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Low threshold nanorod-based plasmonic nanolasers with optimized cavity length
Optics and Laser Technology 111 (2019) 315–322
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Full length article
Low threshold nanorod-based plasmonic nanolasers with optimized cavity length Mohammad Hossein Motavas, Abbas Zarifkar
T
⁎
Department of Communications and Electronics, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
H I GH L IG H T S
two spasers based on one-dimensional arrays formed by three nanorods. • Introducing the lasing threshold and the normalized mode area by utilizing an extra gain medium. • Reducing the large wave vector of the lasing mode by using an extra metallic core. • Reducing • Obtaining more precise lasing threshold by accurate selection of the cavity length.
A R T I C LE I N FO
A B S T R A C T
Keywords: Coupled nanowires Cavity length optimization Nanorod-based plasmonic nanolasers Spaser Hybrid plasmonic waveguide
In this article two optically pumped nanorod-based plasmonic nanolasers which composed of two coupled metalinsulator-semiconductor (MIS) hybrid plasmonic waveguides are investigated. In the first structure, a common metallic nanorod is utilized to construct a semiconductor-insulator-metal-insulator-semiconductor (SIMIS) nanostructure while in the second one, the semiconductor part is shared and a metal-insulator-semiconductorinsulator-metal (MISIM) based plasmonic nanolaser is formed. Simulation results based on the finite element method (FEM) show that the SIMIS structure with nanorods’ radii of 40 nm and insulator layer thickness of more than 12.67 nm has lower threshold and simultaneously lower normalized mode area at the lasing wavelength of 490 nm compared to the previously reported MIS nanostructure with the same parameters. The simulation results for the second proposed structure show that the MISIM based spaser has a lower effective mode index and consequently lower wave number at the wavelength of 490 nm, compared to both SIMIS and MIS based nanocavities. This results in less challenge for coupling to on-chip waveguides. The cavity length of the presented nanorod-based spasers has been optimized by considering the lasing mode propagation distance as the nanocavity length which leads to a better light matter interaction enhancement.
1. Introduction Spaser or surface plasmon amplification by stimulated emission of radiation was suggested by Bergman and Stockman in 2003 [1]. Although spaser originally describes the feedback mechanism based on localized surface plasmons (LSPs), more recently it has also been applied to plasmonic nanolasers based on propagating surface plasmon polaritons (SPPs) [2]. One important advantage of SPP-based nanolasers compared to LSP-based ones, is the capability of using them as inline cavity or waveguide integrated plasmonic nanolasers due to better light coupling to on-chip waveguides. It is shown that utilization of a waveguide-integrated nanoscale plasmon source can realize high coupling efficiency of ∼60% and a small footprint of ∼0.06 μm2 which is suitable for dense integration [3]. The results of the recent researches
⁎
on waveguide-integrated configuration show experimental evidence of lasing emission and its coupling into the propagating modes of a plasmonic waveguide based on V-groove structure [4]. This makes possible on-chip routing of coherent and subdiffraction confined light at room temperature. The first successful laboratory sample of SPP-based spaser or plasmonic nanolaser was created by Oulton et al. at the lasing wavelength of 490 nm [5]. The structure was based on metal-insulatorsemiconductor (MIS) platform in which a semiconductor nanorod is placed on a metal film with an insulator gap. They had also designed a similar structure as a hybrid plasmonic waveguide [6]. The semiconductor nanorod acts as the gain medium and its end facets form a microscale Fabry–Perot (FP) cavity [7]. This nanorod-based MIS platform which supports propagation of the hybrid plasmonic modes, has been widely exploited in many researches during the past decade. In
Corresponding author. E-mail addresses: [email protected] (M.H. Motavas), [email protected] (A. Zarifkar).
https://doi.org/10.1016/j.optlastec.2018.10.010 Received 15 January 2018; Received in revised form 9 June 2018; Accepted 7 October 2018 0030-3992/ © 2018 Elsevier Ltd. All rights reserved.
Optics and Laser Technology 111 (2019) 315–322
M.H. Motavas, A. Zarifkar
and MgF2 as insulator shell is placed between two CdS nanorods as semiconductor gain materials on MgF2 substrate. So, this plasmonic nanolaser has a waveguide structure as semiconductor-insulator-metalinsulator-semiconductor (SIMIS). Such structure can be fabricated by using self-assembly techniques [20]. The distance between two semiconductor nanorods and the metal core is always the same and defined as h. r1 and r2 are the radii of Ag and CdS nanorods, respectively. It is assumed that r2 = r1 + h. The lengths of three nanorods are equal and defined as L. Since Ag is used as metal, MgF2 as insulator, and CdS as semiconductor gain material, the lasing wavelength of the plasmonic nanolaser is 490 nm [5]. At this wavelength, the refractive indices of Ag, MgF2 and CdS are 0.05 + 3.039i, 1.4 and 2.4, respectively. As Oulton et al. mentioned in [5], the mode area and the emission rates depend on the nanorod diameter. Due to the dependence of exciton dynamics to the size of nanorods [25], it is more precise to use a limited range for the nanorods radii in the simulations. So, according to the assumptions and results of references [5,20], we have selected three different values of 40, 50 and 60 nm for the nanorods radii. In the following, we first obtain the electric field distribution at the device cross section and then calculate the propagation characteristics. According to the fact that our SIMIS structure consists of two coupled MIS hybrid plasmonic waveguides sharing a common metallic nanorod, existence of a symmetric and an asymmetric mode is logically expected. Our simulation results confirm this issue as illustrated in Fig. 2. Both modes consist of the surface plasmon (SP) modes at the metallic interface of the core-shell structure coupled to the guided modes of the semiconductor nanorods. Our studies reveal that in such symmetric structures the symmetric hybrid plasmonic mode has much lower propagation loss than the asymmetric one. For example, for h = 5 nm, the symmetric mode has a propagation loss about five times lower than the asymmetric mode. So, we only investigate the symmetric hybrid mode as the main propagating mode in the presented structures. Fig. 2 (a) and (b) shows the 2D Ex field distributions of the symmetric and asymmetric hybrid plasmonic modes, respectively, for r2 = 50 nm and h = 10 nm. Fig. 2 (c) and (d) shows the 1D Ex field distributions of the symmetric and asymmetric modes along the horizontal dash-dotted lines in Fig. 2 (a) and (b), respectively. The standing-wave pattern of the electric field which is formed by the SIMIS based FP cavity is depicted in Fig. 2(e) and (f) for r2 = 50 nm and h = 10 nm in the y = 50 nm plane and the waveguide/air interface is located at z ≈ 0.68 μm. The absolute normalized electric field of the symmetric mode which demonstrates the confinement of the field, is presented in Fig. 3 for h = 5 nm and r2 = 50 nm. As can be seen in Fig. 3 (a), the hybrid mode is mainly localized within the insulator spacer region. Fig. 3 (b) and (c) show the distribution of the x and y components of the electric field
2010, Zhu investigated the modal properties of this MIS structure at the same lasing wavelength [8]. Similar structures with different materials [9] and also various nanorod structures such as core–shell [10,11], multi-quantum-well (MQW) [12,13], hexagonal [14–17], and triangular [18] have also been presented. The idea of using more than one nanorod in a MIS structure was proposed by Bian et al. in 2012 [19]. They designed a long range hybrid plasmonic waveguide based on coupled nanowires with simultaneously subwavelength mode confinement (∼λ2/460 to λ2/35) and long range propagation length (∼33 to 1260 μm) at the wavelength of 1550 nm. Then, they introduced a low threshold plasmonic nanolaser based on two coupled nanorods by using a CdS nanorod as semiconductor gain material alongside a core-shell structure with Ag as metallic core and MgF2 as insulator shell [20]. Their idea and results have inspired many subsequent researches in recent years [21–23]. It is specified that the coupling of long range SPP and dielectric nanorod modes can provide a mode confinement level similar to a hybrid plasmonic mode with one order of magnitude longer propagation length [24]. In this work, we propose and analyze two optically pumped nanorod-based spasers at the lasing wavelength of 490 nm with semiconductor-insulator-metal-insulator-semiconductor (SIMIS) and metal-insulator-semiconductor-insulator-metal (MISIM) nanostructures. These structures can support propagation of LRHPP modes at the telecommunication wavelength [19]. Although there is a compromise between the normalized mode area and the lasing threshold, by utilizing the SIMIS structure, we can achieve a lower normalized mode area and at the same time, a lower threshold compared to MIS nanostructure. On the other hand, the MISIM based nanolaser can realize lower propagation loss and lower effective index compared to both SIMIS and MIS based nanolasers. In the similar simulations on nanorod based plasmonic nanolasers, a constant value is commonly used for the nanocavity length in the plasmonic nanolaser structure [8,13,20,21]. According to the relationship between the propagation length of the hybrid plasmonic mode and the geometric parameters values, it’s not absolutely true to assume a constant value for the cavity length. So, we modify this assumption and accordingly the calculation of the lasing threshold in the presented structures. The remainder of this paper is organized as follows. In Sections 2 and 3, the SIMIS- and MISIM- based nanolasers are introduced and analyzed, respectively. In Section 4, the necessity of optimization of the nanocavity length is discussed and the lasing threshold of two structures are simulated again. Finally, the paper is concluded in Section 5. 2. The SIMIS-based plasmonic nanolaser The three dimensional schematic of the first proposed spaser and its cross-sectional view are shown in Fig. 1 (a) and (b), respectively. As shown in these figures, a core-shell structure with Ag as metallic core
Fig. 1. (a) Three dimensional schematic of the SIMIS based spaser (b) Cross-sectional view of the SIMIS structure. 316
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Fig. 2. (a),(b) 2D electric field distributions of the asymmetric and symmetric hybrid plasmonic modes at xy plane. (c),(d) 1D EX profiles of the hybrid modes along the blue dash-dotted lines (y = 50 nm) for asymmetric and symmetric hybrid plasmonic modes. (e),(f) The standing-wave pattern of the electric field in the xz plane for the asymmetric and symmetric modes. The facet at z ≈ 0.68 μm is the waveguide/air interface. In all figures r2 = 50 nm and h = 10 nm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
this area. In fact, the optical coupling between the SP mode and the fundamental mode of the CdS nanorod provides sufficient optical gain which can compensate the intrinsic losses of the nanocavity. This loss compensation can enhance both the spasing mechanism and the
along the horizontal and vertical dashed lines in Fig. 3 (a), respectively. Fig. 3 (b) illustrates the lateral coupling between the metallic core and the semiconductor nanorods causes significant field enhancement in the gap region and confines the x component of the electric field in
Fig. 3. The normalized electric field of the propagating hybrid plasmonic mode in the SIMIS structure. (a) the cross sectional view, (b) the x component along the y = 50 nm line, and (c) the y component along the x = 50 nm line for r2 = 50 nm and h = 5 nm. 317
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Fig. 4. (a) The mode effective index, (b) propagation loss, (c) confinement factor, and (d) normalized mode area of the hybrid plasmonic mode in the SIMIS structure.
weakens by increasing h and is dominant at relatively small distances. The second one is the confinement of the electric field around the metallic core surface. When h increases, the latter phenomenon is dominant since the core becomes relatively small compared to h. On the other hand, when h increases, the overlap between the hybrid mode and the gain region decreases which in turn causes that the confinement factor be reduced as shown in Fig. 4 (c). The confinement factor reduction shows the decrease of spatial confinement of the hybrid plasmonic mode. This will lead to increment of the effective mode area and consequently the normalized mode area as shown in Fig. 4 (d). However, for larger h values, the electric field is more confined around the metallic core which causes that the normalized mode area to be decreased. The lasing threshold, gth, in the SIMIS structure is related to n eff , α eff , and Γ through [8]: Fig. 5. The lasing threshold of the SIMIS-based spaser as a function of h for L = 30 μm and three different values of r2.
resonance of the FP cavity which is formed by the end facets of the nanorods. The continuation of this process leads to a real surface palsmon amplification and spasing action. In Fig. 3 (c) an abrupt jump at y = 0 is seen. This behavior can be explained as follows. The interface of the MgF2 substrate (n1 = 1.4) and air (n2 = 1) is at y = 0. According to the continuity of the normal → components of the electric flux density (D (x,y)) at the common border → → of air and substrate, we have |D1 (y )| = |D2 (y )| which leads to → → 2 2 ε0 (n1) |E1 (y )|=ε0 (n2) |E2 (y )|. Based on this boundary condition, it is → → deduced that |E2 (y )|/|E1 (y )|=(n1/ n2)2 = 1.96 which causes the mentioned abrupt increase in the y component of the electric field. The calculated hybrid plasmonic mode characteristics including the mode effective index (n eff ), the effective propagation loss (α eff ), the confinement factor (Γ ), and the normalized mode area (A eff /A 0 ) are shown in Fig. 4 (a)–(d), respectively, for three different values of the CdS nanorod radius, r2. The mode effective index and the effective propagation loss are respectively defined as the imaginary and real parts of the hybrid waveguide propagation constant divided by the free space wavevector [8]. A 0 is the free space diffraction-limited mode area and defined as:
A 0 = λ2 /4
(∬ |E|2 dxdy) ∬ |E|4 dxdy
+ ln
( )/L) 1 R
Γ(n eff /nwire )
(3)
where R = (neff − 1)/(neff + 1) is the end facet reflectivity and L is the nanorods length. Fig. 5 depicts the lasing threshold as a function of h for L = 30 μm and three different values of r2. By increasing h, the propagation loss increases and the confinement factor decreases which according to Eq. (3), lead to increment of the lasing threshold. According to the simulation results, by selecting proper geometric parameters (as an example, r2 = 40 nm and h = 7.5 nm), the lasing threshold of the SIMIS-based nanolaser can reach to about 3.28 μm−1, which is relatively lower than the lasing threshold of the MIS-based plasmonic nanolaser in Ref. [20]. For specific dimensions (e.g. r2 = 40 nm and h = 17.5 nm), the SIMIS-based nanolaser has a lower normalized mode area as 0.1, compared to the MIS-based nanolaser. Generally, the SIMIS spaser for r2 = 40 nm and h ≥ 12.67 nm has lower threshold and at the same time lower normalized mode area compared to the MIS spaser. 3. The MISIM-based plasmonic nanolaser The three dimensional schematic of the second proposed plasmonic nanolaser and its cross-sectional view are shown in Fig. 6 (a) and (b), respectively. As shown in this figure, a CdS nanorod is placed between two core–shell structures with Ag as metallic cores and MgF2 as insulator shells on MgF2 substrate. So, this spaser has a waveguide structure as metal-insulator-semiconductor-insulator-metal (MISIM). In this paper, we have investigated the optical properties of this structure and obtain its lasing threshold. As shown in Fig. 6, r1 and r2 are the radii of Ag and CdS nanorods, respectively. The assumption of r2 = r1 + h is still valid and again the lengths of three nanorods are equal and defined as L. This plasmonic nanolaser is investigated at the same lasing wavelength, 490 nm. The symmetric hybrid plasmonic mode of the MISIM nanostructure
(1)
The effective mode area is calculated by:
A eff =
(k α
0 eff
g th =
2
(2)
The confinement factor is defined as the ratio of the electric energy in the CdS nanorod to the total electric energy of the mode [20]. As seen in Fig. 4 (a) and (b), the mode effective index and the effective propagation loss show minimum values at a given h. This behavior can be explained as follows. There are two phenomena that affect n eff and α eff . The first one is the optical coupling between two nanorods which 318
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Fig. 6. (a) Three dimensional schematic and (b) Cross-sectional view of the MISIM-based spaser.
1
x=50nm
y=50nm
100nm
0 (a)
(b)
(c)
Fig. 7. The normalized electric field of the propagating hybrid plasmonic mode in the MISIM structure (a) the cross sectional view, (b) the x component along the y = 50 nm line, and (c) the y component along the x = 50 nm line for r2 = 50 nm and h = 5 nm.
k = nω/c
is shown in Fig. 7. Fig. 7 (b) and (c) shows the distribution of the x and y components of the electric field along the horizontal and vertical dashed lines in Fig. 7 (a), respectively. Fig. 8 (a)–(d) shows the mode effective index, effective propagation loss, confinement factor and the normalized mode area of the hybrid plasmonic mode in MISIM-based nanocavity. Compared to the characteristics of the SIMIS-based nanostructure, the mode effective index and the propagation loss are relatively reduced for the metallic nanorod with the radius of 40 nm and the insulator gap width of below 10.7 nm. Given the fact that the large wave vector of the subdiffraction-limited mode is one of the main reasons of the inefficient coupling of SPP emission to on-chip waveguides [3], reducing the wave number k = 2π / λ can relatively solve this problem. The relationship between the refractive index of a medium, n, and the wave number k is as follows:
(4)
where c is the velocity of light in vacuum and ω is the angular frequency. So, at a given angular frequency, a decrease in the mode effective index is directly related to reduction of the wave number. Accordingly, the lower effective index of the hybrid plasmonic mode in the MISIM-based nanocavity corresponds to a lower wave number and consequently better coupling efficiency in the MISIM structure compared to SIMIS and MIS nanostructures. By comparing Figs. 8 (d) and 4 (d), we conclude that the hybrid plasmonic mode of the MISIM-based nanocavity has larger normalized mode area compared to that of the SIMIS-based plasmonic cavities. It is noticeable that the difference between the normalized mode areas of the MISIM and SIMIS based nanocavities declines and can be neglected for larger radii and lower insulator gap. For example, for r2 = 60 nm and h = 2.5 nm this difference is about 0.002.
Fig. 8. (a) The mode effective index, (b) propagation loss, (c) confinement factor, and (d) normalized mode area of the hybrid plasmonic mode in the MISIM structure. 319
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Fig. 13. The Purcell factor (Fp) of the (a) SIMIS, (b) MISIM based spaser for L = LP.
Fig. 9. The Lasing Threshold of MISIM based spaser as a function of h for L = 30 μm and three different values of r2.
Fig. 14. The Purcell factor (Fp) of the (a) SIMIS, (b) MISIM based spaser for L = 30 µm. Fig. 10. The propagation length of the hybrid plasmonic mode in the (a) SIMIS (b) MISIM based nanocavities.
By obtaining the mode effective index, the effective propagation loss and the confinement factor, the lasing threshold of the MISIM-based spaser is calculated as a function of h for L = 30 μm and three different values of r2 and depicted in Fig. 9. This figure shows that the MISIMbased plasmonic nanolaser have relatively higher lasing threshold compared to SIMIS and MIS based spasers. 4. Optimization of the nanocavity length In this section, we investigate the propagation lengths of the hybrid plasmonic modes of the MISIM and SIMIS based nanocavities. The propagation length of the hybrid plasmonic mode relates to the geometric parameters of the plasmonic nanolasers. The propagation length (LP ) is given by [19]:
Fig. 11. The optimized lasing threshold of (a) SIMIS (b) MISIM based spaser as a function of h for L = LP and three different values of r2.
LP =
λ 1 = 4πα eff 2α eff k
(5)
where λ is the wavelength of the propagating mode. The propagation length is inversely related to the propagation loss which in turn depends on the structural characteristics of the nanocavity such as the insulator gap width (h). According to the simulation results, it is observed that the propagation length changes by altering the value of h as depicted in Fig. 10 (a) and (b). Although such structures can support the propagation of LRHPP modes at the wavelength of 1550 nm with a propagation length more than 1 mm [19], here at the wavelength of 490 nm, the maximum propagation length is 1.73 μm and 1.62 μm for the SIMIS and MISIM structures, respectively, as shown in Fig. 10. Because of the different propagation lengths of the hybrid plasmonic mode for various h values, it’s not accurate to assume a constant value for the cavity length when we are going to sweep the h parameter. In fact, the semiconductor nanorod, as a gain medium, provides enough gain to compensate the overall loss and ensure the lasing action through the spectral and spatial overlap with the SP mode. The transferred resonance energy from the gain medium excites the SP mode at first and then amplifies it. Without such a gain medium, the lasing
Fig. 12. The Lasing threshold of MIS-based spaser as a function of h for L = LP and three different values of r2.
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consisting of two coupled MIS structures. The first one is the SIMIS structure with a common metallic core and the second one is the MISIM with a common gain medium. The presented SIMIS-based spaser shows better threshold and mode effective area in comparison with both MIS and MISIM based nanolasers for suitable choice of the nanorod radius and insulator gap width. Modal analysis of the MISIM structure shows lower mode effective index which results in more efficient coupling to the on-chip waveguides due to lower wave number although this benefit is achieved at the expense of larger normalized mode area and lasing threshold. Also, we optimized the design of the nanocavity by considering the nonocavity length equal to the modal propagation distance in order to consider the lasing action more precisely. This optimization led to a better enhancement of the light matter interaction which is seen as the increase of the Purcell factor.
mode cannot propagate over its propagation length. So, the FP cavity length is limited by the lasing mode propagation length and cannot be adopted larger than the mode’s energy attenuation length or LP. In fact, without a loss compensation mechanism, if we design a FP cavity with a length much larger than the propagation length of the FP lasing mode, a correct feedback mechanism and the standing wave pattern formation cannot be accomplished. One the other hand, presence of the gain medium cannot break the cavity length limitation. This means, when the propagation distance of the lasing mode becomes lower and lower regardless of its reason, we will need more and more pumping energy to achieve enough gain for compensating the intrinsic losses and consequently realize the desirable effective spasing action. In other word, a long plasmonic FP cavity encounters very high metal loss that causes larger absorption of the propagating hybrid modes in travelling between the end facets. Under such condition and without expending too much energy to compensate the losses, the hybrid lasing mode cannot be subjected to the cavity feedback that is necessary for spasing. In this situation we cannot enlarge the FP cavity length as much as we need in order to minimize the mirror loss. It is a fact that decreasing the cavity length is limited by increase of the mirror loss, although it may be relatively compensated by a high-reflectivity coating or an appropriate feedback structure under a proper phase matching condition [26]. So, it’s better to choose the device length L around the energy attenuation length (L ≈ LP) to minimize the high metal loss. Thus, to obtain the optimum spasing action, an optimized value for the cavity length is needed which can be a multiple or fraction of the modal propagation length. In this study, we approximately set the cavity length equal to the propagation length of the hybrid plasmonic modes of the presented nanostructures to focus on the effect of the geometrical changes on the spasing action. With the assumption of L = LP in Eq. (3), we have:
1 + 2ln g th = k 0α eff (
( ))
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1 R
Γ(n eff /nwire )
(6)
Using this relation in the simulations, the lasing threshold of the MISIM and SIMIS based spasers is obtained as shown in Fig. 11 (a) and (b). As can be seen in this figure, the behavior of the threshold curves of both structures have not changed, but the values of the lasing threshold increase and become closer to the reality. We have simulated the proposed MIS structure in Ref. [20] and calculated its threshold using the assumption of L = LP as demonstrated in Fig. 12. By comparing Figs. 12 and 11 (a) and (b) it becomes clear that the SIMIS spaser has a lower threshold for r2 = 40 nm compared to both MIS and MISIM structures. For r2 = 50 nm and r2 = 60 nm, the SIMIS spaser has a lower threshold only for h ≥ 8.63 nm and h ≥ 12.03 nm, respectively. Generally, the MISIM structure has relatively larger threshold compared to both MIS and SIMIS. Thus, we will need more pumping energy to realize the spasing action in this structure. For evaluation of the spasers operation, we can obtain the Purcell factor, which denotes the light matter interaction enhancement, through the relation FP = 3Q/4π 2Vn where Vn is the normalized mode volume [13,27] and Q is the plasmonic cavity’s quality factor [26]. It should be noticed that the normalized mode area that we have reported in the manuscript, is inversely related to the Purcell factor through the relation FP A0 /2πAeff [28]. Figs. 13 and 14 shows the calculated Purcell factor for L = LP and L = 30 μm, respectively. As can be seen in Figs. 13 and 14, by assuming L = LP, the Purcell factor is increased compared to the case of L = 30 µm. This shows that the participating mechanisms of spasing act more effectively, the gain medium can amplify the SPPs much better, and so the spasing can be realized easier. 5. Conclusion In this paper, we proposed and investigated two SPP-based spasers 321
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Opt. Express 18 (2010) 14. [27] E.M. Purcell, Spontaneous emission probabilities at radio frequencies, Phys. Rev. 69 (1946) 681. [28] R.F. Oulton, G. Bartal, D.F.P. Pile, X. Zhang, Confinement and propagation characteristics of subwavelength plasmonic modes, New J. Phys. 10 (2008) 105018.
(2014) 6870. [25] X. Liu, Q. Zhang, G. Xing, Q. Xiong, T.C. Sum, Size-dependent exciton recombination dynamics in single CdS nanowires beyond the quantum confinement regime, J. Phys. Chem. C 117 (2013) 20. [26] S.W. Chang, T.R. Lin, S.L. Chuang, Theory of plasmonic Fabry-Perot nanolasers,
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Full length article
Low threshold nanorod-based plasmonic nanolasers with optimized cavity length Mohammad Hossein Motavas, Abbas Zarifkar
T
⁎
Department of Communications and Electronics, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
H I GH L IG H T S
two spasers based on one-dimensional arrays formed by three nanorods. • Introducing the lasing threshold and the normalized mode area by utilizing an extra gain medium. • Reducing the large wave vector of the lasing mode by using an extra metallic core. • Reducing • Obtaining more precise lasing threshold by accurate selection of the cavity length.
A R T I C LE I N FO
A B S T R A C T
Keywords: Coupled nanowires Cavity length optimization Nanorod-based plasmonic nanolasers Spaser Hybrid plasmonic waveguide
In this article two optically pumped nanorod-based plasmonic nanolasers which composed of two coupled metalinsulator-semiconductor (MIS) hybrid plasmonic waveguides are investigated. In the first structure, a common metallic nanorod is utilized to construct a semiconductor-insulator-metal-insulator-semiconductor (SIMIS) nanostructure while in the second one, the semiconductor part is shared and a metal-insulator-semiconductorinsulator-metal (MISIM) based plasmonic nanolaser is formed. Simulation results based on the finite element method (FEM) show that the SIMIS structure with nanorods’ radii of 40 nm and insulator layer thickness of more than 12.67 nm has lower threshold and simultaneously lower normalized mode area at the lasing wavelength of 490 nm compared to the previously reported MIS nanostructure with the same parameters. The simulation results for the second proposed structure show that the MISIM based spaser has a lower effective mode index and consequently lower wave number at the wavelength of 490 nm, compared to both SIMIS and MIS based nanocavities. This results in less challenge for coupling to on-chip waveguides. The cavity length of the presented nanorod-based spasers has been optimized by considering the lasing mode propagation distance as the nanocavity length which leads to a better light matter interaction enhancement.
1. Introduction Spaser or surface plasmon amplification by stimulated emission of radiation was suggested by Bergman and Stockman in 2003 [1]. Although spaser originally describes the feedback mechanism based on localized surface plasmons (LSPs), more recently it has also been applied to plasmonic nanolasers based on propagating surface plasmon polaritons (SPPs) [2]. One important advantage of SPP-based nanolasers compared to LSP-based ones, is the capability of using them as inline cavity or waveguide integrated plasmonic nanolasers due to better light coupling to on-chip waveguides. It is shown that utilization of a waveguide-integrated nanoscale plasmon source can realize high coupling efficiency of ∼60% and a small footprint of ∼0.06 μm2 which is suitable for dense integration [3]. The results of the recent researches
⁎
on waveguide-integrated configuration show experimental evidence of lasing emission and its coupling into the propagating modes of a plasmonic waveguide based on V-groove structure [4]. This makes possible on-chip routing of coherent and subdiffraction confined light at room temperature. The first successful laboratory sample of SPP-based spaser or plasmonic nanolaser was created by Oulton et al. at the lasing wavelength of 490 nm [5]. The structure was based on metal-insulatorsemiconductor (MIS) platform in which a semiconductor nanorod is placed on a metal film with an insulator gap. They had also designed a similar structure as a hybrid plasmonic waveguide [6]. The semiconductor nanorod acts as the gain medium and its end facets form a microscale Fabry–Perot (FP) cavity [7]. This nanorod-based MIS platform which supports propagation of the hybrid plasmonic modes, has been widely exploited in many researches during the past decade. In
Corresponding author. E-mail addresses: [email protected] (M.H. Motavas), [email protected] (A. Zarifkar).
https://doi.org/10.1016/j.optlastec.2018.10.010 Received 15 January 2018; Received in revised form 9 June 2018; Accepted 7 October 2018 0030-3992/ © 2018 Elsevier Ltd. All rights reserved.
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and MgF2 as insulator shell is placed between two CdS nanorods as semiconductor gain materials on MgF2 substrate. So, this plasmonic nanolaser has a waveguide structure as semiconductor-insulator-metalinsulator-semiconductor (SIMIS). Such structure can be fabricated by using self-assembly techniques [20]. The distance between two semiconductor nanorods and the metal core is always the same and defined as h. r1 and r2 are the radii of Ag and CdS nanorods, respectively. It is assumed that r2 = r1 + h. The lengths of three nanorods are equal and defined as L. Since Ag is used as metal, MgF2 as insulator, and CdS as semiconductor gain material, the lasing wavelength of the plasmonic nanolaser is 490 nm [5]. At this wavelength, the refractive indices of Ag, MgF2 and CdS are 0.05 + 3.039i, 1.4 and 2.4, respectively. As Oulton et al. mentioned in [5], the mode area and the emission rates depend on the nanorod diameter. Due to the dependence of exciton dynamics to the size of nanorods [25], it is more precise to use a limited range for the nanorods radii in the simulations. So, according to the assumptions and results of references [5,20], we have selected three different values of 40, 50 and 60 nm for the nanorods radii. In the following, we first obtain the electric field distribution at the device cross section and then calculate the propagation characteristics. According to the fact that our SIMIS structure consists of two coupled MIS hybrid plasmonic waveguides sharing a common metallic nanorod, existence of a symmetric and an asymmetric mode is logically expected. Our simulation results confirm this issue as illustrated in Fig. 2. Both modes consist of the surface plasmon (SP) modes at the metallic interface of the core-shell structure coupled to the guided modes of the semiconductor nanorods. Our studies reveal that in such symmetric structures the symmetric hybrid plasmonic mode has much lower propagation loss than the asymmetric one. For example, for h = 5 nm, the symmetric mode has a propagation loss about five times lower than the asymmetric mode. So, we only investigate the symmetric hybrid mode as the main propagating mode in the presented structures. Fig. 2 (a) and (b) shows the 2D Ex field distributions of the symmetric and asymmetric hybrid plasmonic modes, respectively, for r2 = 50 nm and h = 10 nm. Fig. 2 (c) and (d) shows the 1D Ex field distributions of the symmetric and asymmetric modes along the horizontal dash-dotted lines in Fig. 2 (a) and (b), respectively. The standing-wave pattern of the electric field which is formed by the SIMIS based FP cavity is depicted in Fig. 2(e) and (f) for r2 = 50 nm and h = 10 nm in the y = 50 nm plane and the waveguide/air interface is located at z ≈ 0.68 μm. The absolute normalized electric field of the symmetric mode which demonstrates the confinement of the field, is presented in Fig. 3 for h = 5 nm and r2 = 50 nm. As can be seen in Fig. 3 (a), the hybrid mode is mainly localized within the insulator spacer region. Fig. 3 (b) and (c) show the distribution of the x and y components of the electric field
2010, Zhu investigated the modal properties of this MIS structure at the same lasing wavelength [8]. Similar structures with different materials [9] and also various nanorod structures such as core–shell [10,11], multi-quantum-well (MQW) [12,13], hexagonal [14–17], and triangular [18] have also been presented. The idea of using more than one nanorod in a MIS structure was proposed by Bian et al. in 2012 [19]. They designed a long range hybrid plasmonic waveguide based on coupled nanowires with simultaneously subwavelength mode confinement (∼λ2/460 to λ2/35) and long range propagation length (∼33 to 1260 μm) at the wavelength of 1550 nm. Then, they introduced a low threshold plasmonic nanolaser based on two coupled nanorods by using a CdS nanorod as semiconductor gain material alongside a core-shell structure with Ag as metallic core and MgF2 as insulator shell [20]. Their idea and results have inspired many subsequent researches in recent years [21–23]. It is specified that the coupling of long range SPP and dielectric nanorod modes can provide a mode confinement level similar to a hybrid plasmonic mode with one order of magnitude longer propagation length [24]. In this work, we propose and analyze two optically pumped nanorod-based spasers at the lasing wavelength of 490 nm with semiconductor-insulator-metal-insulator-semiconductor (SIMIS) and metal-insulator-semiconductor-insulator-metal (MISIM) nanostructures. These structures can support propagation of LRHPP modes at the telecommunication wavelength [19]. Although there is a compromise between the normalized mode area and the lasing threshold, by utilizing the SIMIS structure, we can achieve a lower normalized mode area and at the same time, a lower threshold compared to MIS nanostructure. On the other hand, the MISIM based nanolaser can realize lower propagation loss and lower effective index compared to both SIMIS and MIS based nanolasers. In the similar simulations on nanorod based plasmonic nanolasers, a constant value is commonly used for the nanocavity length in the plasmonic nanolaser structure [8,13,20,21]. According to the relationship between the propagation length of the hybrid plasmonic mode and the geometric parameters values, it’s not absolutely true to assume a constant value for the cavity length. So, we modify this assumption and accordingly the calculation of the lasing threshold in the presented structures. The remainder of this paper is organized as follows. In Sections 2 and 3, the SIMIS- and MISIM- based nanolasers are introduced and analyzed, respectively. In Section 4, the necessity of optimization of the nanocavity length is discussed and the lasing threshold of two structures are simulated again. Finally, the paper is concluded in Section 5. 2. The SIMIS-based plasmonic nanolaser The three dimensional schematic of the first proposed spaser and its cross-sectional view are shown in Fig. 1 (a) and (b), respectively. As shown in these figures, a core-shell structure with Ag as metallic core
Fig. 1. (a) Three dimensional schematic of the SIMIS based spaser (b) Cross-sectional view of the SIMIS structure. 316
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Fig. 2. (a),(b) 2D electric field distributions of the asymmetric and symmetric hybrid plasmonic modes at xy plane. (c),(d) 1D EX profiles of the hybrid modes along the blue dash-dotted lines (y = 50 nm) for asymmetric and symmetric hybrid plasmonic modes. (e),(f) The standing-wave pattern of the electric field in the xz plane for the asymmetric and symmetric modes. The facet at z ≈ 0.68 μm is the waveguide/air interface. In all figures r2 = 50 nm and h = 10 nm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
this area. In fact, the optical coupling between the SP mode and the fundamental mode of the CdS nanorod provides sufficient optical gain which can compensate the intrinsic losses of the nanocavity. This loss compensation can enhance both the spasing mechanism and the
along the horizontal and vertical dashed lines in Fig. 3 (a), respectively. Fig. 3 (b) illustrates the lateral coupling between the metallic core and the semiconductor nanorods causes significant field enhancement in the gap region and confines the x component of the electric field in
Fig. 3. The normalized electric field of the propagating hybrid plasmonic mode in the SIMIS structure. (a) the cross sectional view, (b) the x component along the y = 50 nm line, and (c) the y component along the x = 50 nm line for r2 = 50 nm and h = 5 nm. 317
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Fig. 4. (a) The mode effective index, (b) propagation loss, (c) confinement factor, and (d) normalized mode area of the hybrid plasmonic mode in the SIMIS structure.
weakens by increasing h and is dominant at relatively small distances. The second one is the confinement of the electric field around the metallic core surface. When h increases, the latter phenomenon is dominant since the core becomes relatively small compared to h. On the other hand, when h increases, the overlap between the hybrid mode and the gain region decreases which in turn causes that the confinement factor be reduced as shown in Fig. 4 (c). The confinement factor reduction shows the decrease of spatial confinement of the hybrid plasmonic mode. This will lead to increment of the effective mode area and consequently the normalized mode area as shown in Fig. 4 (d). However, for larger h values, the electric field is more confined around the metallic core which causes that the normalized mode area to be decreased. The lasing threshold, gth, in the SIMIS structure is related to n eff , α eff , and Γ through [8]: Fig. 5. The lasing threshold of the SIMIS-based spaser as a function of h for L = 30 μm and three different values of r2.
resonance of the FP cavity which is formed by the end facets of the nanorods. The continuation of this process leads to a real surface palsmon amplification and spasing action. In Fig. 3 (c) an abrupt jump at y = 0 is seen. This behavior can be explained as follows. The interface of the MgF2 substrate (n1 = 1.4) and air (n2 = 1) is at y = 0. According to the continuity of the normal → components of the electric flux density (D (x,y)) at the common border → → of air and substrate, we have |D1 (y )| = |D2 (y )| which leads to → → 2 2 ε0 (n1) |E1 (y )|=ε0 (n2) |E2 (y )|. Based on this boundary condition, it is → → deduced that |E2 (y )|/|E1 (y )|=(n1/ n2)2 = 1.96 which causes the mentioned abrupt increase in the y component of the electric field. The calculated hybrid plasmonic mode characteristics including the mode effective index (n eff ), the effective propagation loss (α eff ), the confinement factor (Γ ), and the normalized mode area (A eff /A 0 ) are shown in Fig. 4 (a)–(d), respectively, for three different values of the CdS nanorod radius, r2. The mode effective index and the effective propagation loss are respectively defined as the imaginary and real parts of the hybrid waveguide propagation constant divided by the free space wavevector [8]. A 0 is the free space diffraction-limited mode area and defined as:
A 0 = λ2 /4
(∬ |E|2 dxdy) ∬ |E|4 dxdy
+ ln
( )/L) 1 R
Γ(n eff /nwire )
(3)
where R = (neff − 1)/(neff + 1) is the end facet reflectivity and L is the nanorods length. Fig. 5 depicts the lasing threshold as a function of h for L = 30 μm and three different values of r2. By increasing h, the propagation loss increases and the confinement factor decreases which according to Eq. (3), lead to increment of the lasing threshold. According to the simulation results, by selecting proper geometric parameters (as an example, r2 = 40 nm and h = 7.5 nm), the lasing threshold of the SIMIS-based nanolaser can reach to about 3.28 μm−1, which is relatively lower than the lasing threshold of the MIS-based plasmonic nanolaser in Ref. [20]. For specific dimensions (e.g. r2 = 40 nm and h = 17.5 nm), the SIMIS-based nanolaser has a lower normalized mode area as 0.1, compared to the MIS-based nanolaser. Generally, the SIMIS spaser for r2 = 40 nm and h ≥ 12.67 nm has lower threshold and at the same time lower normalized mode area compared to the MIS spaser. 3. The MISIM-based plasmonic nanolaser The three dimensional schematic of the second proposed plasmonic nanolaser and its cross-sectional view are shown in Fig. 6 (a) and (b), respectively. As shown in this figure, a CdS nanorod is placed between two core–shell structures with Ag as metallic cores and MgF2 as insulator shells on MgF2 substrate. So, this spaser has a waveguide structure as metal-insulator-semiconductor-insulator-metal (MISIM). In this paper, we have investigated the optical properties of this structure and obtain its lasing threshold. As shown in Fig. 6, r1 and r2 are the radii of Ag and CdS nanorods, respectively. The assumption of r2 = r1 + h is still valid and again the lengths of three nanorods are equal and defined as L. This plasmonic nanolaser is investigated at the same lasing wavelength, 490 nm. The symmetric hybrid plasmonic mode of the MISIM nanostructure
(1)
The effective mode area is calculated by:
A eff =
(k α
0 eff
g th =
2
(2)
The confinement factor is defined as the ratio of the electric energy in the CdS nanorod to the total electric energy of the mode [20]. As seen in Fig. 4 (a) and (b), the mode effective index and the effective propagation loss show minimum values at a given h. This behavior can be explained as follows. There are two phenomena that affect n eff and α eff . The first one is the optical coupling between two nanorods which 318
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Fig. 6. (a) Three dimensional schematic and (b) Cross-sectional view of the MISIM-based spaser.
1
x=50nm
y=50nm
100nm
0 (a)
(b)
(c)
Fig. 7. The normalized electric field of the propagating hybrid plasmonic mode in the MISIM structure (a) the cross sectional view, (b) the x component along the y = 50 nm line, and (c) the y component along the x = 50 nm line for r2 = 50 nm and h = 5 nm.
k = nω/c
is shown in Fig. 7. Fig. 7 (b) and (c) shows the distribution of the x and y components of the electric field along the horizontal and vertical dashed lines in Fig. 7 (a), respectively. Fig. 8 (a)–(d) shows the mode effective index, effective propagation loss, confinement factor and the normalized mode area of the hybrid plasmonic mode in MISIM-based nanocavity. Compared to the characteristics of the SIMIS-based nanostructure, the mode effective index and the propagation loss are relatively reduced for the metallic nanorod with the radius of 40 nm and the insulator gap width of below 10.7 nm. Given the fact that the large wave vector of the subdiffraction-limited mode is one of the main reasons of the inefficient coupling of SPP emission to on-chip waveguides [3], reducing the wave number k = 2π / λ can relatively solve this problem. The relationship between the refractive index of a medium, n, and the wave number k is as follows:
(4)
where c is the velocity of light in vacuum and ω is the angular frequency. So, at a given angular frequency, a decrease in the mode effective index is directly related to reduction of the wave number. Accordingly, the lower effective index of the hybrid plasmonic mode in the MISIM-based nanocavity corresponds to a lower wave number and consequently better coupling efficiency in the MISIM structure compared to SIMIS and MIS nanostructures. By comparing Figs. 8 (d) and 4 (d), we conclude that the hybrid plasmonic mode of the MISIM-based nanocavity has larger normalized mode area compared to that of the SIMIS-based plasmonic cavities. It is noticeable that the difference between the normalized mode areas of the MISIM and SIMIS based nanocavities declines and can be neglected for larger radii and lower insulator gap. For example, for r2 = 60 nm and h = 2.5 nm this difference is about 0.002.
Fig. 8. (a) The mode effective index, (b) propagation loss, (c) confinement factor, and (d) normalized mode area of the hybrid plasmonic mode in the MISIM structure. 319
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Fig. 13. The Purcell factor (Fp) of the (a) SIMIS, (b) MISIM based spaser for L = LP.
Fig. 9. The Lasing Threshold of MISIM based spaser as a function of h for L = 30 μm and three different values of r2.
Fig. 14. The Purcell factor (Fp) of the (a) SIMIS, (b) MISIM based spaser for L = 30 µm. Fig. 10. The propagation length of the hybrid plasmonic mode in the (a) SIMIS (b) MISIM based nanocavities.
By obtaining the mode effective index, the effective propagation loss and the confinement factor, the lasing threshold of the MISIM-based spaser is calculated as a function of h for L = 30 μm and three different values of r2 and depicted in Fig. 9. This figure shows that the MISIMbased plasmonic nanolaser have relatively higher lasing threshold compared to SIMIS and MIS based spasers. 4. Optimization of the nanocavity length In this section, we investigate the propagation lengths of the hybrid plasmonic modes of the MISIM and SIMIS based nanocavities. The propagation length of the hybrid plasmonic mode relates to the geometric parameters of the plasmonic nanolasers. The propagation length (LP ) is given by [19]:
Fig. 11. The optimized lasing threshold of (a) SIMIS (b) MISIM based spaser as a function of h for L = LP and three different values of r2.
LP =
λ 1 = 4πα eff 2α eff k
(5)
where λ is the wavelength of the propagating mode. The propagation length is inversely related to the propagation loss which in turn depends on the structural characteristics of the nanocavity such as the insulator gap width (h). According to the simulation results, it is observed that the propagation length changes by altering the value of h as depicted in Fig. 10 (a) and (b). Although such structures can support the propagation of LRHPP modes at the wavelength of 1550 nm with a propagation length more than 1 mm [19], here at the wavelength of 490 nm, the maximum propagation length is 1.73 μm and 1.62 μm for the SIMIS and MISIM structures, respectively, as shown in Fig. 10. Because of the different propagation lengths of the hybrid plasmonic mode for various h values, it’s not accurate to assume a constant value for the cavity length when we are going to sweep the h parameter. In fact, the semiconductor nanorod, as a gain medium, provides enough gain to compensate the overall loss and ensure the lasing action through the spectral and spatial overlap with the SP mode. The transferred resonance energy from the gain medium excites the SP mode at first and then amplifies it. Without such a gain medium, the lasing
Fig. 12. The Lasing threshold of MIS-based spaser as a function of h for L = LP and three different values of r2.
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consisting of two coupled MIS structures. The first one is the SIMIS structure with a common metallic core and the second one is the MISIM with a common gain medium. The presented SIMIS-based spaser shows better threshold and mode effective area in comparison with both MIS and MISIM based nanolasers for suitable choice of the nanorod radius and insulator gap width. Modal analysis of the MISIM structure shows lower mode effective index which results in more efficient coupling to the on-chip waveguides due to lower wave number although this benefit is achieved at the expense of larger normalized mode area and lasing threshold. Also, we optimized the design of the nanocavity by considering the nonocavity length equal to the modal propagation distance in order to consider the lasing action more precisely. This optimization led to a better enhancement of the light matter interaction which is seen as the increase of the Purcell factor.
mode cannot propagate over its propagation length. So, the FP cavity length is limited by the lasing mode propagation length and cannot be adopted larger than the mode’s energy attenuation length or LP. In fact, without a loss compensation mechanism, if we design a FP cavity with a length much larger than the propagation length of the FP lasing mode, a correct feedback mechanism and the standing wave pattern formation cannot be accomplished. One the other hand, presence of the gain medium cannot break the cavity length limitation. This means, when the propagation distance of the lasing mode becomes lower and lower regardless of its reason, we will need more and more pumping energy to achieve enough gain for compensating the intrinsic losses and consequently realize the desirable effective spasing action. In other word, a long plasmonic FP cavity encounters very high metal loss that causes larger absorption of the propagating hybrid modes in travelling between the end facets. Under such condition and without expending too much energy to compensate the losses, the hybrid lasing mode cannot be subjected to the cavity feedback that is necessary for spasing. In this situation we cannot enlarge the FP cavity length as much as we need in order to minimize the mirror loss. It is a fact that decreasing the cavity length is limited by increase of the mirror loss, although it may be relatively compensated by a high-reflectivity coating or an appropriate feedback structure under a proper phase matching condition [26]. So, it’s better to choose the device length L around the energy attenuation length (L ≈ LP) to minimize the high metal loss. Thus, to obtain the optimum spasing action, an optimized value for the cavity length is needed which can be a multiple or fraction of the modal propagation length. In this study, we approximately set the cavity length equal to the propagation length of the hybrid plasmonic modes of the presented nanostructures to focus on the effect of the geometrical changes on the spasing action. With the assumption of L = LP in Eq. (3), we have:
1 + 2ln g th = k 0α eff (
( ))
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1 R
Γ(n eff /nwire )
(6)
Using this relation in the simulations, the lasing threshold of the MISIM and SIMIS based spasers is obtained as shown in Fig. 11 (a) and (b). As can be seen in this figure, the behavior of the threshold curves of both structures have not changed, but the values of the lasing threshold increase and become closer to the reality. We have simulated the proposed MIS structure in Ref. [20] and calculated its threshold using the assumption of L = LP as demonstrated in Fig. 12. By comparing Figs. 12 and 11 (a) and (b) it becomes clear that the SIMIS spaser has a lower threshold for r2 = 40 nm compared to both MIS and MISIM structures. For r2 = 50 nm and r2 = 60 nm, the SIMIS spaser has a lower threshold only for h ≥ 8.63 nm and h ≥ 12.03 nm, respectively. Generally, the MISIM structure has relatively larger threshold compared to both MIS and SIMIS. Thus, we will need more pumping energy to realize the spasing action in this structure. For evaluation of the spasers operation, we can obtain the Purcell factor, which denotes the light matter interaction enhancement, through the relation FP = 3Q/4π 2Vn where Vn is the normalized mode volume [13,27] and Q is the plasmonic cavity’s quality factor [26]. It should be noticed that the normalized mode area that we have reported in the manuscript, is inversely related to the Purcell factor through the relation FP A0 /2πAeff [28]. Figs. 13 and 14 shows the calculated Purcell factor for L = LP and L = 30 μm, respectively. As can be seen in Figs. 13 and 14, by assuming L = LP, the Purcell factor is increased compared to the case of L = 30 µm. This shows that the participating mechanisms of spasing act more effectively, the gain medium can amplify the SPPs much better, and so the spasing can be realized easier. 5. Conclusion In this paper, we proposed and investigated two SPP-based spasers 321
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