Complex-k modes of plasmonic chain waveguides

Before we look into realistic plasmonic metals, we study mode properties of chain waveguides made of fictitious non-dispersive 'metal' particles. Three distinct permittivity values (${\epsilon }_{m}=-5,-6.25,\mathrm{and}-25$) are considered, with background medium _b = 2.25. The calculated bands for modes identified (also called modal dispersion curves) are shown in figure 2. Note that when presenting mode dispersion curves of a periodic system made of non-dispersive material, one often uses scaling law—i.e. both frequency and wave number are normalized against some length metric (most often period of structure). In this work we consider dispersive metal materials; for ease of comparison, we choose to normalize angular frequency against ω₀ = 10¹⁵ rad s⁻¹. Wave number is normalized against 2π/a (a is chain period), as conventionally done in photonic-crystal analyses. Wave number k is in general a complex number, which is even true for all-real permittivity values used. Modes with frequency falling in photonic band gaps carry such a complex k value, with its imaginary part denoting its degree of attenuation. As a common practice for periodic systems, dispersion plots in this work only show bands in the first Brillouin zone (BZ), with positive k_r.

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The figure reveals that the chain waveguide in general possesses both T and L modes over the examined frequency range. The dispersions of modes can be quite peculiar as compared to conventional all-dielectric waveguides. The dispersion curves of T modes are seemingly stemming out of the light line of background medium at small k_r values. This feature is shared by modes in dielectric waveguides, and is fundamentally due to the fact that the T mode and the mode corresponding to the light line have the same TEM symmetry. Different from bands in dielectric waveguides, here in the plasmonic case, the T-mode dispersion curves can reach a maximum in frequency and thereafter exhibit a negative slope (represented by dotted-line sections). Negative-slope dispersion curves were previously considered for realizing light propagation with negative group velocity (-GV), and the plateau point in a dispersion curve for achieving 'stopped' light with zero GV (0GV). However, as will be shown in figure 4, dispersion close to 0GV point or with small GV in general is highly susceptible to material loss. Note that at the 0GV position, T modes from +GV and -GV sections converge and become a degenerate pair; the pair branches out towards high frequency with a steep slope, at the same time their k values turning into a complex-conjugate pair (figure 4). This is a signature that the modes from both +GV and −GV sides enter into a photonic band gap regime, as was also noticed in [10]. In contrast, for non-dispersive dielectric photonic structures, extremities of a band occurs at either center or edge of its BZ (or in general high-symmetry wavevector points). The L mode in figure 2 has a more intriguing dispersion profile. Firstly, L mode has no coupling with light line of the background medium. This is fundamentally due to the unique polarization nature of the mode. Secondly, if it does not reside in the radiation cone of the background medium (which we did not look into), the L mode persists over a very broad frequency range, even when frequency approaches zero. This suggests such a nanostructred chain waveguide (especially with small particle permittivity value) can potentially guide light with a very long wavelength. In addition, the guided mode can have a very large effective mode index ${n}_{\mathrm{eff}}={k}_{r}/{k}_{0};$ therefore deep-subwavelength guidance can be ensured.

When the particle permittivity value increases negatively, in general, the dispersion curves shift to higher frequencies. This can be explained by the fact that the particles become more and more metallic at large negative permittivity values; thereby EM field is squeezed more out of the particles. Resonance is sustained by effectively smaller and low-index space surrounding the particles, hence the higher frequencies observed.

EM field patterns of three representative modes are presented in figure 3. Only cases for _m = −6.25 are plotted. For ease of interpretation, the fields are constructed over multiple cells based on Floquet theorem to reveal full-wavelength evolutions. Panels (a) and (b) show the T and L modes at ω = 2.8ω₀, respectively. The frequency corresponds to λ₀ = 673 nm. The plots visualize clearly orientation and coupling of particle dipoles, as well as wavelength of the guided wave (thereof n_eff and mode confinement). For the T mode in figure 3(a), it has a major magnetic field directed along y direction, i.e. H_y component (as shown by the colormap in the figure). Whereas, the L mode in figure 3(b) has its magnetic field curling around the waveguide axis. Comparatively, the L mode is somewhat similar to the so-called transverse-magnetic mode in cylindrical optical fibers. Figure 3(c) shows an L mode at relatively a low frequency of 0.3ω₀, corresponding to λ₀ = 6.28 μm. Deep-subwavelength guidance at the MIR frequency is then achieved. We will discuss the possibility of achieving such guidance using e.g. doped silicon in section 4.

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L mode is heavily affected by particle spacing. A study with a period of 100 nm (particle permittivity remains at −6.25) shows that the L-mode band is markedly blue-shifted, by ∼21% at ${k}_{r}=0.3[2\pi /a]$ (where a = 100 nm), whereas the T-mode band is almost unchanged (increase in frequency less than 2% at the same wave number). The corresponding dispersion curves can be found in 'Supplemental data available online at stacks.iop.org/JPCO/3/115015/mmedia'. Smaller particle tends to increase band frequencies. This is confirmed with a calculation of dispersion curves for a chain waveguide with a = 80 nm and d = 40 nm, again with the same particle permittivity (see also 'Supplemental data').

In the above sub-section, only lossless chain waveguides were studied. As a matter of fact, material losses associated with negative-epsilon materials are usually quite significant. Here we look into the effect of adding an imaginary part to the particle permittivity. Still, the considered particles will have non-dispersive permittivity. Similar to the previous sub-section, the focus is on a chain waveguide with a = 80 nm, d = 60 nm, and ${\epsilon }_{b}=2.25$. A frequency-independent loss tangent of 0.01 is added to the material, i.e. ${\epsilon }_{m}={\epsilon }_{r}+{\epsilon }_{i}i=-6.25-0.062\,5i$. _i is negative for lossy material for the time-harmonic convention used.

In figure 4(a), dispersion curves obtained for the lossless and lossy chain waveguides are compared. As mentioned previously, even with all-real permittivity values, the mode solver obtains modes with complex k, more specifically for modes falling in photonic band gaps. When material loss is introduced, modes at all frequencies carry complex k values. The imaginary part of k (k_i) is plotted in figure 4(b). For a reason to be clarified in the following paragraph, our trace of dispersion usually stops when k_i reaches ∼0.1. There are two observations worth commenting for figure 4. First (already pointed out in last sub-section), when loss is added, dispersion curve close to an originally 0GV point is highly affected. For the T mode, there are two 0GV points, one at the plateau point of its dispersion curve and the other at the BZ boundary. With material loss, the slopes of the dispersion curves at those positions tend to increase rather than to approach zero. At the same time, k_i increases sharply. A direct consequence is that T mode with -GV (dashed black line) becomes highly lossy. The more useful T-mode dispersion curve (remaining +GV section, solid black curve) bends back in k_r before reaching the plateau point, towards the originally degenerate mode-pair branch. The bending starts sooner as material loss increases. This observation suggests that small-GV modes are highly vulnerable to material losses; 0GV is simply not possible. Similar finding on effect of material loss on GV was also described in our previous investigation regarding a light absorbing structure [35]. The second observation is that dispersion-curve sections with negative slopes (dotted lines in figure 4) are associated with modes with amplifying amplitudes as they propagation in +x direction, which is manifested by their positive k_i values. In the current study, such amplifying mode propagation applies to both the upper section of L band and the -GV section of the T band. The effects and consequences of such amplifying modes will be investigated in a separate study.

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Propagation length L_p of a mode (distance for mode intensity decreasing to its 1/e) is calculated from imaginary part of wave number as ${L}_{p}=-1/(2{k}_{i})$, i.e. L_p inversely proportional to k_i. When amplifying modes are concerned, k_i is positive and the distance in absolute value corresponds to a mode's intensity increased to its e times; we can refer to this length particularly as amplification length. For reference, a mode with ${k}_{i}=-0.1\left[\tfrac{2\pi }{a}\right]$ with a = 80 nm has a propagation length of 64 nm. This is already smaller than the waveguide's period, rendering therefore such a waveguide almost useless at the frequency considered. We plot in figure 5 the propagation lengths for modes of the waveguide at two loss tangents (0.1 and 0.01). The plot can be examined together with figure 4. Amplification lengths are plotted in negative values. The T mode is found to be extremely sensitive to frequency. Its L_p can be relatively long (beyond 1000 nm) in a short frequency range; however the long-L_p modes are quite close to the light line and therefore their modal confinements are compromised. The L mode can sustain a consistent level of propagation length over a large frequency range. Comparing the propagation lengths at two loss levels, one finds the propagation length is roughly inversely scaled to the imaginary part of metal permittivity. Subwavelength mode confinement of L modes can be guaranteed as its dispersion curve can stay quite far below the light line, especially when small negative permittivity is used for particles (see figure 2). This observation hints that it would be interesting to look for new materials with small negative permittivities for subwavelength light guiding at especially low frequencies.

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In figure 6 we lump together dispersion curves calculated for a silver chain waveguide (blue curves) and those for a gold chain waveguide (red curves), both with a = 80 nm, d = 60 nm, and _b = 2.25. In general the dispersion curves for the silver chain waveguide appear at higher frequencies (3.4–4.8 ω₀, or 550–400 nm in free-space wavelength, respectively), while the modes for gold chain waveguides appear at 3–3.6 ω₀ (630–520 nm in free-space wavelength, respectively). This is in agreement with the fact that the plasma frequency of silver is higher than that of gold. As suggested by permittivity values, the silver chain waveguide suffers less propagation loss compared to the gold counterpart. For the gold waveguide, the material is so lossy that its T-mode dispersion curve tends to fold back as soon as it stems out of the light line; as a result, the T-mode dispersion curve nearly laps over the background light line and the mode is therefore quite delocalized. The L mode has better confinement but it carries a k_i around 0.1, hence difficult to channel light across a distance larger than a period. The silver waveguide is comparatively less lossy. At ω = 3.91ω₀ and below, the T mode has L_p > 1000 nm, with diverging L_p as frequency decreases (to L_p = 54 μm at ω = 3.4ω₀). The L mode has L_p > 1000 nm when ω < 4.17ω₀ (until 4ω₀ where L_p = 1890 nm). The results suggest that silver chain, especially its T-mode, can be potentially used for channeling light at limited visible wavelength ranges. That said, it should be pointed out that not all guidance is subwavelength; modes with long L_p can be due to that they stay close to the light line, therefore with their fields extended in the background medium. One has to make a careful comparison to other types of waveguides, plasmonic or not, in terms of some figure of merit based on loss and mode confinement. The L mode tends to have deep-subwavelength guidance. However, it has to be used with extreme care, since it only achieves larger than 1000 nm propagation in a less than 20 nm wavelength window, which can be further subject to experimental imperfections.

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It was pointed out in [29] that smaller inter-particle separation can increase plasmonic coupling and therefore increase propagation length of chain mode. Extremely small separation of ∼2 nm was used in a very recent demonstration [20], and the work claimed a L-mode propagation length of 190 nm for a gold chain. In comparison, the gold chain presented in figure 6, which differs predominately with a larger particle separation of 10 nm, can hardly propagate its L mode over one period. To verify the critical role played by inter-particle distance, here a gold chain with a = 44 nm, d = 42 (hence 2 nm gap), _b = 2.25 was modeled with our FEM solver. This structure is very close to what was experimentally characterized in [20]. In section 3, we concluded that a decrease in gap size will decrease mode frequencies (especially L mode), and a decrease in particle size will increase mode frequencies. Here the dominant effect is owing to the extremely small gap size. If we compare modal curves of the current structure to those of the gold chain in figure 6, its T-mode dispersion curve remains almost unchanged, whereas its L-mode dispersion curve down-shifts to 2.65–3.15 ω₀ (710–600 nm in wavelength). Propagation length is found to be as high as 800 nm at 710 nm wavelength, which is reduced to tens of nanometers at short-wavelength end. Still, this propagation length is in average seven times better than that of the gold chain in figure 6. Our numerical finding in general supports the experimental observation of over 350 nm energy transfer in the gold chain presented in [20], although the dispersive nature of the modes was not mentioned in [20]. From our numerical investigation, the deciding factor for achieving longer propagation length (compared to the gold chain in figure 6) is that the L mode appears at a lower frequency where gold material has less absorption.

In figure 1(c), a heterogeneous chain waveguide is illustrated, where two types of metal spheres are interleaved. Such design was motivated by [25]. It was proposed in [25] that by inserting a silver nanoparticle in between two gold nanoparticles one can facilitate more efficient energy transfer between two gold particles. The argument was tested with plasmonic trimer structures, which were realized by a delicate DNA-based self-assembly procedure (method also used in [20]). The loss reduction in energy transfer was argued through reduced bandwidth of observed dark-field scattering spectrum. Energy transfer across a longer chain beyond the trimer structure was not discussed. Here we extend the idea to a hetero-plasmonic chain waveguide, and numerically check its modal properties including propagation length. It is worth noting that, in a finite Au-Ag-Au trimer structure, it was the energy transfer process between two gold particles that was examined and was found to be more efficient compared to two directly coupled gold particles; in an infinite hetero-plasmonic waveguide, energy transfer can happen through hopping between silver particles, which also has to be considered.

The hetero-plasmonic chain waveguide to be studied comprises of simultaneously silver and gold nanoparticles. Our FEM analysis uses a supercell with a period of 80 nm, including a gold particle of diameter 40 nm and a silver particle of diameter 30 nm (thereof a gap size of 5 nm). _b = 2.25. The geometrical parameters as well as the background permittivity are very close to those studied in [25], where gold-to-gold distance was 78 nm and background was assumed to have _b = 2.15. It turns out that the hetero-plasmonic waveguide under study is a very lossy waveguide—key modal features are killed by the presence of heavy material losses. In order to better explain the guidance mechanism, we first calculated its dispersion curves with imaginary parts of gold and silver permittivities reduced by a factor of ten¹ . The resulted dispersion curves for the guided modes are shown in figure 7(a) by the black curves. In order to know the nature of the modes we plot three representative modal fields, one for each dispersion curve, in figure 8. Examination of the mode fields reveals that the two bands around 3.4ω₀ correspond to T and L modes dominantly contributed by resonance in gold particles; and the mode around 4.5ω₀ frequency is a T mode dominantly contributed by resonance in silver particles. The fact that one type of particles is clearly in resonance (dipolar) while the other is not (acting somewhat like spacers) suggests that a hetero-plasmonic waveguide can rather be treated as two superposed 'homo-plasmonic' chain waveguides. Each homo-plasmonic chain contain a single-type particles with relatively large inter-particle spacings.

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In figure 7(b) we present the dispersion curves of gold/silver homo-plasmonic chain waveguides, both with period a = 80 nm but particle diameter d = 40 nm for the gold case and d = 30 nm for the silver case. The gold chain waveguide has T and L bands around 3.5ω₀, corresponding well to the lower set of modes in figure 7(a). The silver chain waveguide has T and L bands around 4.5ω₀ with the T mode corresponding well to the upper T band in figure 7(a). The L mode in silver chain waveguide finds no counterpart in the hetero-plasmonic case. The lack of L mode for the hetero-plasmonic chain waveguide is due to the fact that gold is too lossy at the frequency. The effect of gold can be substantial as the electric field of expected L mode has to majorly pass through the gold particles. If one examines modes' propagation lengths (see 'Supplemental data'), one can conclude that: by superimposing two homo-plasmonic chains into one hetero-plasmonic chain waveguide, the chain modes supported by silver particles suffer higher losses with the L mode disappearing completely; for the gold chain, its T mode becomes a bit lossier, but the L mode can propagate much longer. More specifically, the L mode of gold chain (figure 7(b)) has propagation length ranging from tens of nanometers at its high-frequency end to 400 nm at its low-frequency end (3.470ω₀); the corresponding L mode in hetero-plasmonic chain (figure 7(a), solid black line) has 140 nm at its high-frequency end and ∼1900 nm at the low-frequency end (3.215ω₀). We refer to the gold-dominated L mode in the hetero-plasmonic chain waveguide as L(Au*) mode, and similarly for others.

In [25], the resonant modes were probed through optical scattering spectra. If translated onto our figure 7(a), their observed modes are within the light cone, close to the k_r = 0 axis if it was nearly normal incidence). Their experiment recorded the dominant resonance peak [corresponding to L(Au*) mode] shifted 'from 549 nm for the AuNP (gold nanoparticle) homodimer to 586 nm for the heterotrimer structure'. These two peaks are respectively 3.456ω₀ and 3.214ω₀ in frequency, which project quite well to the curves that we have simulated. In [25], the increase in mode quality was explained through concepts such as 'plasmonic hotspots' and 'quasi-resonant virtual state' of silver particle, however we argue that the increase of mode quality is merely due to the fact that inclusion of silver particles decreases the L(Au*) mode to a lower frequency where gold has smaller absorption loss. Losses of silver material does not adversely affect mode quality of the L mode significantly, since within that frequency gold has a loss tangent in average ten times as large as that of silver.

When metals take their 100% material losses, there is only one gold-dominated L mode remaining, as shown by the green curve in figure 7(a). Unfortunately its propagation length is found to be less than 100 nm.

The phenomenon of enhanced energy transfer is geometry-dependent. Besides the structure examined above, we also looked into a hetero-plasmonic waveguide with gold and silver particles with identical diameter d = 60 nm and a gap size of 20 nm. The gold-dominated L mode disappeared, possibly due to too large separation between the gold particles.

Philosophically, it can be argued that that a hetero-plasmonic chain waveguide is an improved gold-chain waveguide or a deteriorated silver-chain waveguide.

We observed, through FEM analyses, that one can insert dielectric (e.g. silicon or TiO₂) particles in a plasmonic chain to create hybrid plasmonic-dielectric chain waveguide. Whether such configuration has obvious advantage over singly-plasmonic chain waveguide is subject to further investigation.

The initial investigations on plasmonic chain waveguide used air (or vacuum) as background medium [1, 8], while we have so far focused on dielectric (glass) background. To see the effect of background, we calculated dispersion curves of T and L modes of a silver-in-air chain waveguide with a = 80 nm and d = 60 nm (see 'Supplemental data'). Compared to the corresponding curves of the silver-in-glass chain waveguide (figure 6), both bands experience a blue-shift in frequency when air is used. The respective shifts are roughly 26% for the T mode and 18% for the L mode, such that two bands are crossing each other in the air-background case. Direct comparison of propagation losses is not simple, as in two cases both modes are dispersive: propagation loss is highly dependent on frequency for each mode. In general, use of air background does not markedly increase the propagation length of T mode; it actually shortens the propagation length of L mode as a result of more flattened dispersion curve of the mode. The L-mode band becomes more flat as a result of effectively shortened gap size between the particles—the chain mode is therefore made of more localized gap plasmons with less coupling among the 'hot spots'.

Actual nanofabrication of chain waveguides can lead to a background that is more complex than a homogeneous medium. A more realistic version is a chain made of nanoparticles sitting on top of a dielectric substrate. Such geometry is common for chain waveguides made from the template-assisted self-assembly or DNA-linking process. Previously, the author and his colleagues have reported experimental demonstration of arrayed gold nanoparticles [36–38] on a dielectric substrate based on light-induced rapid thermal annealing of lithographically patterned rectangular metal patches. The gold nanoparticles can have perfectly spherical shape, except with flattened bottom, similar to droplets on a surface. The dome shapes are formed as a result of surface tension when they are melted. A chain waveguide made of such metal particles is schematically shown in figure 1(c). Inclusion of a substrate is problematic for previously used theoretical methods [8–13, 16], but not for our FEM approach. We choose silver dome-shaped nanoparticles for the following case study. In passing, we comment that one can have other particle shapes like cubes and rods [22]. The chain waveguide we are to study has the following parameters: a = 80 nm, d = 60 nm, with a quarter of the sphere height truncated in the bottom. The substrate has _s = 2.25. Rest of the background is air/vacuum.

The dispersion curves of modes identified as well as their k_i magnitudes are plotted in figure 9. It is known that when a substrate is included, or when the particles depart from spherical symmetry, the degeneracy of the T-mode pair will be lifted. In other words, two T-mode dispersion curves will appear, one with major electric-field component directed perpendicular to the substrate (T_⊥ mode) and the other with its major polarization parallel to the substrate while perpendicular to waveguide axis (T_∥ mode). The two T-mode dispersion curves are depicted in figure 9(a). T_∥ mode appears at frequency around 4.4ω₀, lower than that of the T_⊥ mode (around frequency 4.9ω₀). The T_∥ mode has L_p < 250 nm, which is due to the mode is closely coupled to radiation mode in the substrate material. The T_⊥ mode carries a negative GV in general; its k_i has a quite large magnitude in average. The L mode exists in a relatively large frequency range with L_p < 800 nm. Overall, the propagation lengths obtained are not encouraging for information transfer purposes. However, such structure has an exposed surface which can be further functionalized for enhanced nonlinear-optics and sensing applications, especially with the L mode exhibiting strong gap-plasmon resonances.

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We mention that the waveguide geometry has not been optimized. It should be possible to tailor the particle shapes as well as to add more thin substrate layers for further tuning of mode characteristics.

The dispersion diagram in figure 2 hints that if one has a small negative permittivity (−10 < _m < 0) at infrared wavelength (e.g. 0.2 ∼ 0.5ω₀, or correspondingly 9.4 ∼ 3.8 μm), deep-subwavelength guidance of EM wave is possible. Such MIR waveguides can be extremely interesting for integrated gas sensors. Here we propose that one can possibly use heavily doped silicon to achieve the desired negative permittivity and consquently a MIR chain waveguide. It is already known that doped semiconductors can become metallic at long EM wavelengths. Drude model can be used for describing permittivity of doped silicon, as $\epsilon (\omega )={\epsilon }_{\infty }-\tfrac{{\omega }_{p}^{2}}{{\omega }^{2}\,+\,i\tfrac{\omega }{\tau }}$, where ${\epsilon }_{\infty }$ is permittivity at high-frequency limit, ω_p is plasma frequency, and τ is collision time of free carriers. Plasma frequency is calculated as ${\omega }_{p}^{2}=\tfrac{{{Ne}}^{2}}{{m}^{* }{\epsilon }_{0}}$ with N doping concentration, e elementary charge, m^* effective mass of free carrier, and ₀ vacuum permittivity. Following the numerical values adopted in [30], we choose n-doped silicon with ${\epsilon }_{\infty }=11.7$ and ${m}^{* }=0.272{m}_{0}$ (m₀ is electron mass). Doping concentration chosen for the following case study is at $N=2\times {10}^{20}{\mathrm{cm}}^{-3}$. Electron collision time can calculated from measured carrier mobility as $\mu =e\tau /{m}^{* }$. μ in [30] was stated as $\mu =50{\mathrm{cm}}^{2}/({\rm{V}}\cdot {\rm{s}})$ at the mentioned doping concentration, which led to τ = 7.73 fs. We find that with this collision time, Si permittivity has an imaginary part comparable to its real part, which is too lossy for making a waveguide. It was reported in [39] that electrons' mobility can increase significantly at lower temperature. At a doping concentration of $1.3\times {10}^{17}{\mathrm{cm}}^{-3}$, μ increases from $\sim 500\,{\mathrm{cm}}^{2}/({\rm{V}}\cdot \ {\rm{s}})$ at room temperature to over $4000\,{\mathrm{cm}}^{2}/({\rm{V}}\cdot \ {\rm{s}})$ at 50 K. Although there were no explicit experimental data for higher doping concentration scenario, here for a theoretical exploration, we assume a certain low temperature can increase μ by a factor of eight for the considered doping concentration. Collision time increases correspondingly to τ = 387 ps. The overall objective of the setting, as inspired by figure 2, is to obtain a relatively small (less than 10) negative permittivity at MIR frequency such that deep-subwavelength MIR guidance can be realized using a chain waveguide made of such doped semiconductor materials. This idea can extended naturally to chain waveguide design at even longer wavelengths.

By using the standard geometry in this work, i.e. a = 80 nm and d = 60 nm, together with the above-mentioned particle material setting, we find an L mode appearing around frequency 0.4ω₀, as shown by figure 10(a). Inset in figure 10 gives the real and imaginary parts of silicon's permittivity at the relevant wavelength range. The background medium has standard _b = 2.25. The imaginary part of the mode's propagation constant (figure 10(b)) is however found to be quite large in general. After conversion, the corresponding propagation length hardly goes beyond 100 nm, with the longest at 96 nm when ω = 0.405ω₀.

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