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Minimal spaser threshold within electrodynamic framework: Shape, size and modes.

Arnold N, Hrelescu C, Klar TA - Ann Phys (2015)

Bottom Line: Here, we derive this result from the purely classical electromagnetic scattering framework.Furthermore, retardation is included straightforwardly into our framework; and the global spectral gain minimum persists beyond the quasi-static limit.We illustrate this with two examples of widely used geometries: Silver spheroids and spherical shells embedded in and filled with gain materials.

View Article: PubMed Central - PubMed

Affiliation: Institute of Applied PhysicsJohannes Kepler University LinzAltenbergerstraße 694040 LinzAustria; Soft Matter PhysicsJohannes Kepler University LinzAltenbergerstraße 694040 LinzAustria.

ABSTRACT

It is known (yet often ignored) from quantum mechanical or energetic considerations, that the threshold gain of the quasi-static spaser depends only on the dielectric functions of the metal and the gain material. Here, we derive this result from the purely classical electromagnetic scattering framework. This is of great importance, because electrodynamic modelling is far simpler than quantum mechanical one. The influence of the material dispersion and spaser geometry are clearly separated; the latter influences the threshold gain only indirectly, defining the resonant wavelength. We show that the threshold gain has a minimum as a function of wavelength. A variation of nanoparticle shape, composition, or spasing mode may shift the plasmonic resonance to this optimal wavelength, but it cannot overcome the material-imposed minimal gain. Furthermore, retardation is included straightforwardly into our framework; and the global spectral gain minimum persists beyond the quasi-static limit. We illustrate this with two examples of widely used geometries: Silver spheroids and spherical shells embedded in and filled with gain materials.

No MeSH data available.


Related in: MedlinePlus

Universal quasi‐static threshold parameters. a) Solid curves (left scale) show the dependence of −ɛM′′/ɛM′′ɛM′ɛM′ on the wavelength λ for Ag (black) and Au (red) with Johnson and Christy values. The threshold gain −ɛG′′ can be obtained from this ratio by multiplying it with ɛG′ of the gain material. Dashed curves (right scale) show the corresponding −1/−1ɛM′ɛM′ values, which are equal to the normalized shape factor N/NɛG′ɛG′ at the threshold. Both ordinates have logarithmic scales. b) Dependences of the threshold amplification β on the background dielectric ɛG′ for 5 local minima of the −[ɛM′′/ɛM′′ɛM′ɛM′](λ) curves for Ag, as color‐coded in the legend.
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andp201500318-fig-0001: Universal quasi‐static threshold parameters. a) Solid curves (left scale) show the dependence of −ɛM′′/ɛM′′ɛM′ɛM′ on the wavelength λ for Ag (black) and Au (red) with Johnson and Christy values. The threshold gain −ɛG′′ can be obtained from this ratio by multiplying it with ɛG′ of the gain material. Dashed curves (right scale) show the corresponding −1/−1ɛM′ɛM′ values, which are equal to the normalized shape factor N/NɛG′ɛG′ at the threshold. Both ordinates have logarithmic scales. b) Dependences of the threshold amplification β on the background dielectric ɛG′ for 5 local minima of the −[ɛM′′/ɛM′′ɛM′ɛM′](λ) curves for Ag, as color‐coded in the legend.

Mentions: Figure 1a shows the dependences N(λ) (dashed) and −ɛG′′(λ) (solid) defined by the relations (11) for silver and gold with dielectric functions from Johnson and Christy 71 and normalized to the real part ɛG′ of the gain material. For ɛG′≠1 the results simply rescale, with larger ɛG′ resulting in larger threshold gain and N values, a fact that should be considered in the selection of a suitable gain medium. The global minima around 1060 nm for silver and around 750 nm for gold can be understood in the following way: To reduce the required gain, N should be decreased, because of −ɛG′′=NɛM′′. The decrease in N by shape‐tuning, e.g., increase in the aspect ratio of nanorods (eq. (6)), or by using thinner shells (eq. (9)), unavoidably red‐shifts the plasmon resonance because of −ɛG′=NɛM′. However, for metals ɛM′′ increases with wavelength. This counteracts the decrease in N and leads to a global minimum. To elaborate on these trends, Supp. Info. 3 provides formulas for a general Drude metal. The full red line in Fig. 1a predicts for example, that small Au spheres, or short rods with the resonances in the range 500<λ<600 nm used in Refs.  19, 20, are expected to have thresholds that exceed the optimal values by a factor of 6–10, and that of Ag by a factor of 20–50. In Fig. 1b we plot the dependence of the amplification β for the five local minima of gain that can be seen in the black solid curve for Ag in Fig. 1a. Note that data in Fig. 1 use bulk values for gold and silver, neglecting surface induced damping. Hence, if the wavelength of minimal necessary gain (defined by the right expression in (11)) can be reached with several different geometries (which all may fulfill the left equality in (11)), the one with the smallest surface to volume ratio should be used to minimize losses by surface damping.


Minimal spaser threshold within electrodynamic framework: Shape, size and modes.

Arnold N, Hrelescu C, Klar TA - Ann Phys (2015)

Universal quasi‐static threshold parameters. a) Solid curves (left scale) show the dependence of −ɛM′′/ɛM′′ɛM′ɛM′ on the wavelength λ for Ag (black) and Au (red) with Johnson and Christy values. The threshold gain −ɛG′′ can be obtained from this ratio by multiplying it with ɛG′ of the gain material. Dashed curves (right scale) show the corresponding −1/−1ɛM′ɛM′ values, which are equal to the normalized shape factor N/NɛG′ɛG′ at the threshold. Both ordinates have logarithmic scales. b) Dependences of the threshold amplification β on the background dielectric ɛG′ for 5 local minima of the −[ɛM′′/ɛM′′ɛM′ɛM′](λ) curves for Ag, as color‐coded in the legend.
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Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4834728&req=5

andp201500318-fig-0001: Universal quasi‐static threshold parameters. a) Solid curves (left scale) show the dependence of −ɛM′′/ɛM′′ɛM′ɛM′ on the wavelength λ for Ag (black) and Au (red) with Johnson and Christy values. The threshold gain −ɛG′′ can be obtained from this ratio by multiplying it with ɛG′ of the gain material. Dashed curves (right scale) show the corresponding −1/−1ɛM′ɛM′ values, which are equal to the normalized shape factor N/NɛG′ɛG′ at the threshold. Both ordinates have logarithmic scales. b) Dependences of the threshold amplification β on the background dielectric ɛG′ for 5 local minima of the −[ɛM′′/ɛM′′ɛM′ɛM′](λ) curves for Ag, as color‐coded in the legend.
Mentions: Figure 1a shows the dependences N(λ) (dashed) and −ɛG′′(λ) (solid) defined by the relations (11) for silver and gold with dielectric functions from Johnson and Christy 71 and normalized to the real part ɛG′ of the gain material. For ɛG′≠1 the results simply rescale, with larger ɛG′ resulting in larger threshold gain and N values, a fact that should be considered in the selection of a suitable gain medium. The global minima around 1060 nm for silver and around 750 nm for gold can be understood in the following way: To reduce the required gain, N should be decreased, because of −ɛG′′=NɛM′′. The decrease in N by shape‐tuning, e.g., increase in the aspect ratio of nanorods (eq. (6)), or by using thinner shells (eq. (9)), unavoidably red‐shifts the plasmon resonance because of −ɛG′=NɛM′. However, for metals ɛM′′ increases with wavelength. This counteracts the decrease in N and leads to a global minimum. To elaborate on these trends, Supp. Info. 3 provides formulas for a general Drude metal. The full red line in Fig. 1a predicts for example, that small Au spheres, or short rods with the resonances in the range 500<λ<600 nm used in Refs.  19, 20, are expected to have thresholds that exceed the optimal values by a factor of 6–10, and that of Ag by a factor of 20–50. In Fig. 1b we plot the dependence of the amplification β for the five local minima of gain that can be seen in the black solid curve for Ag in Fig. 1a. Note that data in Fig. 1 use bulk values for gold and silver, neglecting surface induced damping. Hence, if the wavelength of minimal necessary gain (defined by the right expression in (11)) can be reached with several different geometries (which all may fulfill the left equality in (11)), the one with the smallest surface to volume ratio should be used to minimize losses by surface damping.

Bottom Line: Here, we derive this result from the purely classical electromagnetic scattering framework.Furthermore, retardation is included straightforwardly into our framework; and the global spectral gain minimum persists beyond the quasi-static limit.We illustrate this with two examples of widely used geometries: Silver spheroids and spherical shells embedded in and filled with gain materials.

View Article: PubMed Central - PubMed

Affiliation: Institute of Applied PhysicsJohannes Kepler University LinzAltenbergerstraße 694040 LinzAustria; Soft Matter PhysicsJohannes Kepler University LinzAltenbergerstraße 694040 LinzAustria.

ABSTRACT

It is known (yet often ignored) from quantum mechanical or energetic considerations, that the threshold gain of the quasi-static spaser depends only on the dielectric functions of the metal and the gain material. Here, we derive this result from the purely classical electromagnetic scattering framework. This is of great importance, because electrodynamic modelling is far simpler than quantum mechanical one. The influence of the material dispersion and spaser geometry are clearly separated; the latter influences the threshold gain only indirectly, defining the resonant wavelength. We show that the threshold gain has a minimum as a function of wavelength. A variation of nanoparticle shape, composition, or spasing mode may shift the plasmonic resonance to this optimal wavelength, but it cannot overcome the material-imposed minimal gain. Furthermore, retardation is included straightforwardly into our framework; and the global spectral gain minimum persists beyond the quasi-static limit. We illustrate this with two examples of widely used geometries: Silver spheroids and spherical shells embedded in and filled with gain materials.

No MeSH data available.


Related in: MedlinePlus