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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22269–22284
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Photo-thermal modulation of surface plasmon polariton propagation at telecommunication wavelengths

S. Kaya, J.-C. Weeber, F. Zacharatos, K. Hassan, T. Bernardin, B. Cluzel, J. Fatome, and C. Finot  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22269-22284 (2013)
http://dx.doi.org/10.1364/OE.21.022269


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Abstract

We report on photo-thermal modulation of thin film surface plasmon polaritons (SPP) excited at telecom wavelengths and traveling at a gold/air interface. By operating a modulated continuous-wave or a Q-switched nanosecond pump laser, we investigate the photo-thermally induced modulation of SPP propagation mediated by the temperature-dependent ohmic losses in the gold film. We use a fiber-to-fiber characterization set-up to measure accurately the modulation depth of the SPP signal under photo-thermal excitation. On the basis of these measurements, we extract the thermo-plasmonic coefficient of the SPP mode defined as the temperature derivative of the SPP damping constant. Next, we introduce a figure of merit which is relevant to characterize the impact of temperature onto the properties of bounded or weakly leaky SPP modes supported by a given metal at a given wavelength. By combining our measurements with tabulated values of the temperature-dependent imaginary part of gold dielectric function, we compute the thermo-optical coefficients (TOC) of gold at telecom wavelengths. Finally, we investigate a pulsed photo-thermal excitation of the SPP in the nanosecond regime. The experimental SPP depth of modulation obtained in this situation are found to be in fair agreement with the modulation depths computed by using our values of gold TOC.

© 2013 OSA

1. Introduction

Optical properties of metals are known to be temperature-dependent. Thermo-optical properties of metals have been investigated in details several decades ago, mostly for noble metals [1

1. G. P. Pells and M. Shiga, “The optical properties of copper and gold as a function of temperature,” J. Phys. C (Solid St. Phys.) 2, 1835–1846 (1969). [CrossRef]

5

5. P. Winsemius, F. F. van Kampen, H. P. Lengkeek, and C. G. van Went, “Temperature dependence of the optical properties of Au, Ag and Cu,” J. Phys. F: Metal Phys. 6, 1583–1606 (1976). [CrossRef]

]. When a metal is heated, many phenomenons contribute to the change of its dielectric function. A non-exhaustive list of these effects includes electron-electron scattering, electron-phonon scattering, electron-impurity/defect scattering, lattice expansion and/or change in the Fermi distribution [6

6. R. Rosei and D. W. Lynch, “Thermomodulation spectra of Al, Au and Cu,” Phys. Rev. B 5, 3883–3893 (1972). [CrossRef]

]. The respective contribution of each of these channels may depend upon the metal, the incident photon frequency, the actual temperature and the amplitude of the temperature change. The temperature-dependent dielectric function of noble metals can be described in the framework of the Drude model [7

7. T. Holstein, “Theory of transport phenomena in an electron-phonon gas,” Ann. Phys. 29, 410 (1964). [CrossRef]

9

9. J. A. McKay and J. A. Rayne, “Temperature dependence of the infrared absorptivity of the noble metals,” Phys. Rev. B 13, 673–685 (1976). [CrossRef]

] at least for energies significantly below the interband absorption edge of the metal of interest [9

9. J. A. McKay and J. A. Rayne, “Temperature dependence of the infrared absorptivity of the noble metals,” Phys. Rev. B 13, 673–685 (1976). [CrossRef]

]. An explicit form of this model can be found in refs. [10

10. C. S. Moreira, A. M. N. Lima, H. Neff, and C. Thirstrup, “Temperature-dependent sensitivity of surface plasmon resonance sensors at gold-water interface,” Sensor and Actuators B 134, 854–862 (2008). [CrossRef]

12

12. M. Liu, M. Pelton, and P. Guyot-Sionnest, “Reduced damping of surface plasmons at low temperatures,” Phys. Rev. B 79, 035 418 (2009).

].

The temperature dependence of noble metal optical properties has been reconsidered recently in the context of surface plasmon modes excitation. Indeed, noble metals, are to date the most common materials used to investigate the properties of surface plasmon polariton (SPP) sustained by either extended structures such as thin films or localized structures such as nanoparticles. The thermal properties of nanoparticles have been mainly investigated in the context of localized heat sources relying on resonantly excited nanoparticles (for a recent review see [13

13. G. Baffou and R. Quidant, “Thermo-Plasmonics: using metallic nanostructures as nanosources of heat,” Laser Photon. Rev. 7, 171–187 (2013). [CrossRef]

]). On the other hand, the impact of temperature onto localized surface plasmon resonances has been also investigated recently for single gold nanoparticles [12

12. M. Liu, M. Pelton, and P. Guyot-Sionnest, “Reduced damping of surface plasmons at low temperatures,” Phys. Rev. B 79, 035 418 (2009).

], gold nanoparticles embedded into a dielectric matrix [14

14. M. Rashidi-Huyeh and B. Palpant, “Counterintuitive thermo-optical response of metal-dielectric nanocomposite materials as a result of local electromagnetic enhancement,” Phys. Rev. B 74, 075 405 (2006). [CrossRef]

,15

15. B. Palpant, M. Rashidi-Huyeh, B. Gallas, S. Chenot, and S. Fisson, “Highly dispersive thermo-optical properties of gold nanoparticles,” Appl. Phys. Lett. 90, 223 105 (2007). [CrossRef]

] and closely packed gold nanorods forming a metamaterial [16

16. J.-S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low temperature plasmonics of metallic nanosctructures,” Nanolett. 12, 1561–1565 (2012). [CrossRef]

]. For crystalline gold nanoparticles, a very good agreement of the experimental data with the results of the temperature-dependent Drude model has been found [12

12. M. Liu, M. Pelton, and P. Guyot-Sionnest, “Reduced damping of surface plasmons at low temperatures,” Phys. Rev. B 79, 035 418 (2009).

] whereas significant deviations from this model were noticed in the case of evaporated thin films [17

17. R. T. Beach and R. W. Christy, “Electron-electron scattering in the intraband optical conductivity of Cu, Ag and Au,” Phys. Rev. B 16, 5277–5284 (1977). [CrossRef]

,18

18. G. R. Parkins, W. E. Lawrence, and R. W. Christy, “Intraband optical condutuctivity σ(ω, T) of Cu, Ag, Au: Contribution from electron-electron scattering,” Phys. Rev. B 23, 6408–6416 (1981). [CrossRef]

]. When excited onto a thin film, SPP modes can be used for bio-sensing purposes and in this respect the role of temperature onto the sensitivity of SPP based sensors has been investigated as well [10

10. C. S. Moreira, A. M. N. Lima, H. Neff, and C. Thirstrup, “Temperature-dependent sensitivity of surface plasmon resonance sensors at gold-water interface,” Sensor and Actuators B 134, 854–862 (2008). [CrossRef]

, 19

19. S. K. Ozdemir and G. Turhan-Sayan, “Temperature effects on surface plasmon resonance: Design considerations for an optical temperature sensor,” J. Lightwave Technol. 21, 805 (2003). [CrossRef]

]. Thermal modulation of thin film SPPs excited in the Kretschmann-Raether configuration at visible frequencies has also been reported [20

20. A. Passian, A. L. Lereu, E. T. Arakawa, A. Wig, T. Thundat, and T. L. Ferrell, “Modulation of multiple photon energies by use of surface plasmons,” Opt. Lett. 30, 41–43 (2005). [CrossRef] [PubMed]

, 21

21. A. L. Lereu, A. Passian, J. P. Goudonnet, T. Thundat, and T. L. Ferrell, “Optical modulation processes in thin films based on thermal effects of surface plasmons,” Appl. Phys. Lett. 86, 154 101 (2005). [CrossRef]

]. However, so far, only little has been done to assess quantitatively thermo-optical properties of thin film SPP modes at telecommunication wavelengths (1500nm–1600nm). Beyond a fundamental interest, such a study is needed to characterize the thermo-optical properties of the metal when used as plasmonic waveguides operated in the telecom band.

In this work we focus onto the properties of gold, however the approach we describe in the following can be applied to any metal sustaining SPPs with sufficient propagation length. Many thermo-optic (TO) plasmon-based devices have been reported in the literature recently [22

22. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “In-line extinction modulator based on long-range surface plasmon polaritons,” Opt. Commun. 244, 455–459 (2005). [CrossRef]

29

29. H. Fan and P. Berini, “Thermo-optic characterization of long-range surface plasmon in Cytop,” Appl. Opt. 52, 162–170 (2013). [CrossRef] [PubMed]

]. In these works, the thermo-optic functionality is provided by a high TO coefficient dielectric material whose index is affected by a temperature change which in turn modifies the field distribution of the plasmon mode traveling at the dielectric/metal interface. In these configurations, the TO active part is a dielectric material and so far, except very few works [11

11. X.-Y. Zhang, T. Zhang, A.-M. Hu, X.-J. Xue, P.-Q. Wu, and Q.-Y. Chen, “Tunable microring resonator based on dielectric-loaded surface plasmon polariton waveguides,” J. Nanosci. Nanotechnol. 11, 10 520–10 524 (2011).

], the role of TO properties of metal in plasmonic structures, at telecom frequencies, has not been considered in details. At telecom wavelengths where gold dielectric function is dominated by free-electron contribution, an increase of the temperature leads to an increase of the ohmic losses through the increase of the electron-phonon scattering rate. In essence, this process is not different from the increase of the static (dc) resistivity of metals for an increasing temperature [30

30. C. Kittel, Introduction to solid state physics, 8th ed. (John Wiley and Sons, 2005).

]. The thermally-controlled losses in the metal leads to a SPP mode temperature dependent propagation length which can be used to evaluate the thermo-plasmonic properties of the metal of interest.

The study is organized as follow. We investigate the TO modulation of thin film SPPs traveling at a gold/air interface by operating a specific fiber-to-fiber detection scheme described in the second section. A photo-thermal heating of the metal film is performed by using either a modulated continuous-wave (cw) laser or a nanosecond Q-switched laser. In the third section, we introduce the thermo-plasmonic coefficient of the SPP mode defined as the temperature derivative of the SPP damping constant. We show that this coefficient can be obtained from the experimental depth of modulation of the SPP signal provided that the temperature distribution along the propagation path is known. The computation scheme of the temperature distribution generated by the photo-thermal heating is described in section four along with the procedure leading to the evaluation of gold TOC extracted from the thermo-plasmonic coefficient. Finally, we compare in section five, the thermo-modulation of the SPP signal modeled by using our gold TO coefficients to the experimental depth of modulation produced by a pulsed nanosecond photo-thermal excitation.

2. Experimental set-up

The experimental set-up used for the measurement of SPP photo-thermal modulation is schematically shown in Fig. 1(a). The set-up is built-up around an inverted microscope equipped with an immersion oil objective featuring a large numerical aperture (NA=1.49). The signal collected by the immersion oil objective can be directed either on a visible CCD camera or on a sensitive InGaAs camera. The samples mounted onto the microscope stage are excited at telecom frequencies by a lensed fiber generating a slightly focused spot with a typical radius of 20μm. The input focuser is tilted at an angle of 30° with respect to the optical axis of the microscope. A similar focuser is used for output detection. An arrangement comprising a low numerical aperture long working distance objective is placed above the microscope stage to focus the pump beam (free-space wavelength=532nm). The pump laser can be either a continuous-wave laser modulated by a highly efficient 20ns rise/fall-time electro-optics modulator or a Q-switched laser generating nanosecond pulses. The samples fabricated to investigate photo-thermal modulation of SPP propagation are comprised of a 80nm-thick gold film evaporated onto a 170μm-thick glass substrate. Dielectric input and output grating couplers shown in Fig. 1(b) have been fabricated at the surface of the samples by electron beam lithography using a negative tone resist and following the principles given in ref. [31

31. M. G. Nielsen, J.-C. Weeber, K. Hassan, J. Fatome, C. Finot, S. Kaya, L. Markey, O. Albrektsen, S. I. Bozhevolnyi, G. Millot, and A. Dereux, “Grating couplers for fiber-to-fiber characterizations of stand-alone dielectric loaded surface plasmon waveguide components,” J. Lightwave Technol. 30, 3118–3125 (2012). [CrossRef]

]. The grating couplers are separated by a distance of 200μm and are excited by an incoherent near-infrared amplified spontaneous source peaked at 1530nm whose spectrum is displayed in Fig. 1(c). The choice of a broadband incoherent light source in the context of this work is dictated by the fact that we want to minimize potential thermally induced changes of the grating coupling efficiency. Indeed, the thermally-induced mismatch with the Bragg condition is expected to be attenuated for a broadband source compared to the case of a highly coherent incident light. The near-infrared signal detected by the output focuser can be directed onto an optical spectrum analyzer (OSA) or a fast (rise time of 2ns) sensitive InGaAs photodiode. The signal of the photodiode is monitored by a 1GHz bandwidth oscilloscope triggered by the signal of the visible photodiode receiving a tiny fraction of the pump beam. The coupling efficiency of the gratings has been characterized by recording the spectrum of the transmitted SPP signal when the input and output focusers are adjusted at optimum positions on the gratings. This spectrum has been subsequently compared to the reference spectrum obtained by detecting the specular reflection of the incident infrared spot onto a bare gold film area. With this definition of the reference situation, insertion losses including in and out grating coupling losses and SPP propagation losses are below 18dB at 1530nm. We note also that the insertion losses increase with the wavelength indicating that the main contribution to these losses are the coupling losses and not the propagation losses.

Fig. 1 (a) Schematic view of the experimental set-up. (b) Bird-eye view of the input and output grating couplers implemented at the surface of a thin gold film. (c) Spectrum of the incoherent light-source used for the excitation of the SPP modes. (d) Typical insertion losses for SPP signal fiber-to-fiber transmission.

Our configuration allows us to record simultaneously leakage radiation (LR) images [32

32. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steiberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B 149, 220–229 (2008). [CrossRef]

] of a SPP mode traveling between the input and output grating couplers and the corresponding fiber-to-fiber signal. LR images of the gold/air SPP jet excited by the input grating coupler shown on the scanning electron microscope image of Fig. 2(a) are displayed in Figs. 2(b) and (c) with the visible pump being respectively off and on. The illumination of the input grating by the incident spot leads to a SPP jet with a transverse cross-cut of the intensity matching a gaussian profile given by I(x, y) = I(x, 0) exp2y2wjet2 with a waist wjet =9.5μm (see Fig. 2(d)). The pump spot visible onto Fig.2(c) can be observed with the InGaAs camera provided that the long-pass filter placed in front of the camera is removed. In this situation, the visible pump light can be observed with the InGaAs sensor at the expense of a sufficiently long exposure time. By varying the pump beam power and monitoring the resulting signal onto the InGaAs camera, we have verified the linearity of the InGaAs sensor making the use of the InGaAs camera convenient for measurement of the pump spot size. The pump spot shown in Fig. 2(c) and on the inset of Fig. 2(e) corresponds to the pump spot used during the experiments conducted with the cw illumination. As shown in Fig. 2(e), the spot is gaussian with a 16μm half-width at 1/e2 of the maximum intensity.

Fig. 2 (a) Scanning electron microscope image of the grating couplers showing the respective location of the input infrared and visible pump spot (scale bar=100μm). (b) (resp.(c)) Leakage radiation image of the SPP jet generated by the input grating coupler with the pump spot off (resp. on). (d) Solid line: Cross-cut of the SPP jet. The jet can be approximated by a gaussian profile (dashed line) with a waist of 9.5μm. (e) Solid-line: Cross-cut of the intensity distribution of the pump spot. The pump spot can be approximated by a gaussian beam (dashed line) with a waist of 16μm.

3. Photo-Thermal modulation of SPP propagation

Figure 3(a) shows the ac coupled SPP signal detected by the InGaAs photo-diode along with the visible photo-diode signal when a photo-thermal cw excitation modulated at a frequency of 1kHz (duty cycle=50%) is applied on the SPP propagation path. For an incident power of 100mW (power of the pump beam measured after the 5x objective and without modulation), the relative depth of modulation |ΔI/I| is found to be 4.5%. In all the following, the relative depth of modulation is defined as ratio of the SPP signal modulation ΔI during the photo-thermal pumping to the SPP signal I in the cold state in absence of pumping. Knowing the depth of modulation ΔII obtained for known parameters (pump power, pump spot size, plasmon jet width...), a model for SPP propagation modulation is needed to correlate the thermo-optical characteristics of the SPP mode to the experimental value of ΔII. We consider the situation depicted in Fig. 3(b). A gold/air plasmon jet centered on the y-axis is traveling along x-axis toward the positive values of x. The plasmon jet has a gaussian transverse profile (along the y-axis) with a waist wjet. We assume that the gold film supporting the SPP mode is locally heated creating a known temperature distribution Tf (x, y) in the plane of the film. More precisely, Tf (x, y) denotes the average temperature of the film over the penetration depth of the SPP mode field. At telecom wavelengths, this penetration depth does not exceed of a few tenths of nanometers. Owing to the vanishing temperature gradient of the gold film over such a thickness, the z dependence for temperature Tf can be safely neglected. For wjet large enough compared to the SPP wavelength, the width of the spatial frequencies spectrum of the plasmon jet can be neglected in the first approximation and the electric field of the SPP jet in the plane of the film verifies:
Eh(x,y)=Eh(xi,y)exp(iΦ(x,xi,y))
(1)
where the superscript h indicates that the film is in the ”hot” state. The function Φ(x, xi, y) accounts for the accumulation of phase and damping along the distance (xxi) and is given by:
Φ(x,xi,y)=xixkspp(x,y)dx
(2)
The function kspp(x′, y) denotes the complex SPP wavevector which depends upon the observation point through the temperature distribution Tf (x′, y). The skin depth of gold in the infrared depends linearly on temperature over a range of more than 1000K [2

2. K. Ujihara, “Reflectivity of metals at high temperatures,” J. Appl. Phys 43, 2376–2382 (1972). [CrossRef]

]. Thus, for a sufficiently small temperature change in the range of one or two hundreds of Kelvins, the SPP wave-vector is expected to depend linearly on the temperature as well in such a way that:
kspp(x,y)=kspp+Tkspp(Tf(x,y)Troom)
(3)
and
kspp(x,y)=kspp+Tkspp(Tf(x,y)Troom)
(4)
where k′spp and k″spp are respectively the phase and damping constant of the SPP mode in the ”cold” state when the film is at room temperature Troom and where Tk′spp and Tk″spp are the partial derivative of respectively the phase and damping constant with respect to the temperature. By introducing equations (3) and (4) into (1), the field at x = xo can be written:
Eh(xo,y)=Eh(xi,y)exp(iΦ(xo,xi,y))
(5)
where Φ(xo, xi, y) is given by:
Φ(xo,xi,y)=(kspp+ikspp)(xoxi)+(Tkspp+iTkspp)xixoΔT(x,y)dx
(6)
where ΔT (x′, y) = Tf (x′, y) − Troom.

Fig. 3 (a) Solid-line: SPP signal modulation recorded with the IR photo-diode in the case of a photo-thermal excitation with a cw illumination (100mW) modulated at a frequency of 1kHz (duty-cycle=50%). Dashed-line: control signal of the visible photo-diode showing the modulation of the cw pump beam. (b) Schematic view of the SPP jet propagation. The input (resp. output) grating coupler is located at x = xi (resp. x = xo).

4. Thermo-optical coefficients of gold at telecom frequencies

4.1. Gold TO coefficients characterization from gold/air SPP thermo-modulation

4.2. Temperature distribution of the gold film under modulated cw illumination

Fig. 4 (a) Schematic view of the system considered for the finite-difference computation of the temperature distribution along the gold thin film under photo-thermal excitation. The computation window is 100μm long (along r coordinate) and 50μm high (in the Z direction). A Neumann (N) boundary condition Tr=0 is applied at r = 0 due to the rotational symmetry around Z-axis whereas Dirichlet (D) conditions T = Troom are used for the three other boundaries. (b) Electric intensity distribution of the interference pattern created by the interaction of the normally incident and back-reflected pump beams. The power carried by the incident beam is 100mW. (c) Heat-source density distribution corresponding to the electric intensity distribution shown in (b). (d) Temperature in the thin film at the focal point of the gaussian beam as a function of time in the case of a 1kHz modulation frequency (duty-cycle=50%). (e) Spatial temperature distribution at the surface of the thin film at t=500μs.

4.3. Thermo-plasmonic figure of merit and gold TOC at telecom frequencies

The effective computation of gold TOC is conducted by considering the experimental situation where the pump beam has a power of 100mW. When modulated at 1kHz with a duty-cycle of 50%, the photo-thermal modulation of the SPP signal traveling through the heated area leads to a modulation depth of |ΔII|=4.5% (see Fig. 3(a)). From this modulation depth and the temperature distribution along the thin film at the end of the heating period (t =500μs), we evaluate (from equation (12)) the thermo-plasmonic coefficient Tk″spp of the gold/air SPP mode at 2.90×10−6μm−1.K−1. Next, for the reference situation we consider (pump beam power of 100mW), the maximum temperature of the film at the end of the heating period is around 510K. Thus, the temperature derivative Tε″ for the gold film is expected to be 0.01425 K−1 according to the results reported in ref. [1

1. G. P. Pells and M. Shiga, “The optical properties of copper and gold as a function of temperature,” J. Phys. C (Solid St. Phys.) 2, 1835–1846 (1969). [CrossRef]

] for temperatures ranging from 470 to 670K. Knowing both Tk″spp and Tε″, the TO coefficients of the gold film can be obtained from equations (17) and (16) provided that the refractive index of gold at room temperature is known. In all the following we use a complex refractive index of gold of 0.515+ i10.65 extracted from tabulated values reported in ref. [43

43. P. B. Jonhson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

] for a free-space wavelength of 1.53μm. This choice is motivated by the fact that the refractive index of gold thin films was measured at telecom frequencies [44

44. W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys. 103, 073 713 (2008). [CrossRef]

] and found to be in good agreement with the values reported in ref. [43

43. P. B. Jonhson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

]. For example, at a free space wavelength of 1.55μm, the refractive index of thin films with a typical thickness around 60nm was measured at 0.505+i10.7 deviating by less than 3% for n and κ compared to the value 0.52+i10.8 reported in ref. [43

43. P. B. Jonhson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

]. Given the refractive index of gold at room temperature, the linear system formed by equations (17) and (16) can be solved for Tn and Tκ leading to TO coefficients displayed in Table 1. Our TO coefficients are compared to the linear TO coefficients obtained from the temperature-dependent Drude model for gold described in ref. [11

11. X.-Y. Zhang, T. Zhang, A.-M. Hu, X.-J. Xue, P.-Q. Wu, and Q.-Y. Chen, “Tunable microring resonator based on dielectric-loaded surface plasmon polariton waveguides,” J. Nanosci. Nanotechnol. 11, 10 520–10 524 (2011).

]. Although of same order of magnitude, we note however that our TO coefficients are significantly different from those given in ref. [11

11. X.-Y. Zhang, T. Zhang, A.-M. Hu, X.-J. Xue, P.-Q. Wu, and Q.-Y. Chen, “Tunable microring resonator based on dielectric-loaded surface plasmon polariton waveguides,” J. Nanosci. Nanotechnol. 11, 10 520–10 524 (2011).

]. In particular, Tn is more than two-fold smaller in our case as compared to the model developed in ref. [11

11. X.-Y. Zhang, T. Zhang, A.-M. Hu, X.-J. Xue, P.-Q. Wu, and Q.-Y. Chen, “Tunable microring resonator based on dielectric-loaded surface plasmon polariton waveguides,” J. Nanosci. Nanotechnol. 11, 10 520–10 524 (2011).

]. Actually, our values compare more favorably with the linear extrapolation at 1.53μm of the gold TOC (obtained from the Kramer-Kronig analysis of ref. [1

1. G. P. Pells and M. Shiga, “The optical properties of copper and gold as a function of temperature,” J. Phys. C (Solid St. Phys.) 2, 1835–1846 (1969). [CrossRef]

] data) given in ref. [14

14. M. Rashidi-Huyeh and B. Palpant, “Counterintuitive thermo-optical response of metal-dielectric nanocomposite materials as a result of local electromagnetic enhancement,” Phys. Rev. B 74, 075 405 (2006). [CrossRef]

] at a wavelength of 1.24μm.

Table 1. Comparison of the thermo-optical coefficients of gold extracted from the measured SPP depth of modulation and from the temperature-dependent Drude model for gold given in ref. [11]. The results of ref. [14] are obtained from a linear extrapolation (see text).

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Figure 5(a) shows the modulation depth computed from equation (12) by using either our TO values or the model of ref. [11

11. X.-Y. Zhang, T. Zhang, A.-M. Hu, X.-J. Xue, P.-Q. Wu, and Q.-Y. Chen, “Tunable microring resonator based on dielectric-loaded surface plasmon polariton waveguides,” J. Nanosci. Nanotechnol. 11, 10 520–10 524 (2011).

] for different incident peak power pump beam modulated at 1kHz with a 50% duty cycle. Whatever the incident pump power, the temperature-dependent Drude model for gold leads to a depth of modulation which is overestimated by about 60% compared to our experimental values or computed values. The experimental oscilloscope traces of the thermo-modulated SPP signal displayed in Fig. 5(b)–(d) are compared to the modeled traces computed by using our gold TOC values. Apart for a small deviation for times larger than 200μs after the beginning of the heating cycle and for an incident power of 100mW, we observe a quite good agreement between the computed and experimental profiles suggesting that our computations allow us to extract not only the amplitude of the photo-thermal modulation but also the characteristic time for this modulation. Although of great practical importance, the dynamic of photo-thermal SPP modulation is however out of the scope of the present work and is analyzed in details down to the nanosecond regime for confined plasmonic waveguides in ref. [37

37. J.-C. Weeber, T. Bernardin, M. G. Nielsen, K. Hassan, S. Kaya, J. Fatome, C. Finot, A. Dereux, and N. Pleros, “Nanosecond thermo-optical dynamics of polymer loaded plasmonic waveguides,” Submitted for publication, (2013).

].

Fig. 5 (a) Comparison of the experimental modulation depth (open circles) and the modulation depth computed using either our values for gold TOC (open squares) or the temperature-dependent refractive index given in ref. [11] (open diamonds). (b) (resp. (c) and (d)) Comparison of the experimental (solid line) and computed SPP modulation (dashed line) as a function of time in the case of a modulated cw excitation (1kHz, duty-cycle=50% for an incident power of 100mW (resp. 73mW and 59mW).

When considering the results displayed in Fig. 5(a), it is worth to comment on the amplitude of the SPP signal photo-thermal modulation. We have found that a temperature rise as large as 200K (for a pump power of 100mW) generates a SPP modulation amplitude as small as 4.5%. This result can be understood from the fact that the gold/air interface mode we consider is only very little confined within the metal and thus features a long propagation distance and accordingly a weak thermo-plasmonic Tk″spp coefficient. For SPP modes with losses dominated by ohmic losses (bounded or a weakly leaking modes), the damping constant Lspp=12kspp is inversely proportional to the SPP field confinement into the metal. In this respect, the quantity Lspp × Tk″spp can be viewed as a figure of merit (FOM) characterizing the thermo-optical sensitivity of the metal for plasmonic applications at a given frequency. Relying on the modulation obtained for the pump power of 100mW, we find in our case a FOM for gold at telecom wavelengths of:
Lspp×Tkspp=8.0×104K1
(22)
where Lspp =276μm calculated for the gold/air SPP mode has been used to evaluate the numerical value of the FOM. As long as thermal modulation of surface plasmon modes is concerned, the use of the FOM defined above is an interesting practical approach. Indeed, knowing this FOM from measurements performed on a given SPP mode, one can obtained at least a coarse evaluation of the thermo-modulation depth for an other SPP mode provided that the propagation distances for the two modes are known in the cold state. For example, by using our gold TOC values, we calculate that the depth of modulation for the glass/gold interface SPP mode (Lspp=81μm) should be around 14.6% when the pump power is 100mW. For the same excitation conditions, this depth of modulation is evaluated at 15.3% when the calculation is performed by using the FOM value for gold given above.

5. Photo-thermal SPP modulation in the nanosecond regime

The TO coefficients of gold have been obtained from the measurements of the SPP signal thermo-modulation created by a modulated cw source. The goal of this last section is to test the reliability of our gold TOC values in the context of a nanosecond pulsed illumination. The experiments of this last section have been performed simply by replacing the cw laser and the electro-modulator shown in Fig. 1 by a 532nm Q-switched laser. The laser has a repetition rate of 8.3kHz and delivers 6.5μJ pulses with a width (full width at half maximum) of 0.59ns as specified by the manufacturer. In all the following, we assume that the pulses have a gaussian temporal profile. In order to prevent thin film damaging, the pump spot has been enlarged compared to the case of cw excitation. As shown in Figs. 6(a) and (b), the pump spot we use now has a gaussian spatial profile with a waist of 50μm. Figure 6(c) shows the oscilloscope trace of the SPP signal recorded for an average pump power of 14mW. The modulation of the SPP signal exhibits an asymmetric shape with an amplitude of 11.7% resulting from the abrupt rise of the film temperature over the duration of the pulse and the much slower cooling after the end of the pulse. Figure 6(d) displays the modulation depth ΔII as a function of the average pump power. Note that these experiments were conducted by increasing the pump power meaning that low pump powers were investigated first. The modulation depth depends linearly on the pump power until a threshold around 12.2mW whereas the point obtained for 14mW departs significantly from the linear behavior. This result is attributed to a change of the structural properties of the metal film. This conclusion has been confirmed by a subsequent electron microscope characterization showing a clear change of the grain size of the film at the location of the pump spot center indicative of a possible partial melting of the film in this area. Disregarding the result obtained at 14mW, we obtain a modulation efficiency corresponding to the slope of the linear fit displayed in Fig. 6(d) of 0.66 %.mW−1.

Fig. 6 (a) Leakage radiation image showing the infrared plasmon jet and the nanosecond pump spot. (b) Optical image of the pump spot. The experimental intensity profile (solid line) of the pump beam is close to a gaussian beam (dashed line) with a waist of 50μm. (c) Oscilloscope trace showing the modulation of the SPP signal under excitation with the Q-switched nanosecond laser. The average power is 14mW. (d) Depth of modulation of the SPP signal as a function of the average power of the pulsed pump beam.

Fig. 7 (a) Temperature of the thin film at the center of the pump beam in the case of an incident pulse with an average power of 12.2mW a repetition rate and a pulse duration corresponding to the experimental situation. The incident pulse reaches its maximum at t=5ns as shown on the inset displaying the power of the pulse as a function of time. (b) (resp. (c)) Comparison of the experimental (solid line) and computed (dashed line) SPP photo-thermal modulation for an average incident power of 12.2mW (resp. 10.6mW). The computed profile accounts for the 2ns rise time of the infrared photo-diode response. (d) Comparison of the experimental (dashed line) depth of modulation and the depth of modulation (solid line) computed using our gold TOC values.

6. Conclusion

Acknowledgments

This work has benefited from the financial support of the Agence Nationale de la Recherche through the P2N program project MASSTOR (Grant number ANR-11-NANO-022).

References and links

1.

G. P. Pells and M. Shiga, “The optical properties of copper and gold as a function of temperature,” J. Phys. C (Solid St. Phys.) 2, 1835–1846 (1969). [CrossRef]

2.

K. Ujihara, “Reflectivity of metals at high temperatures,” J. Appl. Phys 43, 2376–2382 (1972). [CrossRef]

3.

R. Rosei, F. Antonangeli, and U. M. Grassano, “d bands position and width in gold from very low temperature thermomodulation measurements,” Surf. Sci. 37, 689–699 (1973). [CrossRef]

4.

P. Winsemius, M. Guerrisi, and R. Rosei, “Splitting of the interband absorption edge in Au: Temperature dependence,” Phys. Rev. B 12, 4570–4572 (1975). [CrossRef]

5.

P. Winsemius, F. F. van Kampen, H. P. Lengkeek, and C. G. van Went, “Temperature dependence of the optical properties of Au, Ag and Cu,” J. Phys. F: Metal Phys. 6, 1583–1606 (1976). [CrossRef]

6.

R. Rosei and D. W. Lynch, “Thermomodulation spectra of Al, Au and Cu,” Phys. Rev. B 5, 3883–3893 (1972). [CrossRef]

7.

T. Holstein, “Theory of transport phenomena in an electron-phonon gas,” Ann. Phys. 29, 410 (1964). [CrossRef]

8.

R. N. Gurzhi, “Mutual electron correlation in metal optics,” Sov. Phys. JETP 8, 673 (1959).

9.

J. A. McKay and J. A. Rayne, “Temperature dependence of the infrared absorptivity of the noble metals,” Phys. Rev. B 13, 673–685 (1976). [CrossRef]

10.

C. S. Moreira, A. M. N. Lima, H. Neff, and C. Thirstrup, “Temperature-dependent sensitivity of surface plasmon resonance sensors at gold-water interface,” Sensor and Actuators B 134, 854–862 (2008). [CrossRef]

11.

X.-Y. Zhang, T. Zhang, A.-M. Hu, X.-J. Xue, P.-Q. Wu, and Q.-Y. Chen, “Tunable microring resonator based on dielectric-loaded surface plasmon polariton waveguides,” J. Nanosci. Nanotechnol. 11, 10 520–10 524 (2011).

12.

M. Liu, M. Pelton, and P. Guyot-Sionnest, “Reduced damping of surface plasmons at low temperatures,” Phys. Rev. B 79, 035 418 (2009).

13.

G. Baffou and R. Quidant, “Thermo-Plasmonics: using metallic nanostructures as nanosources of heat,” Laser Photon. Rev. 7, 171–187 (2013). [CrossRef]

14.

M. Rashidi-Huyeh and B. Palpant, “Counterintuitive thermo-optical response of metal-dielectric nanocomposite materials as a result of local electromagnetic enhancement,” Phys. Rev. B 74, 075 405 (2006). [CrossRef]

15.

B. Palpant, M. Rashidi-Huyeh, B. Gallas, S. Chenot, and S. Fisson, “Highly dispersive thermo-optical properties of gold nanoparticles,” Appl. Phys. Lett. 90, 223 105 (2007). [CrossRef]

16.

J.-S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low temperature plasmonics of metallic nanosctructures,” Nanolett. 12, 1561–1565 (2012). [CrossRef]

17.

R. T. Beach and R. W. Christy, “Electron-electron scattering in the intraband optical conductivity of Cu, Ag and Au,” Phys. Rev. B 16, 5277–5284 (1977). [CrossRef]

18.

G. R. Parkins, W. E. Lawrence, and R. W. Christy, “Intraband optical condutuctivity σ(ω, T) of Cu, Ag, Au: Contribution from electron-electron scattering,” Phys. Rev. B 23, 6408–6416 (1981). [CrossRef]

19.

S. K. Ozdemir and G. Turhan-Sayan, “Temperature effects on surface plasmon resonance: Design considerations for an optical temperature sensor,” J. Lightwave Technol. 21, 805 (2003). [CrossRef]

20.

A. Passian, A. L. Lereu, E. T. Arakawa, A. Wig, T. Thundat, and T. L. Ferrell, “Modulation of multiple photon energies by use of surface plasmons,” Opt. Lett. 30, 41–43 (2005). [CrossRef] [PubMed]

21.

A. L. Lereu, A. Passian, J. P. Goudonnet, T. Thundat, and T. L. Ferrell, “Optical modulation processes in thin films based on thermal effects of surface plasmons,” Appl. Phys. Lett. 86, 154 101 (2005). [CrossRef]

22.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “In-line extinction modulator based on long-range surface plasmon polaritons,” Opt. Commun. 244, 455–459 (2005). [CrossRef]

23.

G. Gagnon, N. Lahoud, G. Mattiussi, and P. Berini, “Thermally activated variable attenuation of long-range surface plasmon polariton waves,” J. Lightwave Technol. 24, 4391–4409 (2006). [CrossRef]

24.

O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Theoretical analysis of thermally tunable microring resonator filters made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys. 106, 093 109 (2009). [CrossRef]

25.

K. Hassan, J.-C. Weeber, L. Markey, A. Dereux, O. Pitilakis, and E. E. Kriezis, “Thermo-optic plasmo-photonic mode interference switches based on dielectric loaded waveguides,” Appl. Phys. Lett. 99, 241 110 (2011). [CrossRef]

26.

A. Pitilakis and E. E. Kriezis, “Longitudinal 2×2 switching configurations based on thermo-optically addressed dielectric-loaded plasmonic waveguides,” J. Lightwave Technol. 29, 2636–2646 (2011). [CrossRef]

27.

J.-C. Weeber, K. Hassan, L. Saviot, A. Dereux, C. Boissière, O. Durupthy, C. Chaneac, E. Burov, and A. Pastouret, “Efficient photo-thermal activation of gold nanoparticle-doped polymer plasmonic switches,” Opt. Express 20, 27 636–27 649 (2012). [CrossRef]

28.

J. Gosciniak and S. I. Bozhevolnyi, “Performance of thermo-optics components based on dielectric loaded surface plasmon polariton waveguides,” Scientific report 3, 1803 (2013).

29.

H. Fan and P. Berini, “Thermo-optic characterization of long-range surface plasmon in Cytop,” Appl. Opt. 52, 162–170 (2013). [CrossRef] [PubMed]

30.

C. Kittel, Introduction to solid state physics, 8th ed. (John Wiley and Sons, 2005).

31.

M. G. Nielsen, J.-C. Weeber, K. Hassan, J. Fatome, C. Finot, S. Kaya, L. Markey, O. Albrektsen, S. I. Bozhevolnyi, G. Millot, and A. Dereux, “Grating couplers for fiber-to-fiber characterizations of stand-alone dielectric loaded surface plasmon waveguide components,” J. Lightwave Technol. 30, 3118–3125 (2012). [CrossRef]

32.

A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steiberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B 149, 220–229 (2008). [CrossRef]

33.

H. Raether, Surface Plasmons on Smooth and Rough Surface and on Gratings (Springer-Verlag, Berlin, 1988).

34.

N. B. Dahotre and S. P. Harimkar, Laser Fabrication and Machining of Materials (Springer, New-York, 2008).

35.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, 1999).

36.

J.-C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E 62, 7381–7388 (2000). [CrossRef]

37.

J.-C. Weeber, T. Bernardin, M. G. Nielsen, K. Hassan, S. Kaya, J. Fatome, C. Finot, A. Dereux, and N. Pleros, “Nanosecond thermo-optical dynamics of polymer loaded plasmonic waveguides,” Submitted for publication, (2013).

38.

G. V. Miloshevsky, V. A. Sizyuk, M. B. Partenskii, A. Hassanein, and P. C. Jordan, “Application of finite difference methods to membrane-mediated protein interactions and to heat and magnetic field diffusion in plasmas,” J. Comp. Phys. 212, 25–51 (2006). [CrossRef]

39.

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations (SIAM, Philadelphia, 2007).

40.

M. J. Latif, Heat Conduction, 3rd ed. (Springer-Verlag, Berlin Heidelberg, 2009), pp377.

41.

G. Chen and P. Hui, “Thermal conductivities of evaporated gold films on silicon and glass,” Appl. Phys. Lett. 74, 2942–2944 (1999). [CrossRef]

42.

D. Canchal-Arias and P. Dawson, “Measurement and interpretation of mid-infrared properties of single crystal and polycrystalline gold,” Surf. Sci. 577, 95–111 (2005). [CrossRef]

43.

P. B. Jonhson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

44.

W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys. 103, 073 713 (2008). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics
(350.5340) Other areas of optics : Photothermal effects

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 15, 2013
Revised Manuscript: August 30, 2013
Manuscript Accepted: September 4, 2013
Published: September 13, 2013

Citation
S. Kaya, J.-C. Weeber, F. Zacharatos, K. Hassan, T. Bernardin, B. Cluzel, J. Fatome, and C. Finot, "Photo-thermal modulation of surface plasmon polariton propagation at telecommunication wavelengths," Opt. Express 21, 22269-22284 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22269


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References

  1. G. P. Pells and M. Shiga, “The optical properties of copper and gold as a function of temperature,” J. Phys. C (Solid St. Phys.)2, 1835–1846 (1969). [CrossRef]
  2. K. Ujihara, “Reflectivity of metals at high temperatures,” J. Appl. Phys43, 2376–2382 (1972). [CrossRef]
  3. R. Rosei, F. Antonangeli, and U. M. Grassano, “d bands position and width in gold from very low temperature thermomodulation measurements,” Surf. Sci.37, 689–699 (1973). [CrossRef]
  4. P. Winsemius, M. Guerrisi, and R. Rosei, “Splitting of the interband absorption edge in Au: Temperature dependence,” Phys. Rev. B12, 4570–4572 (1975). [CrossRef]
  5. P. Winsemius, F. F. van Kampen, H. P. Lengkeek, and C. G. van Went, “Temperature dependence of the optical properties of Au, Ag and Cu,” J. Phys. F: Metal Phys.6, 1583–1606 (1976). [CrossRef]
  6. R. Rosei and D. W. Lynch, “Thermomodulation spectra of Al, Au and Cu,” Phys. Rev. B5, 3883–3893 (1972). [CrossRef]
  7. T. Holstein, “Theory of transport phenomena in an electron-phonon gas,” Ann. Phys.29, 410 (1964). [CrossRef]
  8. R. N. Gurzhi, “Mutual electron correlation in metal optics,” Sov. Phys. JETP8, 673 (1959).
  9. J. A. McKay and J. A. Rayne, “Temperature dependence of the infrared absorptivity of the noble metals,” Phys. Rev. B13, 673–685 (1976). [CrossRef]
  10. C. S. Moreira, A. M. N. Lima, H. Neff, and C. Thirstrup, “Temperature-dependent sensitivity of surface plasmon resonance sensors at gold-water interface,” Sensor and Actuators B134, 854–862 (2008). [CrossRef]
  11. X.-Y. Zhang, T. Zhang, A.-M. Hu, X.-J. Xue, P.-Q. Wu, and Q.-Y. Chen, “Tunable microring resonator based on dielectric-loaded surface plasmon polariton waveguides,” J. Nanosci. Nanotechnol.11, 10 520–10 524 (2011).
  12. M. Liu, M. Pelton, and P. Guyot-Sionnest, “Reduced damping of surface plasmons at low temperatures,” Phys. Rev. B79, 035 418 (2009).
  13. G. Baffou and R. Quidant, “Thermo-Plasmonics: using metallic nanostructures as nanosources of heat,” Laser Photon. Rev.7, 171–187 (2013). [CrossRef]
  14. M. Rashidi-Huyeh and B. Palpant, “Counterintuitive thermo-optical response of metal-dielectric nanocomposite materials as a result of local electromagnetic enhancement,” Phys. Rev. B74, 075 405 (2006). [CrossRef]
  15. B. Palpant, M. Rashidi-Huyeh, B. Gallas, S. Chenot, and S. Fisson, “Highly dispersive thermo-optical properties of gold nanoparticles,” Appl. Phys. Lett.90, 223 105 (2007). [CrossRef]
  16. J.-S. G. Bouillard, W. Dickson, D. P. O’Connor, G. A. Wurtz, and A. V. Zayats, “Low temperature plasmonics of metallic nanosctructures,” Nanolett.12, 1561–1565 (2012). [CrossRef]
  17. R. T. Beach and R. W. Christy, “Electron-electron scattering in the intraband optical conductivity of Cu, Ag and Au,” Phys. Rev. B16, 5277–5284 (1977). [CrossRef]
  18. G. R. Parkins, W. E. Lawrence, and R. W. Christy, “Intraband optical condutuctivity σ(ω, T) of Cu, Ag, Au: Contribution from electron-electron scattering,” Phys. Rev. B23, 6408–6416 (1981). [CrossRef]
  19. S. K. Ozdemir and G. Turhan-Sayan, “Temperature effects on surface plasmon resonance: Design considerations for an optical temperature sensor,” J. Lightwave Technol.21, 805 (2003). [CrossRef]
  20. A. Passian, A. L. Lereu, E. T. Arakawa, A. Wig, T. Thundat, and T. L. Ferrell, “Modulation of multiple photon energies by use of surface plasmons,” Opt. Lett.30, 41–43 (2005). [CrossRef] [PubMed]
  21. A. L. Lereu, A. Passian, J. P. Goudonnet, T. Thundat, and T. L. Ferrell, “Optical modulation processes in thin films based on thermal effects of surface plasmons,” Appl. Phys. Lett.86, 154 101 (2005). [CrossRef]
  22. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “In-line extinction modulator based on long-range surface plasmon polaritons,” Opt. Commun.244, 455–459 (2005). [CrossRef]
  23. G. Gagnon, N. Lahoud, G. Mattiussi, and P. Berini, “Thermally activated variable attenuation of long-range surface plasmon polariton waves,” J. Lightwave Technol.24, 4391–4409 (2006). [CrossRef]
  24. O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Theoretical analysis of thermally tunable microring resonator filters made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys.106, 093 109 (2009). [CrossRef]
  25. K. Hassan, J.-C. Weeber, L. Markey, A. Dereux, O. Pitilakis, and E. E. Kriezis, “Thermo-optic plasmo-photonic mode interference switches based on dielectric loaded waveguides,” Appl. Phys. Lett.99, 241 110 (2011). [CrossRef]
  26. A. Pitilakis and E. E. Kriezis, “Longitudinal 2×2 switching configurations based on thermo-optically addressed dielectric-loaded plasmonic waveguides,” J. Lightwave Technol.29, 2636–2646 (2011). [CrossRef]
  27. J.-C. Weeber, K. Hassan, L. Saviot, A. Dereux, C. Boissière, O. Durupthy, C. Chaneac, E. Burov, and A. Pastouret, “Efficient photo-thermal activation of gold nanoparticle-doped polymer plasmonic switches,” Opt. Express20, 27 636–27 649 (2012). [CrossRef]
  28. J. Gosciniak and S. I. Bozhevolnyi, “Performance of thermo-optics components based on dielectric loaded surface plasmon polariton waveguides,” Scientific report3, 1803 (2013).
  29. H. Fan and P. Berini, “Thermo-optic characterization of long-range surface plasmon in Cytop,” Appl. Opt.52, 162–170 (2013). [CrossRef] [PubMed]
  30. C. Kittel, Introduction to solid state physics, 8th ed. (John Wiley and Sons, 2005).
  31. M. G. Nielsen, J.-C. Weeber, K. Hassan, J. Fatome, C. Finot, S. Kaya, L. Markey, O. Albrektsen, S. I. Bozhevolnyi, G. Millot, and A. Dereux, “Grating couplers for fiber-to-fiber characterizations of stand-alone dielectric loaded surface plasmon waveguide components,” J. Lightwave Technol.30, 3118–3125 (2012). [CrossRef]
  32. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steiberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B149, 220–229 (2008). [CrossRef]
  33. H. Raether, Surface Plasmons on Smooth and Rough Surface and on Gratings (Springer-Verlag, Berlin, 1988).
  34. N. B. Dahotre and S. P. Harimkar, Laser Fabrication and Machining of Materials (Springer, New-York, 2008).
  35. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, 1999).
  36. J.-C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E62, 7381–7388 (2000). [CrossRef]
  37. J.-C. Weeber, T. Bernardin, M. G. Nielsen, K. Hassan, S. Kaya, J. Fatome, C. Finot, A. Dereux, and N. Pleros, “Nanosecond thermo-optical dynamics of polymer loaded plasmonic waveguides,” Submitted for publication, (2013).
  38. G. V. Miloshevsky, V. A. Sizyuk, M. B. Partenskii, A. Hassanein, and P. C. Jordan, “Application of finite difference methods to membrane-mediated protein interactions and to heat and magnetic field diffusion in plasmas,” J. Comp. Phys.212, 25–51 (2006). [CrossRef]
  39. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations (SIAM, Philadelphia, 2007).
  40. M. J. Latif, Heat Conduction, 3rd ed. (Springer-Verlag, Berlin Heidelberg, 2009), pp377.
  41. G. Chen and P. Hui, “Thermal conductivities of evaporated gold films on silicon and glass,” Appl. Phys. Lett.74, 2942–2944 (1999). [CrossRef]
  42. D. Canchal-Arias and P. Dawson, “Measurement and interpretation of mid-infrared properties of single crystal and polycrystalline gold,” Surf. Sci.577, 95–111 (2005). [CrossRef]
  43. P. B. Jonhson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
  44. W.-J. Lee, J.-E. Kim, H. Y. Park, S. Park, M.-S. Kim, J. T. Kim, and J. J. Ju, “Optical constants of evaporated gold films measured by surface plasmon resonance at telecommunication wavelengths,” J. Appl. Phys.103, 073 713 (2008). [CrossRef]

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