the pumped area. Such behavior is similar to the experimental one, see Fig. 3c in Ref. 20 The phase increases from the center to the boundaries of the pump region, because the laser operates in the modes with a nonzero Bloch wavevector. Indeed, the profile of the in-plane electric field in the active medium can be phenomenologically described as ∼ exp (ikBx x + gx) − exp (−ikBx x − gx), where kBx is the Bloch wavevector of the lasing mode (see Fig. 4b) and g is a gain of the active medium. 41 If the laser operates in the modes with zero Bloch wavevector, then the phase of the in-plane electric field experienses a π jump in the center of the surface of the plasmonic DFB laser and is constant on both sides of the center (see green dotted line in Fig. 8b). However, if the laser operates in the modes with nonzero Bloch wavevector, then the phase of the in-plane electric field also experiences a π jump in the center of the surface of the plasmonic DFB laser and increases to the boundaries of the pumped area (see red dashed line in Fig. 8b). This is the case that is arisen due to mode cooperation. Such qualitative arguments are confirmed by our numerical simulations (see blue solid line in Fig. 8b) and the experiment. 20 Such phase distribution of the in-plane electric field results in the fact that when half of the radiating surface of the two-dimensional plasmonic DFB laser is blocked, the radiation pattern has maximum at the nonzero angle, see Fig. 9. This result is consistent with the 18


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experimental measurements, see Fig. 3g in Ref. 20

Figure 8: x-polarization of the in-plane electric field: the amplitude (a) and phase (b) profiles. The blue solid line shows the phase profile obtained by our theory; the red dashed and green dotted lines show the phase profiles obtained by the phenomenological expression for Ek with nonzero and zero Bloch wavevectors. 1.0 Intensity Ha.u.L

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0.8 0.6 0.4 0.2 0.0 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 Θx HradL

Figure 9: Comparison of intensity profiles versus θx angle when the radiating surface is opened (blue solid line) and the half of the surface is blocked (red dashed line).

Discussion As it has been mentioned in Ref. 20 the radiation pattern of two-dimensional plasmonic DFB laser is broadened with respect to the prediction of standard Kogelnik theory. 37 In this theory, the laser modes, their eigenfrequencies, and generation thresholds for the onedimensional DFB laser are calculated by the coupled-mode theory. The radiation pattern 19


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of the DFB laser in such a model coincides with the radiation pattern of the mode with the lowest generation threshold, i.e. with the radiation pattern of the dark mode at the edge of the band gap in the considered case. Therefore, the radiation angle predicted by the standard theory does not agree with the experiment for the two-dimensional system, Fig. 2f in. 20 Moreover, the amplitude and phase distributions of the in-plane electric field observed in 20 essentially deviate from the predictions of the Kogelnik theory 20,37 (cf. Fig. 3a and 3c in Ref. 20 ). In Ref., 20 the authors have extended the standard model for DFB lasers by considering inhomogeneity of the gain and refractive index of the active medium in the pumped area. They demonstrated that the position dependent gain and refractive index with Gaussian profile can explain the experimental data. However, it remains unclear why this behavior is observed only in two-dimensional systems and does not occur in standard DFB lasers. In our paper, based on the self-consistent approach for considering nonlinear phenomena in complex plasmonic structures and active media, we demonstrate that the main reason of widening of the radiation pattern is mode cooperation which follows from non-orthogonality of the modes in the pumped area. This effect manifests only for the two-dimensional plasmonic DFB laser. It should be underlined that the finite size of a pump beam is a necessary condition for the observation of mode cooperation. In its turn, the finite size of the pump beam introduces spatial inhomogeneity in the system: the pump is zero outside the pumped area and the constant value inside one. This inhomogeneity of the pump beam should obviously result in position dependence of the gain and refractive index. Our numerical simulations show that mode cooperation leads to such distributions of refractive index (see Fig. 5S and Note S6 in Supp. Inf.) that are similar to suggested in the Ref. 20 Thus, mode cooperation clarifies the existence these refractive index distributions. In real experiments there is rather Gaussian profile of the pump beam than rectangular one. The calculations with Gaussian profile show that no qualitative differences appear in comparison with rectangular one (cf. Fig. 3S and 4S in the Supp. Inf.). 20


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Conclusions We have presented numerical simulations and analytical analysis of the dynamics of a twodimensional plasmonic DFB laser. We have found the dependence of photon number, the radiation pattern, and the laser output on the pump intensities. We have shown that in the two dimensional plasmonic DFB laser, when the pumped area is less than the whole size of the laser, the nonlinear interaction between modes of the plasmonic structure via the active medium leads to a new effect, namely, mode cooperation. This effect arises due to non-orthogonality of the laser modes in the active medium and is the opposite of the mode competition in usual lasers. Since the pump beam occupies only part of the surface of the plasmonic DFB laser, the laser modes are non-orthogonal within the pumped area of the active medium. The nonorthogonality of laser modes leads to the fact that the total intensity of the electric field in the pumped area depends not only on the amplitudes of the electric field of the modes, but also on the different phases between them. When the electric fields of the different modes oscillate in phase within the pumped area, the total intensity of the electric field increases. In this case, the rate of the stimulated emission also grows that leads to a decrease of the generation threshold for the modes oscillating in phase. Such behavior we named as mode cooperation. However, only modes with the same frequencies can oscillate in phase for a long time. For this reason, mode cooperation is observed for the modes with frequency degeneracy. In a two-dimensional DFB laser, there is a frequency degeneracy of laser modes with Bloch wavevectors with identical absolute values, but with different directions. The highest frequency degeneracy order is achieved by the modes with nonzero Bloch wavevectors that lie in the allowed band. As a result, mode cooperation is more pronounced for modes with nonzero Bloch wavevectors that lie in the allowed band. Due to mode cooperation, these modes have the lowest generation threshold and the highest amplitudes. Two observed effects follow from mode cooperation. The first one is a dramatic increase 21


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in the laser radiation by crossing the pump threshold. This is explained by the fact that the spontaneous emission is primarily emitted into the dark modes below the threshold while above the threshold, due to the mode cooperation, the stimulated emission is mainly radiated into the bright modes. The radiation output in such lasers has a clear threshold. The second effect is a widening of the radiation pattern. This is due to the dependence of the radiation pattern on the Bloch wavevector of the mode. The dark mode at the edge of a band gap has zero Bloch wavevector and does not radiate. The neighboring modes radiate within a narrow cone at angles close to the normal direction. Below the threshold, spontaneous emission dominates, the photon number is maximum in the dark mode at the edge of the band gap and slowly decreases when the mode wavelength shifts to the allowed band. Above the threshold, the lasing takes place in the bright modes which have nonzero Bloch wavevectors. In this case, above the threshold the radiation pattern has a maximum at a nonzero angle. Since the mode cooperation shifts the lasing from the dark mode to the bright modes, this widens the radiation pattern, see Fig. 7b. We indicated above that in 20 it was demonstrated that the measured radiation pattern of a plasmonic DFB laser is wider than the one which is expected from the standard distributed feedback (DFB) theory for one-dimensional systems with finite size. 37 Our numerical simulations show that this distinction arises owing to mode cooperation, something which does not appear in the one-dimensional case.

Acknowledgement Authors thank Yu. E. Lozovik for helpful discussions.

Authors’ contribution All authors contributed equally to this work.

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Supporting Information Available The following files are available free of charge. • Filename: supporting information Derivation of Eqs. 1; analysis of mode structure of the plasmonic DFB laser; calculation of the coupling constant between the field and the active medium; details of numerical simulations; comparison of one- and two-dimensional systems; detailed comparison with experiment. 20

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We show that the nonlinear interaction between the modes of a periodic plasmonic structure through the active medium leads to mode cooperation. This new effect results in broadening of the radiation pattern above the generation threshold, which has been observed in recent experiments.

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Mode cooperation in two-dimensional plasmonic distributed-feedback

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