Temperature-induced hysteresis in amplification and attenuation of surface-plasmon-polariton waves

The propagation of surface-plasmon-polariton (SPP) waves at the planar interface of a metal and a dielectric material was investigated for a dielectric material with strongly temperature-dependent constitutive properties. The metal was silver and the dielectric material was vanadium multioxide impregnated with a combination of active dyes. Depending upon the volume fraction of vanadium multioxide, either attenuation or amplification of the SPP waves may be achieved; the degree of attenuation or amplification is strongly dependent on both the temperature and whether the temperature is increasing or decreasing. At intermediate volume fractions of vanadium multioxide, for a fixed temperature, a SPP wave may experience attenuation if the temperature is increasing but experience amplification if the temperature is decreasing.


Introduction
The planar interface of a plasmonic material and dielectric material guides the propagation of surface-plasmonpolariton (SPP) waves [1][2][3]. As the propagation of SPP waves is acutely sensitive to the constitutive properties of the plasmonic and dielectric materials involved, these surface waves are widely exploited in optical sensing applications [4]. The prospect of harnessing dielectric materials whose constitutive properties are strongly temperature dependent opens up possibilities of further applications for SPP waves in reconfigurable and multifunctional devices [5][6][7][8].
At visible wavelengths, vanadium dioxide is a dissipative dielectric material whose constitutive properties are acutely sensitive to temperature over the range 25°C-80°C [9][10][11][12][13]. Indeed, the crystal structure of vanadium dioxide is monoclinic at temperatures below 58°C and tetragonal at temperatures above 72°C [14], with both monoclinic and tetragonal crystals coexisting at intermediate temperatures. Furthermore, the temperatureinduced monoclinic-to-tetragonal transition is hysteretic. The electromagnetic response of vanadium dioxide is characterized by its (complex-valued) relative permittivity ò VO , with { } Re 0 VO >  and { } Im 0 VO >  at visible wavelengths. The value of ò VO depends upon temperature; also, over the range 25°C-80°C, it depends upon whether the material is being heated or cooled. Parenthetically, the dissipative dielectric material-to-metal phase transition [14] that vanadium dioxide exhibits at free-space wavelength λ 0 > 1100 nm [15] is not relevant to our study.
For optical applications, thin films of vanadium dioxide may often be desired [16,17]. Such thin films are conveniently fabricated by a vapor deposition process. However, depending upon the processing conditions and thickness of the film, the deposition process may result in significant proportions of vanadium oxides other than vanadium dioxide being present in such films. Accordingly, in the absence of definitive stoichiometric evidence, we shall refer these films as being composed of vanadium multioxide. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Losses due to the dissipative nature of vanadium multioxide represent a potential impediment for optical applications. However, these losses may be overcome by mixing vanadium multioxide with an active material. Rhodamine dyes provide a class of suitable active materials that are commonly used to overcome losses at optical wavelengths in otherwise dissipative metamaterials [18,19]. The use of active materials to amplify SPP waves is a well-established practice [20][21][22][23].
Therefore, in the following, we investigate the temperature dependence of SPP waves guided by the interface of (i) a homogenized mixture of vanadium multioxide and rhodamine dyes, and (ii) a plasmonic material which is taken to be silver. In particular, the thermal hysteresis is explored for both amplified and attenuated SPP waves. The canonical boundary-value problem is considered in which SPP waves are guided by the interface z = 0; the plasmonic material occupies the half-space z < 0 and the dielectric material occupies the halfspace z > 0.
Plots of the real and imaginary parts of ò VO are provided in figure 2 for the temperature range [ ] 25 C, 80 C   .
These values were derived by extrapolation of experimentally-determined values which were found at λ 0 = 800 nm for both heating and cooling phases, following the method described in [24]; and using values determined by ellipsometry at 25°C and 95°C for λ 0 = 710 nm. The hysteresis phenomenon displayed in  Re VO  between heating and cooling phases is approximately 0.9 and the maximum difference in { } Im VO  between heating and cooling phases is approximately 0.11. A homogenized mixture of vanadium multioxide, characterized by the relative permittivity ò VO and volume fraction f VO , and a combination of rhodamine dyes, characterized by the relative permittivity ò rho and volume fraction f rho = 1 − f VO , occupies the half-space z > 0. The relative permittivity of the homogenized mixture, namely ò mix , is estimated using the Bruggeman homogenization formalism [25,26]. Accordingly, ò mix is extracted from the Bruggeman equation Since the Bruggeman equation (3) is quadratic in ò mix , it is readily solved by means of the quadratic formula. The electromagnetic response properties of vanadium multioxide is assumed to be unchanged by the gain in the rhodamine dyes, but the foregoing equation clearly shows the gain to affect the electromagnetic response properties of the mixture of vanadium multioxide and rhodamine dyes. Plots of the real and imaginary parts of ò mix versus temperature are presented in figure 3 for f VO = 0.2, 0.5, and 0.8, for both heating and cooling phases. The real part of ò mix is positive valued across the entire temperature range for all volume fractions considered. When f VO = 0.2, { } Im 0 mix <  across the entire temperature range; therefore, the homogenized mixture is effectively an active dielectric material for f VO across the entire temperature range; therefore, the homogenized mixture is effectively a dissipative dielectric material for f VO = 0.8. When f VO = 0.5, { } Im 0 mix <  at low temperatures (less than 63°C for the heating phase and less than 32°C for the cooling phase), and { } Im 0 mix >  at high temperatures. Therefore, for f VO = 0.5, the homogenized mixture is effectively an active material at low temperatures and effectively a dissipative material at high temperatures.
Whereas the Bruggeman formalism has been widely used to provide estimates of the relative permittivity of homogenized mixtures that agree well with experimentally determined values [27][28][29][30], it is illuminating to compare the estimates provided in figure 3 with estimates provided by another much-used homogenization formalism, namely the Maxwell Garnett formalism [26]. The basis for the Maxwell Garnett formalism is somewhat different to that of the Bruggeman formalism: For the Maxwell Garnett formalism, inclusions of relative permititivity ò inc are distributed with a volume fraction f inc in a host medium of relative permittivity ò host . The relative permittivity of the homogenized mixture is estimated as   (3). The most conspicuous differences are observed in the estimates of the imaginary part of the relative permittivity of the homogenized mixture, especially for f VO = 0.2; in this regime the differences are no more than 2%.
The relative permittivity of the plasmonic material that occupies the half-space z < 0, namely silver, was taken to be ò Ag = − 23.40 + 0.39i. Note that ò Ag at λ 0 = 710 nm is sufficiently insensitive to temperature over the range 25°C < T < 80°C that its temperature dependence need not be considered here [31].

Surface-plasmon-polariton waves
For the canonical boundary-value problem, the wave number of the SPP wave is given by [32] ( ) q k , 6 0 mix Ag mix Ag = +

   
wherein k 0 = 2π/λ 0 is the free-space wave number. Notice that equation (6) holds regardless of the sign of { } Im mix  [33]. The real part of q is inversely proportional to the phase speed of the SPP wave, while the imaginary part of q is a measure of the SPP wave's attenuation rate, with { } q Im 0 < signifying amplification and { } q Im 0 > signifying attenuation. The real and imaginary parts of q are plotted against temperature for the range [ ] 25 C, 80 C   in figure 5 for both heating and cooling phases. The volume fractions considered are f VO = 0.2, 0.5, and 0.8. The real part of q is positive valued across the entire temperature range for all volume fractions considered. Since, at each temperature, { } q Re is greater for the heating phase than for the cooling phase, SPP waves propagate at a lower phase speed for the heating phase than for the cooling phase. When f VO = 0.2, { } q Im 0 < across the entire temperature range; therefore, the SPP wave is amplified at all temperatures for f VO = 0.2 and the degree of amplification is greater if the temperature is increasing rather than decreasing. When f VO = 0.8, { } q Im 0 > across the entire temperature range; therefore, the SPP wave is attenuated at all temperatures for f VO = 0.8 and the degree of attenuation is greater if the temperature is decreasing rather than increasing. When f VO = 0.5, { } q Im 0 < at low temperatures (less than 63°C for the heating phase and less than 32°C for the cooling phase), and { } q Im 0 > at high temperatures. Therefore, for f VO = 0.5, at a given temperature, whether the SPP wave is amplified or attenuated depends upon whether the temperature is increasing or decreasing. In particular, at f VO = 0.5, the SPP wave is neither attenuated nor amplified at (i) T = 63°C if the temperature is increasing; and (ii) T = 32°C if the temperature is decreasing. which quantify the differences in the real and imaginary parts of q/k 0 between the heating phase, i.e. | q k 0 heat , and the cooling phase, i.e. | q k 0 cool , over the temperature range 25°C < T < 80°C. Plots of H R and H I versus f VO are presented in figure 6. As expected, both H R and H I vanish in the limit f VO → 0, and both attain their maximum values in the limit f VO → 1. For intermediate values of f VO , both H R and H I increase monotonically as f VO increases. Furthermore, whereas H R is almost a linearly increasing function of f VO , H I increases rapidly with f VO when f VO < 0.6 but saturates for f VO > 0.6.

Closing remarks
The propagation of SPP waves at the planar metal/dielectric interface can be controlled by temperature by choosing a dielectric material whose constitutive properties are strongly temperature dependent and which is impregnated with an active dye. Specifically, if the dielectric material is a homogenized mixture of vandium multioxide and rhodamine dyes and the metal is silver, then either attenuation or amplification of the SPP waves may be achieved, depending upon the volume fraction of vanadium multioxide. The degree of attenuation or amplification is strongly dependent on both the temperature and whether the temperature is increasing or decreasing. At intermediate volume fractions of vanadium multioxide, for a fixed temperature, a SPP wave may experience attenuation if the temperature is increasing but experience amplification if the temperature is decreasing. This thermal hysteresis in amplification and attenuation of SPP waves may be usefully exploited in applications involving reconfigurable and multifunctional devices, as well as those involving temperature sensing.
The canonical boundary-value problem considered here provides insights into the physics underlying SPP wave propagation at the planar metal/dielectric-mixture interface and highlights the role of temperatureinduced hysteresis in the dielectric properties of the non-active (i.e. passive) component of the dielectric mixture. In a more realistic scenario, the dye close to the metal interface may be dielectrically nonuniform in the direction normal to the interface due to position-dependent dipole lifetime and pump irradiance. Theoretical consideration of those phenomenons for a homogeneous dye layer (i.e. f VO = 0) requires careful approximations [34]. These phenomenons are even more difficult to incorporate in theoretical studies when f VO > 0, and require suitable experimental data that is currently unavailable. It would also be desirable to take into account temperature corrections arising from the power density profile of the SPP wave, but this too requires currently unavailable experimental data. Regardless, the results presented here clearly show that thermal hysteresis will affect amplification and attenuation of SPP waves when the dielectric mixture contains active component materials.
Lastly, photobleaching of rhodamine dyes [35] depending upon the power density profile of the SPP waves [36] is a potential issue to be addressed. Photobleaching could be ameliorated by reducing the excitation intensity, and thereby extending the life of the dye molecules, but this would reduce the SPP wave's energy. Accordingly, a careful balance must be found. Rhodamine was chosen as a widely-used representative dye for our study. Alternative dyes that are less susceptible to photobleaching and more photostable, such as Alexa Fluor 633 [37] or Atto 655 [38], could be used.