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Spectral narrowing in a sub-wavelength solid-state laser David Hernandez-Pinilla, Javier Cuerda, Pablo Molina, Mariola O Ramirez, and Luisa E Bausa ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.9b00836 • Publication Date (Web): 16 Aug 2019 Downloaded from pubs.acs.org on August 18, 2019
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Spectral narrowing in a sub-wavelength solid-state laser David Hernández-Pinilla, Javier Cuerda, Pablo Molina, Mariola O Ramírez and Luisa E. Bausá∗ Dept. Física de Materiales, Instituto de Materiales Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049-Madrid, Spain *Author to whom correspondence should be addressed:
[email protected] Abstract The association of metallic nanostructures and solid-state gain media has recently led to the emergence of functional platforms for nanolasing with relevant features such as chemical and thermal stability and the absence of photo-bleaching. In this work, we demonstrate that the incorporation of plasmonic nanoparticle chains on a Nd3+ doped solid-state laser platform enables nanoscale laser emission at the technologically relevant spectral region of telecom (λe =1385 nm), outside the spectral response of the plasmonic resonance of the nanoparticle chains. In addition, we experimentally show that the plasmonic chains improve the monochromaticity of the solidstate laser, thus attaining spectral narrowing of the laser line by up to 38% depending on the pump wavelength. The effect of the plasmon driven absorption enhancement in the vicinities of the Ag nanoparticle chains is analyzed by tuning the optical pump in a broad spectral range (580 - 890 nm) at different Nd3+ transitions that overlap the spectral response of the longitudinal plasmonic mode of the chain. Theoretical insight shows that the plasmon-induced narrowing of the laser linewidth close to the nanoparticle chains can be related to the modification of the effective pump rate within the active medium, which is produced by the excitation of the localized plasmonic mode. The work constitutes a step forward in the development of high-quality and highly integrated photonic devices featuring narrowband laser emission in ultra-small volumes operating in a technologically relevant spectral range.
KEYWORDS: plasmonic nanolaser, line narrowing, Nd3+ solid-state laser, Ag nanoparticle chains, LiNbO3. 1
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Achieving ultimate control of light at the nanoscale is nowadays the focus of an intense research. Novel applications, such as precise spectroscopies, biological and chemical sensing, optical nanodevices for information and quantum storage, and communication are motivating the search for stable and high-quality photonic devices featuring narrowband laser emission in technologically relevant spectral ranges. Plasmon-assisted lasers based on the association of optical gain media with metallic nanostructures are central to this aim, since they provide sub-wavelength confined modes together with reduced physical device dimensions. A diversity of nanolaser designs combining metallic nanostructures with organic dyes or semiconductors media have been reported in the last years.1-10 In this context, solid-state lasers (SSLs) associated with plasmonic nanostructures have demonstrated to act as functional platforms for nanolasing, providing a strong threshold reduction and a significant increase in the slope efficiency for the nanoscale operation with respect to the conventional bulk configuration.11 Moreover, the combination of plasmonic nanostructures with SSLs has given rise to the emergence of new functionalities at nanometric spatial regions, such as dual-wavelength operation in the near infrared region,12 or simultaneous multi-wavelength emission at different spectral ranges via different parametric frequency conversion processes.13 These achievements show the capacity of plasmonic nanostructures to produce drastic modifications in the optical features of SSLs, while keeping their intrinsic advantages such as a high chemical stability and a high thermal frequency stability. In this work, we go a step further in the search for novel features on plasmon-assisted SSLs. On one side, we demonstrate the possibility to extend the operating wavelength of plasmon-assisted SSLs to a technologically relevant spectral region for telecom (E-band) and bio-imaging.14-15 On
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the other hand, we demonstrate that the interaction of plasmonic nanostructures with the active medium can induce a laser linewidth narrowing by up to 38 % for lasing operation at the nanoscale. The interplay of the localized surface plasmon (LSP) modes with the absorption and emission bands of optically-pumped gain media dramatically determines the laser performance of a given device at nanometric scale. The most common route exploiting plasmonic nanostructures to achieve optical gain at the nanoscale involves the tuning of the LSP frequency to the emission wavelength of the active medium.16-17 However, the work devoted to analyze the influence of the optical absorption enhancement provided by the plasmonic field on the laser properties is scarce. 18-19
In this work, we carry out an experimental and a theoretical study on the effect of the
absorption enhancement produced by plasmonic Ag nanoparticle (NP) chains in the laser performance of a SSL at the nanoscale. In particular, we experimentally show that the near-field absorption enhancement provided by the plasmonic nanostructures allows the reduction of the laser linewidth with respect to the case of a conventional Nd3+ based solid state laser, thus attaining record-narrow laser linewidth for a plasmon-assisted nanolaser. The theoretical insight shows that the reduced linewidth in the vicinities of the NP chains is related to the modification of the effective pumping rate within the active medium, which is produced by the excitation of a localized plasmonic mode. Our study was carried out in a plasmon-assisted nanolaser consisted of a Nd3+ doped LiNbO3 SSL on which chains of closely spaced metallic NPs were deposited. This system is of particular interest in the context of this work due to several reasons. On one hand, the plasmonic response of the Ag NP chains features a spectrally broad longitudinal plasmonic mode, which exhibits very low damping, and a strong radiative character in the visible and near infrared spectral range.20 This mode efficiently overlaps a variety of Nd3+ absorption bands in which the optical pump can be tuned. Accordingly, the effect of the absorption enhancement on the laser emission linewidth at 3
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the nanoscale can be analyzed by studying the influence of the optical pump at different wavelengths inside the spectral response of the LSP. Regarding the laser emission wavelength, we go a step further in the search for novel features on plasmon-assisted SSLs and demonstrate their operation at 1385 nm. Unlike other Nd3+ laser wavelengths associated with different optical transitions,21 the Nd3+ emission at 1385 nm is located well outside the spectral region of the plasmonic resonance of the NP chain. In this way, it is possible to disentangle the effect of the absorption enhancement on the nanolaser performance. Finally, due to the relevant electro-optic, acousto-optic and nonlinear properties of LiNbO3 the use of lithium niobate as a functional platform could afford the integration of nanolasers with additional photonic functions such as intensity modulation, frequency conversion or waveguiding, onto the same optical chip.22-23 The work constitutes a step forward towards the integration of high optical quality emitters with ultra-small volumes at one of the spectral windows of minimal attenuation of fiber optics, which could be of potential interest for applications including quantum networks, sensing or computing.24-26
Results and discussion Figure 1a shows a schematic representation of the system under study. It consists of a Nd3+ doped periodically poled LiNbO3 laser crystal on which millimetric long chains of closely spaced Ag NPs were formed via ferroelectric lithography. The fabrication procedures of both the SSL crystal platform and the NP chains are detailed in the Materials and Methods Section. The average diameter and separation distance of the Ag NPs were 50 nm and 2 nm, respectively.27 When placed together by few nanometers, these metallic NPs couple their LSPs, further increasing the electric
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near-field enhancement in their vicinities with respect to the case of isolated NPs. The spectral response (far field scattering cross-section spectrum) of the Ag NP chain is shown in Figure1b (blue line) for a plane wave polarized parallel to the chain axis. A broad band centered at the visible spectral region characterizes this longitudinal plasmonic mode of the Ag NP chain. As previously reported, this mode displays a dominant radiative character, with a negligible absorption crosssection compared to the scattering one.13 Figure 1b also shows the good overlap between the 4f-4f Nd3+ optical absorption bands and the plasmonic response of the Ag NP chain. The different Nd3+ transitions used for optical pumping in this work have been marked with colored arrows and identified in the energy level scheme. In our experiments, lasing takes place in a 4 level scheme at 1385 nm, outside the spectral response of the plasmonic chain through the 4F3/2 → 4I13/2 transition, regardless the optical pump wavelength. For the nanolasing experiments, the hybrid plasmonic-SSL system was placed inside an optical Fabry-Pèrot (FP) resonator formed by two plane-parallel mirrors separated around 1 mm. The cavity was formed by mirrors that exhibited high losses at the commonly used 4F3/2 → 4I11/2 laser transition of Nd3+ (around 1,06 µm) and high reflectance at the 4F3/2 → 4I13/2 transition to obtain the required feedback for lasing at 1385 nm. Laser gain experiments were performed by spatially resolved confocal microscopy under optical pump of different Nd3+ absorption bands matching the LSP resonance of the Ag NP chain (see Figure 1b). A 20x objective lens (NA = 0.45) was used to focus the pump beam to around 2µm-diameter spot onto the sample. The laser radiation was collected in backscattering geometry with the same objective. In all the experiments, the incoming pump beam was polarized parallel to the plasmonic chain. Figure 1a shows, as an example, the spatial map of lasing intensity obtained under pumping at 808 nm (4I9/2 → 4F5/2 + 2H9/2 Nd3+ transition) with a pump power of 30 mW/µm2. The spatial distribution of the laser intensity shows maximum values in the vicinities of the Ag NP chains, confirming the sub-wavelength character 5
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of the lasing radiation and the threshold reduction induced by the plasmonic nanostructures with respect to the regions without Ag NP chains. This threshold reduction almost vanishes for a pump beam polarized perpendicular to the chain axis in agreement with previous works.11, 13 The lasing emission spectrum is depicted in Figure 1c (red line), where it has been superimposed with the spontaneous emission spectrum (black line) for comparison. It is characterized by a sharp peak at λe ≈ 1385 nm associated with the 4F3/2 (1) → 4I13/2 (3) Stark transition of Nd3+ ions in LiNbO3, and demonstrates the possibility of achieving stable lasing at the nanoscale in the telecom optical Eband from a hybrid plasmonic-LiNbO3 solid-state multifunctional platform. Similar spatial distributions (sub-wavelength spatial confinement) of the lasing emission were obtained for all the pumping wavelengths employed in this work. The lasing performance in the vicinities of the Ag NP chains is shown in the inset of Figure 1c together with that obtained in the absence of plasmonic nanostructures. A significant threshold reduction together with an increase in the slope efficiency were achieved in agreement with the near field confinement produced by the Ag NPs close to the surface of the considered crystal.11 Once nanolasing at 1385 nm has been demonstrated, we analyze the effect of the plasmonic Ag chains on the degree of monochromaticity of the laser emission. To that aim, a comparative analysis between the laser linewidth obtained in the vicinities of the Ag NP chains and the one obtained in regions lacking plasmonic nanostructures has been carried out. Figure 2a shows a detail of the spatial distribution of the integrated lasing intensity at 1385 nm over a micrometric zone that contains five Ag NP chains on the Nd3+:LiNbO3 crystal surface. The spatial map has been obtained under pumping at 808 nm (4I9/2 → 4F5/2 + 2H9/2 transition) with a pump power that is above the threshold at both the proximities of the plasmonic Ag NP chains and the bare Nd3+:LiNbO3 regions. Therefore, the scanning gain microscopy experiments show lasing in the whole analyzed region, i.e the proximities of the plasmonic Ag NP chains and the bare Nd3+:LiNbO3 areas. The regions of 6
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maximum intensity correspond to the nanolasing signal around the Ag NP chains, while those of minimum intensity correspond to lasing in the regions lacking the plasmonic chains. The analysis of the laser linewidth in the same area is displayed in Figure 2b. There, the spatial distribution of the laser linewidth reduction with respect to the laser linewidth of the bare regions has been depicted. A good correlation can be established between the spatial maps of Figure 2a and b. As observed, a systematic line-narrowing is achieved in the proximities of the NP chains replicating the spatial distribution of the nanolasing emission. Figure 2c compares the lasing spectra of both configurations. The results provide experimental evidence on the ability of the plasmonic chains to improve the monochromaticity of the SSL. Under the aforementioned conditions, namely, by optical pumping at the low energy side of the plasmonic resonance (p= 808 nm), a reduction of approximately 10% in the laser linewidth is obtained. However, the laser linewidth can be further narrowed by tuning the optical pump wavelength from the sideband towards the maximum of the plasmonic resonance. Figure 3 shows the influence of the pump wavelength, p, on the laser linewidth in the vicinities of the Ag NP chains compared to the laser linewidth obtained in the absence of plasmonic chains. As observed, for the bare configuration the laser linewidth remains constant within the analyzed pump power range when varying the pump wavelength along the different Nd3+ absorption bands. This indicates that thermal losses associated with the quantum defect, qd = p/e, do not play a significant role on the laser linewidth. However, under the presence of the plasmonic chain, a systematic decrease of the laser linewidth is observed as the pump wavelength approaches the maximum of the plasmonic response, regardless of the absorption transition of Nd3+ employed for pumping. Specifically, the laser linewidth is reduced by 30 %, decreasing from 0.29 cm-1 (bare configuration) down to 0.21 cm-1 when the optical pump
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is tuned around the maximum of the plasmonic resonance (p = 587 nm). The corresponding lasing spectra are shown in Figure 3b. Obtaining physical insight into the fundamental origin of plasmon-induced laser line-narrowing involves evaluating the influence of the Ag NP chains on the FP cavity. To that purpose, we have carried out full-wave linear 3-dimensional simulations using the finite element method (FEM), to obtain the near-field profiles of the electromagnetic (EM) fields of the passive cavity mode at the pump and laser wavelengths (see Materials and Methods). In this way, the responses of both the bare and plasmon-assisted configurations are compared. Figure 4a displays the near E−field distribution for the plasmon-assisted cavity mode computed at the laser wavelength λe = 1385 nm. As seen, it is characterized by a series of maxima with a regular spacing of λe/2n (n is the refractive index associated to each medium) along the direction of propagation, as it corresponds to the standing-wave condition provided by the highly reflective mirrors (R 0.98).28 In fact, we have confirmed that the plasmon-assisted cavity mode and the bare FP cavity mode at λe, show very similar E−field profiles. That is, the latter remains practically unaltered by the LSP mode of the chain since the emission wavelength is spectrally located well apart from the influence of the plasmonic mode (see Figure 1b showing the LSP of the NP chain peaking at λLSP = 590 nm). Further, at λe = 1385 nm, the penetration depth of EM fields into the metal is extremely low, preventing the formation of plasma oscillations and hence efficient lightmatter coupling.29 This suggests that the decay rate of the EM field energy at the laser wavelength is not affected by the plasmonic local field-enhancement nearby the Ag NP chain. Therefore, the temporal confinement of the passive resonator in the presence of Ag NP chains remains practically unchanged from that of the FP cavity.
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In contrast, the incorporation of the plasmonic chain introduces a noticeable effect in the near electric field (E-field) distribution at the absorption wavelength due to the excitation of the plasmonic longitudinal mode. Fig 4b shows, as an example, the E-field distribution at the pump wavelength of a = 808 nm (4F3/2 → 4I13/2 Nd3+ transition). On one hand, the plasmonic chain leads to the subwavelength spatial confinement of the E-field around the Ag NP chain. On the other hand, the interaction of the LSP mode with the pump radiation leads to a diffractive pattern in the LiNbO3 crystal. This spatial non-uniformity of the E−field inside the gain medium, together with the high field-confinement associated with the plasmonic localized mode of the chain results in an improved pump efficiency, which allows a higher effective gain to be delivered to the lasing mode with respect to that provided by the active medium in the absence of Ag NPs. The pump rate that is effectively transferred (via the active medium) to the lasing mode is given by the expression: Rp = Ka∫Vad3r|Ea(r)|2|Ee(r)|2 , where |Ea,e(r)| are the E-field distributions at the absorption and emission frequencies respectively, and Ka = 0nLNBca/(ħa∫Vad3r|Ee(r)|2), where 0 and c are the dielectric permittivity and the speed of light in vacuum, nLNB is the refractive index of the LiNbO3 medium, a represents the absorption cross section of the Nd3+ ions in LiNbO3, a = 2c/a is the absorption angular frequency, and Va denotes the active medium volume.28 Hence, in order to compute the pump efficiency enabled by the plasmonic chain compared to the case of the bare configuration, one must evaluate the spatially-dependent quantity G(r) = |Ea(r)|2|Ee(r)|2, which is identified as the optical gain distribution inside the active medium.28 Figure 4c shows the optical gain spatial distribution for the case of the bare FP cavity (i.e., in the absence of plasmonic chains). A regular laser emission pattern is found as a result of both the fringes of the FP cavity mode arising at λe = 1385 nm and the uniform intensity profile created by the pump field in the absence of Ag NP chain.
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Conversely, the effect of the plasmonic Ag NP chain on the pump field distribution is responsible for a more complex pattern on the gain profile due to the efficient excitation of the longitudinal LSP of the chain. As observed in Figure 4d, the optical gain spatial distribution of the plasmonassisted configuration is a combination of the diffraction features shown in Figure 4b, modulated by the regular FP fringes that dominate the cavity mode of Figure 4a. This nanoscale-assisted mechanism at the absorption frequency results in an optical gain spatial distribution highly localized in a region close to the Ag NP chain, which decays with depth into the active medium. In addition, the presence of the plasmonic Ag NP chain leads to the lateral confinement of the pump beam below the chain, which explains the confinement of laser emission along the Ag NP chains in agreement with the experiments (see Figure 1a, top panel). Figure 4e displays a rescaled map of Figure 4d, in which the minimum is fixed to the maximum of panel 4c. By these means, we isolate the influence of the Ag NP chain noticing that the main contribution to the optical gain that enables lasing action from the vicinities of the Ag NP chain is characterized by a single high local-field intensity area close to the LiNbO3 surface. In fact, the simulations reveal the hybrid photonic-plasmonic performance of the laser resulting from both the excitation of a LSP mode of the plasmonic chain at the absorption wavelength and the feedback imposed by the cavity mirrors. As verified experimentally, exploiting the available absorption bands of the Nd3+:LiNbO3 platform that overlap the longitudinal plasmonic mode (see Figure 1b), allows for further improvement of the lasing performance. The optical gain profile near the crystal surface is represented in Figure 5a for different pump wavelengths. As shown, decreasing the pump wavelength from λa = 886 to 587 nm causes a further enhancement of the optical gain in the vicinities of the plasmonic nanostructures, in agreement with the spectral response of the longitudinal plasmonic mode of the Ag NP chains. This result is also accompanied by a higher degree of confinement, leading to nanolaser emission from an increasingly ultra-small volume. 10
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Figure 5b (red points) depicts the experimental values of the laser linewidth narrowing as a function of the pump wavelength, obtained as a ratio of the corresponding bare and plasmon-assisted laser linewidths from Figure 3a. A monotonic decrease in the laser linewidth is obtained as the pump wavelength is tuned into the plasmonic resonance. On the other hand, for the pump power range employed in our work (from 0 to about twice the pump power at threshold), we did not observe any significant variation on the measured laser linewidth, thus indicating that laser narrowing is mainly due to the increasingly shrinking gain confinement (see Figure S1 in the Supporting Information). In order to quantitatively address this dependence of linewidth narrowing on the pump wavelength, we implemented a semi-analytical simple theory based on a laser rate-equation analysis. Within our approach, the fundamental laser performance of our photonic-plasmonic cavity relies on the temporal confinement (through the quality factor of the passive resonator Q), on the spatial field confinement (through the energy confinement factor of the cavity mode, Γ), and on the pumping rate Rp, which, as discussed, accounts for the influence of the plasmonic chain at the pumping energy. In particular, we found that the laser linewidth, for high enough pump power, is given by: (∆ν)laser = ξ/Q2ΓRp (see Materials and Methods), where ξ is a factor that contains the quantities related to the active medium. As computed from the simulations of Figure 4a the Q−factor of the passive bare FP resonator does not differ significantly from that obtained for the plasmonicassisted FP cavity (Q 105 at the laser wavelength). On the other hand, since the E-field distribution of the cavity mode at λe=1385 nm is not influenced by the presence of the plasmonic NP chain (see Figure 4a), we also obtained similar values for the energy confinement factor. Overall, these results point out that the FP mode sustained by the passive cavity at λe = 1385 nm is not affected by the longitudinal LSP mode of the plasmonic chain. In contrast, the presence of the Ag NP chain
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strongly affects the absorption within the active medium resulting in a higher effective pumping rate (see Figs. 4b and 4d). This causes a lower threshold oscillation of the plasmonic-assisted laser cavity, as compared to the bare laser operation (see inset of Figure 1c). Using the expression above, we find that the ratio of the laser linewidth of the plasmon-assisted cavity (∆ν)pl to that of the bare laser (∆ν)b yields: (∆ν)b/(∆ν)pl ≈ Rppl/Rpb. We obtained the effective pumping rate accumulated for the plasmon-assisted laser by spatially integrating the optical gain of Figure 5a over the active medium region in which the plasmon-assisted laser takes place. For the bare laser operation we integrated over the portion of the active medium in which the critical population inversion, needed for lasing, is achieved (see Supporting Information and Figure S2). Figure 5b (black squares) shows the calculated values of the linewidth reduction for different values of p. We found increasing values of the effective pumping rate with decreasing pump wavelength for the plasmon-assisted architecture, while we obtained approximately constant values for the bulk laser operation, accounting for the observed reduction of the nanolasing linewidth shown in Figure 3. The excellent quantitative agreement of our theoretical model with the experimental results (Figure 5b, red squares with error bars) corroborates that plasmonenhanced pumping is the main physical mechanism behind the observed narrowband lasing action at the nanoscale.
Summary and Conclusions To summarize, we have extended the operation of a plasmon-assisted SSL to the technologically relevant E telecom band, with nanolasing emission at λe =1385 nm. The sub-wavelength confinement results into an improved laser performance with respect to the standard bulk operation featuring, among others, an improved monochromaticity of the laser linewidth. Further, unlike
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most of previous works related to plasmonic nanolasing, in our case the lasing transition is spectrally located well outside the spectral response of the plasmonic nanostructures, thus allowing to isolate the influence of the absorption enhancement on the nanolasing performance. By tuning the pumping wavelength into the longitudinal plasmonic mode sustained by Ag chains, we demonstrate a monotonic narrowing of the spectral linewidth, attaining a record laser linewidth of about 0.2 cm-1 for a plasmonic nanolaser. The results are explained in terms of the increasing values of the effective pumping rate when the optical absorption is tuned from the sideband towards the maximum of the plasmonic resonance allowing nanolaser operation with increasingly spatial confinement in ultra-small volumes. The results are demonstrated by using a widely used optoelectronic material LiNbO3, which offers multi-functionality to the nanolasing device. Nevertheless, due to the fundamental nature of the nanoscale amplifying mechanism, we expect that our conclusions may be generalized to other plasmon-assisted laser configurations involving a wide variety of optically active materials. The work constitutes a step towards the development of high-quality multifunctional integrated photonic nanodevices featuring narrowband laser emission in ultra-small volumes and operating at technologically relevant spectral ranges. The potential applications include the fields of information and quantum storage, sensing, telecommunications, optical circuitry and bioimaging.
Materials and Methods Sample preparation The plasmonic Ag NP chains were formed via ferroelectric lithography on the polar surface of a 0.7 mm thick Nd3+ doped periodically poled LiNbO3 (PPLN) crystal. The Nd3+ concentration in the crystal was 0.1 at % relative to Nb5+. The plasmonic nanostructures were obtained by illuminating the PPLN crystal at 253.6 nm with an UV Mercury lamp for 10 minutes at 65º C while the crystal 13
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was immersed in a 0.01M AgNO3 solution. More details on the crystal, on the photodeposition process and on the resultant Ag NP chains can be found elsewhere.30
Laser experiments Laser experiments were performed in a laser scanning confocal microscope by means of a 20x objective lens (NA = 0.45). The pumping beam consisted of a laser beam polarized parallel to the plasmonic chain, which impinges only on the central part of the objective lens, covering a surface < 20 % of the lens. Thus, the presence of out-of-plane components that could come from the objective lens is minimized. The system was provided with a two axis XY motorized platform with 0.2 µm spatial resolution. A cw laser emitting at 587 nm (Genesis MX STM, Coherent) and a cwTi:Sapphire laser (Spectra Physics) (tuned at 750, 808 and 886 nm) were used as excitation sources. Stimulated and spontaneous emission were collected in backscattering geometry. A Horiba iHR 550 monochromator and a cryogenic-cooled InGaAs Symphony II were used for detection. During the lasing experiments, the crystal was placed in a FP resonator formed by two identical planeparallel mirrors. Both input (R1) and output (R2) mirrors were highly reflective (R 0.98) at the laser wavelength (λe = 1385 nm), and displayed a high transmittance (T > 75%) at the pump wavelengths.
Semianalytical theory In order to obtain a theoretical prediction of the plasmon-assisted laser linewidth narrowing, we implement a simple semianalytical theory based on the following rate equations:31-32 𝑑𝑛 𝑑𝑡 𝑑𝑠 𝑑𝑡
= 𝑅𝑝 ― 𝑅𝑛𝑟(𝑛) ― 𝑅𝑠𝑝(𝑛) ― 𝑅𝑠𝑡(𝑛)𝑠 =―
𝑠 + 𝛤𝑅𝑠𝑡(𝑛)𝑠 + 𝛤𝛽𝑠𝑝𝑅𝑠𝑝(𝑛) 𝜏𝑝
(1) (2)
where n is the population inversion density at the lasing transition and s is the emitted photon density; Rnr = n/τ21nr is the non-radiative decay rate, and Rsp = n/ τ21sp is the spontaneous emission decay-rate, with , where the βsp factor determines the degree of coupling of 14
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the spontaneous emission to the lasing mode,33 and 21sp=100 μs is the spontaneous emission lifetime of the considered laser transition.34 Rst = vgg(n) is the stimulated emission decay-rate, with vg being the material group velocity of the active medium, and the gain coefficient g(n) is given by: g(n) = σen, where σe = 2 × 10-20 cm2 is the emission cross-section of the active medium.35 From expression (2), it can be seen that the stimulated emission balances the losses of the cavity without gain, given by the photon lifetime τp = Q/ωe, with Q being the quality factor of the passive resonator and ωe the emission frequency of the active medium. Hence, we can define the active cavity lifetime ’p as:36 1 𝜏′𝑝
=
1 + 𝛤𝑅𝑠𝑡(𝑛) 𝜏𝑝
(3)
Under CW-pumping and above the lasing threshold, the system is expected to reach a steady-state for long times. Therefore, setting ds/dt = 0 in eq. (2), we obtain the following expression for the laser linewidth (FWHM):
(𝛥𝜈)𝑙𝑎𝑠𝑒𝑟 =
1 2𝜋𝜏′𝑝
=
𝛤𝛽𝑠𝑝𝑛𝑠𝑠 2𝜋𝜏𝑠𝑝 21𝑠𝑠𝑠
(4)
where sss is the photon density in the steady-state. The steady-state population inversion density nss can be derived similarly, by fixing dn/dt = 0 in (1). Substituting the resulting steady-state expressions with 𝛽𝑠𝑝
