Abstract
When light is brought to a standstill, its interaction with gain media increases dramatically due to a singularity in the density of optical states. Concurrently, stopped light engenders an inherent and cavityfree feedback mechanism, similar in effect to the feedback that has been demonstrated and exploited in largescale disordered media and random lasers. Here we study the spatial, temporal and spectral signatures of lasing in planar gainenhanced nanoplasmonic structures at nearinfrared frequencies and show that the stoppedlight feedback mechanism allows for nanolasing without a cavity. We reveal that in the absence of cavityinduced feedback, the subwavelength lasing mode forms dynamically as a phaselocked superposition of quasi dispersionfree waveguide modes. This mechanism proves remarkably robust against interface roughness and offers a new route towards nanolasing, the experimental realization of ultrathin surface emitting lasers, and cavityfree active quantum plasmonics.
Introduction
The lasing principle, one of the key discoveries of quantumelectrodynamics, has had tremendous impact on optics research and technology. In a development not too dissimilar to the revolution in transistorbased digital electronics, lasers have undergone a continuous miniaturization that culminated in the recent experimental demonstration of subwavelength nanolasing^{1,2,3,4,5,6}. This evolution was achieved through innovative design concepts as well as advancements in fabrication techniques, which together led to breakthroughs in the two key ingredients that make a laser: light amplification and feedback. Lowdimensional semiconductor gain materials such as quantum wells and dots proved as important in shrinking lasers as did microcavity resonator designs^{7}, for example, in the form of microdisks^{8}, micropillars^{9}, defectmode photonic crystals^{10,11} and photonic nanowires. Advancing beyond traditional cavity concepts, recent nanolasers employ plasmonic resonances for feedback^{12,13,14,15}, allowing them to concentrate light into mode volumes that are no longer limited by diffraction. The use of localized surface plasmon resonances as coldcavity modes, however, is only one route to lasing on subwavelength scales. Lasing, in fact, does not require modes predefined by geometry but merely a feedback mechanism. To illustrate this point, one may consider the situation in a random laser where feedback arises from multiple random scattering events that lead to the spontaneous formation of closed optical paths within the disordered medium^{16,17}. Here we show that cavityfree lasing is not restricted to micrometre sizes in disordered media (using random feedback) but can even be realized on subwavelength scales using stopped light (SL) to provide local feedback.
The fundamental principle behind SL lasing is to trap and amplify photons in space just at the point of their emission^{18}. For this, a slow light regime needs to be entered, which could be achieved, for example, in photonic crystal waveguides via coherent backscattering^{19,20}, through symmetry breaking in axially uniform waveguides^{21,22}, or in metamaterials^{23} through the plasmonic analogue of electromagnetically induced transparency^{24}. Planar plasmonic heterostructures present an even simpler design in which the stopping of light arises from the balancing of opposing energy flows in layers of materials with positive (dielectric) and negative (metal) permittivities. This design approach offers precise control over the group velocity without the need of structuring along the direction of propagation^{25,26,27}. At the SL point, the overall energy flow effectively cancels, forming a characteristic closedloop vortex^{28} (Fig. 1a), which results in a strong enhancement of the local density of optical states (LDOS) that is only limited by dissipative loss. Combined with gain, this SL feedback mechanism can then lead to coherent amplification of the trapped photons via stimulated emission processes.
Above the threshold, where the gain exceeds the losses, a localized SL lasing mode forms, whose mode volume is only limited by the waveguide dispersion. To enable subwavelength localization of the lasing mode, a broad spectrum of kmodes (k denoting the wavevector) needs to be supported by the gain (Fig. 1c). This can be achieved by introducing two (or more) SL points between which a monotonous behaviour of the dispersion is enforced. Adjusting the layer thicknesses and choice of materials allows us to simultaneously maximize the modal gain and minimize the dispersion (Fig. 1c). Although being conceptually simple, this principle can be applied universally to render the dispersion of the SL laser flat by structural optimization using, for example, genetic algorithms (see Methods).
In the following, we first determine the dispersion properties of an SL lasing system that is based on a leakymode metaldielectric waveguide design. Next, we study the transient SL lasing behaviour when a finite gain section is introduced, observing ultrafast relaxation dynamics and highβ characteristics that are caused by a bundling of passive kmodes into a single SL lasing mode. For the same structure, we offer numerical evidence that the SL points are retained up to a certain threshold value of the surface roughness, beyond which light increasingly concentrates in plasmonic hotspots on a deep subwavelength scale. Finally, we identify the modelocking mechanism that leads to the selflocalization of light on subwavelength scales and verify its applicability to both the SL nanolaser and an alternative SL spasing design.
Results
Dispersionless SL in metaldielectric waveguides
We demonstrate SL nanolasing at telecom (nearinfrared) wavelengths using the example structure of a planar metaldielectricmetal stack with a top metal layer of thickness t=500 nm and a waveguide core of height h=290 nm on top of a thick metal substrate (Fig. 2a). This is a principle study, but for specificity, the dielectric permittivities of the core and cladding materials are chosen to be characteristic of III–V semiconductors (such as InGaAsP) and transparent conductive oxides (indium tin oxide) (see Methods), respectively. This particular choice can, for example, for a direct comparison with experiment, be adapted, particularly to take advantage of recent advances in improving the properties of plasmonic materials^{29}. The choice of structure and materials gives rise to a (weakly) leaky TM_{2} mode in the dispersion diagram (Fig. 2b) which, by design, features two SL points that lie close in frequency: SL_{1} at ω_{1}/2π=193.8 THz (λ_{1}=1,546.9 nm) and k_{1}=0, and SL_{2} at ω_{2}/2π=193.78 THz (λ_{2}=1,547.06 nm) and k_{2}=1.42 μm^{−1}. With both SL points situated inside the light cone, we expect the SL lasing mode to couple to freespace radiation perpendicular to the plane. The emission of light can then be controlled within limits by adjusting the thickness of the top metal cladding and amounts, for the given geometry, to around 3%–4% of the total energy lost per cycle. We initially focus on this photonic (that is, leaky) design, but will demonstrate later that it is equally possible to realize plasmonic (that is, spasing^{12}) designs, where multiple SL points fall onto a bound plasmonic mode.
Having established the dispersionless character of the SL mode using a passive mode theory (Fig. 2b), we proceed to find the threshold conditions for SL lasing by performing a smallsignal gain analysis. We assume that the gain medium fills the dielectric core of the structure, except for a 10nm buffer next to the metal, which is introduced to account for gain quenching in close vicinity to the metal. The analytic dispersion equations are solved for varying levels of inversion (see Supplementary Table 1 for the gain material parameters). As the two SL points retain their positions and effectively pin down the dispersion between them, no significant change of the modal dispersion is observed with increasing inversion (Fig. 2c). In contrast, the modal loss shows a strong dependence on the inversion and changes sign at ΔN_{th}≈0.13 (Fig. 2c, bottom). At this point, the mode becomes undamped and, owing to the flatness of the band, experiences an almost uniform level of gain over a broad range of kvalues (up to k≈5 μm^{−1} where the dispersion ω(k) ultimately loses the gain support). As the calculated loss rate includes both dissipative and radiative (leakage through the cladding layer) contributions, we can associate ΔN_{th} with the threshold inversion required for lasing.
Dynamics of cavityfree SL lasing
To gain insight into the transient dynamics and characteristics of SL lasing on subwavelength scales, we perform firstprinciples timedomain simulations, taking into account the nonlinearity of the gain material as well as the spatiotemporal interplay between lightfield and inversion (Fig. 3). The gain material, described by fourlevel MaxwellBloch equations^{14} with parameters that match those used in the semianalytical mode calculations above, is initially confined to a 400nmwide section, approximately four times smaller than the vacuum wavelength at the SL points. By design, the gain section can be pumped optically using a propagating waveguide mode at a higher frequency (Supplementary Fig. 1 and Supplementary Table 2). At a pump level well above threshold, the transient field and inversion dynamics (Fig. 3a) display ultrafast relaxation oscillations with a period of around 10 ps. The steadystate inversion settles at ~0.5, a value significantly higher than the threshold inversion predicted by our semianalytical calculations (Fig. 2b). This increase in the inversion is the result of a reduced confinement of the fields to the now finite gain section (confinement factor Γ≈0.31, as defined in the Supplementary Discussion), additional losses along the waveguide due to group velocity dispersion and spatial hole burning effects caused by the inhomogeneity of the localized SLmode profile (Fig. 3b). The laser output energy follows a familiar Sshaped input–output curve (Fig. 3c) with a threshold pump rate r_{p}≈1.43 ps^{−1}. Spectral narrowing of the emission line (insets of Fig. 3c) above the lasing threshold is due to the buildup of phase coherence in the SL mode, while below threshold we observe amplified spontaneous emission with a spectrum that matches the spectral density of states of the TM_{2} mode at k=0.
Owing to a vanishing group velocity of the lasing TM_{2} mode, the LDOS is strongly enhanced around the frequency of the SL points (Supplementary Fig. 2). The TE_{2} mode, which shares the SL point at k=0 with the TM_{2} mode (Fig. 1), offers a competing channel for spontaneous emission but its LDOS reaches only about onetenth of the value of the TM_{2} mode due to its stronger dispersion. Consequently, 90% of the spontaneous emission are channelled into the dispersionless TMpolarized lasing mode, which is confirmed by the spontaneous emission factor obtained from the threshold curve, β=γ_{c}/(γ_{c}+Γr_{p})≈0.9 (ref. 30) (total loss rate γ_{c}≈3.71 ps^{−1}; Supplementary Fig. 3). In contrast to microcavity lasers, where spontaneous emission channels can be suppressed via cavity design, the concentration of the spontaneous emission into the localized SL lasing mode is a result of the large density of kmodes that fall within the bandwidth of the gain. The observed highβ characteristics and picosecond relaxation oscillations of cavityfree SL lasing can potentially allow for the design of thresholdless plasmonic laser diodes that can be modulated with THz speeds.
Resilience to surface roughness
It has been shown that even small structural imperfections can have a significant impact on the propagation characteristics near SL points in slow light devices^{20}. The increase of scattering at interfaces not only imposes limitations on the achievable group index but can also result in Anderson localization of light, as has been demonstrated for photonic crystal waveguides^{31}. In metal dielectric heterostructures, however, SL arises from negative energy flow inside the metallic cladding and not from backscattering at geometric features^{32}. It is therefore important to determine the sensitivity of the SL feedback mechanism to structural imperfections that are likely to be present in any experimental realization.
We study the influence of surface roughness on SL lasing by applying a normally distributed variation of surface roughness (height in y position) at the interfaces between the core and the cladding layers along the length of the waveguide with a fixed step size of 10 nm in x direction. The autocorrelation of the resulting roughness profiles follows a Gaussian distribution with an associated correlation length of ~8.5 nm. SL lasing is investigated in structures with roughness heights of rootmeansquare (r.m.s.) values between 0 and 3 nm. For each r.m.s. height, the total loss rate is averaged over ten lasing configurations (shown in Fig. 4).
We find that SL lasing remains resilient to surface roughness with almost no variation in the average loss rate below r.m.s. values <1.5 nm. Above this r.m.s. value, however, both the average loss rate and s.d. steadily rise. In addition, for r.m.s. heights above 2.5 nm, an increasing number of simulations do not show SL lasing; for example, ~50% of the configurations at 3 nm r.m.s. surface roughness do not lase. The higher losses arise primarily from the scattering of the modal fields at the rough interfaces and an increased absorption of energy at hotspots, surface irregularities over which the SL lasing mode localizes as demonstrated in the mode profiles for different levels of r.m.s. roughness (insets of Fig. 4). This remarkable resilience of SL points to surface roughness can also be observed in the passive metaldielectric heterostructure (Supplementary Fig. 4) by measuring the rise of the average energy velocity with the interfacial roughness.
Discussion
Our analysis of the lasing dynamics shows that the lasing mode localizes inplane to the subwavelength gain region even though there is no cavity that would enforce this behaviour. To explain the mechanism that governs the formation of the subwavelength modes, we consider two distinct cases: SL nanolasing in the previously introduced leakymode design and SL spasing using an alternative plasmonicmode design that allows for plasmon amplification.
In contrast to conventional singlemode lasing, where the lasing frequency and mode profile are determined by the cavity, the SL lasing mode is characterized by a closedloop energy vortex (Fig. 3c) and forms dynamically by phaselocking the continuum of kmodes in the vicinity of the SL points. During the transient phase, the lasing mode acquires a spectral composition that maximizes the effective gain G(ω)∝∫dk (ω−ω_{0}) · D(ω, k) · sinc^{2}(kw/2) =∫dk g(ω, k) by balancing the spatial confinement of the lasing mode to the gain section against its spectral overlap with the gain line (ω−ω_{0}). The density of states D(ω, k)∝γ(k)/[(ω(k)−ω)^{2}+γ^{2}(k)] of the (TM_{2}) waveguide mode plays a critical role in this process as it links and controls the spatial and spectral overlap. The close agreement of the analytically predicted frequencies ω_{SLL} and kspectra, g(ω_{SLL}, k), with those retrieved from full spatiotemporal simulations (Fig. 5c) confirms this principle: the frequency ω_{SLL} of the SL lasing mode is selected by maximization of G(ω), and its mode shape is determined by g(ω_{SLL}, k). The results further suggest that there is no fundamental limit for the spatial compression of the mode, although a higher localization requires an increasingly flat dispersion to maintain a high spectral overlap.
With bound plasmonic modes one can exploit a wider range of wavevectors and achieve considerably smaller effective mode volumes. We optimized the waveguide dispersion (Fig. 6b and Supplementary Fig. 5) to exhibit two SL points in the plasmonic TM_{1} mode at ω_{1}/2π=128.14 THz (λ_{1}≈2,339.6 nm) at k_{1}=7.41 μm^{−1} and ω_{2}/2π=123.89 THz (λ_{2}≈2,419.7 nm) at k_{2}=30.34 μm^{−1}. There is no outcoupling to propagating freespace waves and the modal loss is approximately three times larger than the total loss of the TM_{2} waveguide mode due to the higher field localization. Figure 6b shows the predicted coupling strength g(ω,k) for a plasmonic SL lasing structure. We choose to tune the emission line of the gain medium to the frequency of the second SL point ω_{2} (Supplementary Table 3) and consequently shift the peak of the the sinc^{2}function to k_{2}. The broad spectrum of supported wavevectors allows for localization of the SL lasing mode down to gain sections with a width of w=200 nm. For these small gain sections, the maximum of the field profile aligns over the gain, effectively locking the amplitude and phase of the complementary modes with positive and negative phase velocities in space. As the spectrum of wavevectors is centred at a nonzero wavevector k_{2}, the lasing mode becomes spatially modulated in propagation direction with nodes every π/k_{2}. For wider gain sections, the positions of the nodal planes can move in time depending on the contributions to the lasing mode from the counterpropagating waveguide modes. This behaviour is a direct consequence of the cavityfree property of SL lasing, where forward and backward modes become uncoupled due to the absence of backreflections at geometric boundaries (compare with, for example, Fabry–Perot lasers), and will be studied in more detail in a future publication.
In both cases, the SL lasing/spasing mode emerges from the mode locking of passive modes via the gaincoupling factor g(ω,k). Excellent agreement is observed between the semianalytical analysis and the numerically extracted spectra for both the lasing frequency and the wavevector spectrum over a range of gain section widths (Fig. 6c). Indeed, the results confirm that SL lasing is enabled by a universal mechanism of SLassisted mode formation that optimizes the net gain in these structures without being limited by diffraction.
With the planar structure being isotropic in two dimensions, the generalization to three dimensions is straightforward. By introducing a small cylinder of gain material (or, for example, a quantum dot) into the waveguide core layer or by spatially selective excitation of a homogeneous gain layer, one can conceivably construct a ‘photonicdot’ laser that, evanescently pumped by a nearfield tip, would receive feedback by formation of a threedimensional SL vortex. In the vicinity of the metal layer, we observe very high gain coupling factors due to the combined action of a vertical plasmonic confinement and the inplane stopping of light, which together have an impact on the local density of states. The presented metaldielectric SL structures are therefore predestined for achieving strong coupling in plasmonic structures, as recently predicted theoretically^{33}. SL lasing may thus open the door to cavityfree nanolasing, ultrathin lasing surfaces and cavityfree quantum electrodynamics. Through a flexible (multipoint) localization of lossless or amplified surface plasmon polaritons, the SL scheme also provides an entry point to quantum gain in quantum plasmonics^{33} and quantum fluids of light^{34}.
Methods
Complexfrequency dispersion analysis
The dispersion and loss curves for the transverse magnetic (transverse electric)polarized modes of the planar waveguide were calculated semianalytically using a modified transfer matrix method^{35}, which allows for extraction of the complexω modes that remain valid solutions at the SL points. Let us consider a planar metaldielectric stack structure of N layers of thickness d_{i} (i=1…N) that are stacked up in y direction and extend infinitely in the x–z plane. For isotropic media, the field component ?_{z}, orthogonal to the propagation direction x, can be written as
where m_{i} is the layer permittivity (permeability) for ?_{z}=H_{z} (?_{z}=E_{z}). Momentum conservation dictates that the y component of the wavevector depends on the x component k (also called propagation constant). The boundary conditions between individual layers translate into
where the matrices M_{i} are given by
The bound mode profile leads to the initial condition in the superstrate, while in the substrate the condition can be expressed as
where define the decay constants in the cladding layers.
For plasmonic and active layer structures, the characteristic function F(ω, k^{2}) is a complexvalued function. Its roots lie on a hyperplane and can be found efficiently using a combination of the argument principle and the Newton–Raphson methods.
In practice, when solving the characteristic equation one considers one of the arguments (ω or k^{2}) as a real parameter and the other as complex eigenvalue. For real ω, the complexk eigenvalue describes a mode with spatially decaying (or increasing) amplitude, a solution that resembles those obtained by cw mode injection into the waveguide. These solutions do not exist at the SL points, as the corresponding mode cannot propagate. Instead, one needs to consider complexω modes (with a fixed, real k). As these modes decay in time (rather than space), they do not show the typical back bending and mode splitting of the complexk dispersion curves at the SL points.
The roots of the dispersion function can be classified according to their behaviour in the cladding layers. The bound modes, characterized by an exponential decay in the super and substrate (Re[γ_{1}], Re[γ_{N}] > 0), are associated with a discrete spectrum of eigenvalues. In contrast, the radiative continuum is described by modes with a purely oscillatory behaviour (Re[γ_{1}],Re[γ_{N}]=0)^{36}. The third class of modes, socalled leaky modes, show a divergent behaviour in the cladding. These modes, despite not being part of the complete basis set, are observable as resonances in the reflection spectrum. For the (weakly leaky) structure under investigation, leaky modes accurately capture the changes to the modal dispersion and the additional radiative loss caused by the coupling of the waveguide mode to the continuum of free space modes.
MaxwellBloch Langevin timedomain simulations
The timeresolved simulations were performed using the finitedifference timedomain method in combination with auxiliary differential equations that describe the material dispersion of the metal cladding and the nonlinear, spatially resolved polarization response of the gain media. Using the MaxwellBloch formalism we modelled the gain medium as a fourlevel system with pump (0→3) and emission (2→1) transitions^{37} that we associate with polarization densities P_{a}(r, t) and P_{e}(r, t). The dynamics of the polarization densities can be mapped on a set of differential equations
which describe the dynamics of light–matter interaction via the dipole coupling of the electric field to the electronic transitions. The resonance frequencies ω_{i} (and transition frequencies ), the spectral halfwidths γ_{i}, the phenomenological coupling constants (that are connected to the emission and absorption crosssections σ_{0,i}), as well as the real part of the host refractive index are determined by the choice of gain material.
The temporal evolution of the occupation densities N_{0} to N_{3}, which in turn determine the inversions ΔN_{e}(r,t)=N_{2}(r,t)−N_{1}(r,t) and for emission and absorption, is given by the set of equations
The lifetimes τ_{jk} account for nonradiative relaxation processes between the levels. When considering a spatially homogeneous and constant local pump rate , the term connected to the polarization density of the absorption transition, (ℏ ω_{r,a})^{−1}(∂P_{a}/∂t+Γ_{a}P_{a})E, can be replaced by .
Dissipationinduced fluctuations of the polarizations and carrier densities are included in the MaxwellBloch equations via Langevin noise terms^{37}. The noise terms are given by F_{e}(r, t), F_{a}(r, t) and F_{00}(r, t) to F_{33}(r, t), where N_{cell} is the number of fourlevel systems per unit volume cell of the finitedifference timedomain grid. Below the lasing threshold the noise acts as source for amplified spontaneous emission giving rise to the recorded background spectra (Fig. 3b). Above the lasing threshold, the noise acts as a seed aiding the transient buildup of the coherent lasing fields through feedback and stimulated emission processes in the gain medium.
Structural and material parameters
The planar metaldielectric stack structures comprises an optically thick metal substrate, a thin dielectric core layer and a second (thin) metal cladding layer. Further control over the modal dispersion is achieved by introducing a dielectric layer on top of the stack structure when necessary^{26}. In our analysis, we assume a constant permittivity of ε_{s}=11.68 for the dielectric layers, a value that is typical for III–V semiconductors (such as InGaAsP). The dispersive permittivity of the metal layers, in this case a transparent conductive oxide, is described using a Drude model. In accordance with experimental data^{38}, we use ω_{p}=3.13·10^{15} rads^{−1} for the plasma frequency of indium tin oxide. The collision rate γ is assumed to be reduced by a factor of 10 to 1.07·10^{13}rads^{−1} as could be achieved by fabricating films of very high quality and cooling them to low temperatures^{5}. In comparison to metals such as silver or gold, the plasma frequency for, e.g., indium tin oxide can be controlled through doping and can be shifted into the visible wavelength range to around 600 nm making this material a suitable option for SL lasing in the nearinfrared.
Genetic optimization of broadband dispersionless modes
The principle of subwavelength SL lasing relies on the formation of a flat band in the modal dispersion that maximizes the range of wavevectors contained within the frequency bandwidth of the gain. The flatness of the band can be achieved by introducing two or more SL points, which enforce a monotonous variation of the dispersion in between them. We employ a genetic algorithm technique to isolate structures of a given design that exhibit two SL points and the smallest possible frequency variation in a selected mode.
Every planar metaldielectric stack structure is uniquely characterized by the number of layers, the layer thicknesses and their permittivities, forming an ordered parameter set, the socalled chromosome of the genetic algorithm. On each iteration of the algorithm, we hold a population of N chromosomes, which are evaluated with respect to a fitness function that returns a single numerical value representing suitability of the structure. The population is sorted to this order and a small percentage of the fittest chromosomes are passed to the next generation unchanged. The remaining places in the new population are filled by ‘tournament selection’, where each tournament is a subset of the current population whose members are chosen at random. The two fittest chromosomes from each tournament are then mutated by randomly perturbing parameters of the chromosome and optionally crossed over by swapping sections of the parameter list between the two tournament winners. The two new chromosomes are added to the next population, and the process of tournament selection is repeated until the population set is complete. The progress of each iteration can be monitored by considering the fittest chromosome in the population and the algorithm is terminated once this chromosome surpasses a threshold fitness.
For the fitness function, we first evaluate the dispersion relation of a given chromosome (that is, structure) and perform a root find for two SL points. If the calculation fails during this step, the function returns to zero, otherwise we determine the average band velocity and return its reciprocal value
The structure with the highest value of f is deemed fittest. It is common to initialize the population with chromosomes representing structures already known to exhibit two SL points, which may in turn be found by optimization using a different, appropriate fitness function.
Additional information
How to cite this article: Pickering, T. et al. Cavityfree plasmonic nanolasing enabled by dispersionless stopped light. Nat. Commun. 5:4972 doi: 10.1038/ncomms5972 (2014).
References
Hill, M. T. et al. Lasing in metalliccoated nanocavities. Nat. Photon. 1, 589–594 (2007).
Oulton, R. F. et al. Plasmon lasers at deep subwavelength scale. Nature 461, 629–632 (2009).
Noginov, M. A. et al. Demonstration of a spaserbased nanolaser. Nature 460, 1110–1112 (2009).
Ma, R.M., Oulton, R. F., Sorger, V. J., Bartal, G. & Zhang, X. Roomtemperature subdiffractionlimited plasmon laser by total internal reflection. Nat. Mater. 10, 110–113 (2011).
Khajavikhan, M. et al. Thresholdless nanoscale coaxial lasers. Nature 482, 204–207 (2012).
Lu, Y.J. et al. Plasmonic nanolaser using epitaxially grown silver film. Science 337, 450–453 (2012).
Vahala, K. J. Optical microcavities. Nature 424, 839–846 (2003).
McCall, S. L., Levi, A. F. J., Slusher, R. E., Pearton, S. J. & Logan, R. A. Whisperinggallery mode microdisk lasers. Appl. Phys. Lett. 60, 289–291 (1992).
Iga, K., Koyama, F. & Kinoshita, S. Surface emitting semiconductor lasers. IEEE J. Quantum Electron. 24, 1845–1855 (1988).
Painter, O. et al. Twodimensional photonic bandgap defect mode laser. Science 284, 1819–1821 (1999).
Altug, H., Englund, D. & Vuckovic, J. Ultrafast photonic crystal nanocavity laser. Nat. Phys. 2, 484–488 (2006).
Bergman, D. J. & Stockman, M. I. Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems. Phys. Rev. Lett. 90, 027402 (2003).
Berini, P. & De Leon, I. Surface plasmonpolariton amplifiers and lasers. Nat. Photon. 6, 16–24 (2011).
Hess, O. et al. Active nanoplasmonic metamaterials. Nat. Mater. 11, 573–584 (2012).
Khurgin, J. B. & Sun, G. Comparative analysis of spasers, verticalcavity surfaceemitting lasers and surfaceplasmonemitting diodes. Nat. Photon. 18, 10–14 (2014).
Cao, H. Lasing in random media. Wave Random Media 13, R1–R39 (2003).
Wiersma, D. S. The physics and applications of random lasers. Nat. Phys. 4, 359–367 (2008).
Hess, O. & Tsakmakidis, K. L. Metamaterials with quantum gain. Science 339, 654–655 (2013).
Krauss, T. F. Slow light in photonic crystal waveguides. J. Phys. D Appl. Phys. 40, 2666 (2007).
Baba, T. Slow light in photonic crystals. Nat. Photon. 2, 465–473 (2008).
Ibanescu, M. et al. Anomalous dispersion relations by symmetry breaking in axially uniform waveguides. Phys. Rev. Lett. 92, 063903 (2004).
Ibanescu, M., Johnson, S. G., Roundy, D., Fink, Y. & Joannopoulos, J. D. Microcavity confinement based on an anomalous zero groupvelocity waveguide mode. Opt. Lett. 30, 552–554 (2005).
Zhang, S., Genov, D. A., Wang, Y., Liu, M. & Zhang, X. Plasmoninduced transparency in metamaterials. Phys. Rev. Lett. 101, 047401 (2008).
Boyd, R. W. & Gauthier, D. J. Controlling the velocity of light pulses. Science 326, 1074–1077 (2009).
Economou, E. N. Surface plasmons in thin films. Phys. Rev. 182, 539–554 (1969).
Karalis, A., Joannopoulos, J. & Soljačić, M. Plasmonicdielectric systems for highorder dispersionless slow or stopped subwavelength light. Phys. Rev. Lett. 103, 043906 (2009).
Lu, C.Y., Chang, S.W., Chuang, S. L., Germann, T. D. & Bimberg, D. Metalcavity surfaceemitting microlaser at room temperature. Appl. Phys. Lett. 96, 251101 (2010).
Tsakmakidis, K. L., Boardman, A. D. & Hess, O. ‘Trapped rainbow’ storage of light in metamaterials. Nature 450, 397–401 (2007).
Naik, G. V., Shalaev, V. M. & Boltasseva, A. Alternative plasmonic materials: beyond gold and silver. Adv. Mater. 25, 3264–3294 (2013).
Ma, R.M., Oulton, R. F., Sorger, V. J. & Zhang, X. Plasmon lasers: coherent light source at molecular scales. Laser Photon. Rev. 7, 1–21 (2012).
Mookherjea, S., Park, J. S., Yang, S.H. & Bandaru, P. R. Localization in silicon nanophotonic slowlight waveguides. Nat. Photon. 2, 90–93 (2008).
Tsakmakidis, K. L., Pickering, T. W., Hamm, J. M., Page, A. F. & Hess, O. Completely stopped and dispersionless light in plasmonic waveguides. Phys. Rev. Lett. 112, 167401 (2014).
Tame, M. S. et al. Quantum plasmonics. Nat. Phys. 9, 329–340 (2013).
Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).
Kwon, M.S. & Shin, S.Y. Simple and fast numerical analysis of multilayer waveguide modes. Opt. Commun. 233, 119–126 (2004).
Kocabas, S. E., Veronis, G., Miller, D. A. B. & Fan, S. Modal analysis and coupling in metalinsulatormetal waveguides. Phys. Rev. B 79, 035120 (2009).
Pusch, A., Wuestner, S., Hamm, J. M., Tsakmakidis, K. L. & Hess, O. Coherent amplification and noise in gainenhanced nanoplasmonic metamaterials: a MaxwellBloch Langevin approach. ACS Nano 6, 2420–2431 (2012).
Noginov, M. A. et al. Transparent conductive oxides: Plasmonic materials for telecom wavelengths. Appl. Phys. Lett. 99, 021101 (2011).
Acknowledgements
We thank Kosmas L. Tsakmakidis, Rupert Oulton and Federico Capasso for inspiring discussions. This research was supported by the Leverhulme Trust and the Engineering and Physical Sciences Research Council (EPSRC) UK. This study was partially support by the Air Force Office of Scientific Research (AFOSR), and the European Office of Aerospace Research and Development (EOARD) is also acknowledged.
Author information
Affiliations
Contributions
T.P., J.M.H. and O.H. developed the concept and conceived the study. T.P. and J.M.H. designed and performed the complex frequency dispersion analysis. T.P., A.F.P. and S.W. performed the MaxwellBloch Langevin timedomain simulations. A.F.P. implemented and ran the genetic optimization. O.H. supervised the work. All authors discussed the results and wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 15, Supplementary Tables 13, Supplementary Discussion and Supplementary References. (PDF 1083 kb)
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Pickering, T., Hamm, J., Page, A. et al. Cavityfree plasmonic nanolasing enabled by dispersionless stopped light. Nat Commun 5, 4972 (2014). https://doi.org/10.1038/ncomms5972
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms5972
Further reading

Stoppedlight nanolasing in optical magicangle graphene
Nature Nanotechnology (2021)

Universal lasing condition
Scientific Reports (2021)

Dielectric slotted nanodisk laser with ultralow pump threshold by anapole excitation
Applied Physics B (2020)

Bandedge engineering for controlled multimodal nanolasing in plasmonic superlattices
Nature Nanotechnology (2017)

Anapole nanolasers for modelocking and ultrafast pulse generation
Nature Communications (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.