Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites

Bin Zhao; Lei Gao

doi:10.1364/OE.17.021433

1. Introduction

It is known that when a light beam is totally reflected from an interface between two different media, the Goos-Hänchen shift (GHS) for the beam takes place [1–3]. Goos-Hänchen shift was widely studied in recent years because of its potential applications in many aspects such as integrated optics [4], optical waveguide switch [5], and optical sensors [6] etc. Although the GH shift was initially found in the total reflection regime, it was later extended to the partial reflection case [7–9]. In addition, the GH shifts were found to be large positive or negative for both reflected and transmitted beams in different media or structures, such as dielectric slabs [10–13], metal surfaces [14,15], dielectric-chiral surface [16–19], resonant absorbers [20], multilayered structures [21], photonic crystals [22], and negative refractive materials [23–26].

On the other hand, based on the GH effect, it is possible to manipulate the spatial beam position. For instance, Hou et al introduced the defect with third-order nonlinear dielectric response in one-dimensional photonic crystals to control the bistable lateral shift [27]. Wang et al [28] presented a proposal to control the GH shift of a light beam, which is injected into a cavity configuration containing two-level atomic medium by adjusting the intensity and detuning of the control field. Later, electric control of the lateral shift of the reflected beam was proposed in a structure of symmetrical metal-cladding waveguide due to the electro-optic and piezoelectric effects [29]. Chen et al reported the possibility of constructing an optical sensor for temperature monitoring based on the Goos-Hänchen effect. They observed significant variation of negative GH shifts with the temperature for p-polarized incident light at almost grazing incidence onto the metal [6].

In this paper, we would like to investigate a tunable Goos-Hänchen lateral shift in metal/dielectric composite system. Here, the composite material consists of metal nanoparticles and dielectric host medium. We shall manipulate the GHS by tuning both the temperature and the volume fraction of metal particles. Actually, the temperature effect must be taken into account [30], because for practical or technological applications, the materials are generally at elevated temperatures. In addition, for metal/dielectric composites, there may exist the percolation threshold fraction f_c, above (below) which the composite exhibits metallic (dielectric) behavior [31]. As a consequence, the dependence of GHS on the temperature from metal/dielectric composites may take quite different behavior from that of the pure metal. And the adjustment of the volume fraction may be helpful to get deep understanding the loss-induced transition of the Goos-Hänchen shift for metals and dielectrics [32].

The paper is organized as follows. In Section 2, we formulate the temperature-dependent permittivity of the metal/dielectric composites within Bruggeman effective medium theory, and then GHS at the surface of metal/dielectric composites is analyzed. Numerical results for temperature-dependent GHS are shown in Section 3. In Section 4, to confirm our theoretical results, we take one step forward to perform numerical simulations for Gaussian beams, and finite-element simulations with COMSOL Multiphysics. A summary of our results and conclusion is given in Section 5.

2. Theoretical analysis

We assume that the incident light is from the air with the permittivity ε ₁=1 onto the surface of metal/dielectric composite material at the angle θ from the normal, as shown in Fig. 1. The composite media consist of spherical metal nanoparticles with volume fraction f and permittivity ε_m and dielectric host with volume fraction 1-f and permittivity ε_d. We consider the Goos-Hänchen shift D of the incident wave reflected from the interface between an air-composite interface. For simplicity, we aim at the p-polarized case, for which the magnetic field is perpendicular to the incident plane.

We take two steps to investigate the GHS for the metal/dielectric composites. In the first step, we derive temperature-dependent effective permittivity of the metal/dielctric composites. Since the permittivity of the dielectric host is generally independent of (or weakly dependent on) the temperature, we need to describe the temperature-dependent permittivity of metal component.

We choose a Drude model to describe the permittivity of certain simple and noble metals within the appropriate ranges of light frequencies. Hence the permittivity of the metal such as Ag can be written as [30, 33],

ε_{m} (T) = 1 - \frac{ω_{p}^{2} (T)}{ω [ω + {iω}_{c} (T)]},

where ω_c(T) and ω_p(T) are, respectively, the collision and plasmon frequencies. ω_p(T) in Eq. (1) is assumed to be the form,

ω_{p} (T) = ω_{p} (T_{0}) {[1 + γ (T - T_{0})]}^{- 1 ⁄ 2},

where γ is the bulk expansion coefficient of the metal, and T ₀ is a reference temperature taken to be the room temperature 300K.

Fig. 1. Geometry indicating the GHS, defined as the light beam is incident from air (medium 1) onto a metal-dielectric composite (medium 2) surface.

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On the other hand, ω_c(T) includes two parts due to the contributions from electron-electron and phonon-electron interactions, and is written as,

ω_{c} (T) = \frac{ω_{p}^{2} (T) ε_{0}}{σ (0)} \frac{[\frac{1}{10} + {(\frac{T}{T_{θ}})}^{5} \int_{0}^{T_{θ} ⁄ T} \frac{y^{4} dy}{e^{y} - 1}]}{\int_{0}^{1} \frac{y^{5}}{(e^{y} - 1) (1 - e^{−y})} dy} + \frac{1}{12} π^{3} \frac{Γ Δ}{ħ E_{F}} {[(k_{B} T)}^{2} + {(\frac{ħ ω}{2 π})}^{2}],

where T_θ is the Debye temperature and σ (0) is DC conductivity at T=T_θ with the unit Ω^-1 m ^-1. In addition, the dimensionless quantities Γ and Δ, respectively, represent the constants giving the average the Fermi surface of the scattering probability and the fractional umklapp scattering [30, 33].

The dielectric component in medium 2 (composite medium) is assumed to be MgF ₂, whose permittivity has a negligible temperature dependence,

ε_{d} = {(1.36975 + \frac{35.821}{10 λ - 1492.5})}^{2}

where the incident wavelength λ is given in nanometers.

We make use of Bruggeman effective medium theory to derive the temperature-dependent effective permittivity ε_e(T) of a metal-dielectric composite, that is,

f \frac{ε_{m} (T) - ε_{e} (T)}{ε_{m} (T) + 2 ε_{e} (T)} + (1 - f) \frac{ε_{d} - ε_{e} (T)}{ε_{d} + 2 ε_{e} (T)} = 0,

and its solution admits

ε_{e} (T) = \frac{1}{4} {ε_{m} (3 f - 1) + ε_{d} (2 - 3 f) \pm \sqrt{8 ε_{d} ε_{m} + {[ε_{m} (3 f - 1) + ε_{d} (2 - 3 f)]}^{2}}} .

Note that a percolation threshold f_c=1/3 is predicted, above (below) which the medium 2 (composite materials) is metallic (dielectric) [31, 34]. As a consequence, the Goos-Hänchen shift is expected to take intriguing behavior for different volume fractions and different temperatures.

In the second step, We adopt the standard Fresnel formula to derive the reflection coefficient R and the corresponding phase δ (θ) for the incident angle θ [7] at the interface between the air and the composite material. They are given as,

R (θ) = \frac{ε_{e} \cos θ - \sqrt{ε_{1} (ε_{e} - ε_{1} \sin^{2} θ)}}{ε_{e} \cos θ + \sqrt{ε_{1} (ε_{e} - ε_{1} \sin^{2} θ)}},

and

δ (θ) = Im {\ln [\frac{ε_{e} \cos θ - \sqrt{ε_{1} (ε_{e} - ε_{1} \sin^{2} θ)}}{ε_{e} \cos θ + \sqrt{ε_{1} (ε_{e} - ε_{1} \sin^{2} θ)}}]} .

As a consequence, we adopt the stationary-phase method to derive the generalized expression [3],

D = - \frac{1}{k} \frac{d δ (θ)}{d θ},

where k=2π/λ is the incident wave number.

3. Theoretical Results

Without loss of generality, we take Ag/MgF ₂ as the composite to study effects of both temperature and volume fraction on Goos-Hänchen shift for the reflected beam from the interface between air and the composite.

Since the Goos-Hänchen shift is strongly dependent on the phase of the refraction coefficient from the air-composite interface, we first investigate the reflectivity as a function of the incident angle for different temperatures in Fig. 2. For f=0.1< f_c=1/3, the composite material exhibits the dielectric behavior [ε_e(T=300K)=1.14+0.348i, ε_e(T=600K)=1.16+0.286i, and ε_e(T=1200K)=1.25+0.184i]. As a consequence, with increasing θ, the reflectivity is decreased (is almost zero, see the insert) at first, then takes a minimum at the Brewster angle, and is increased up to 1 at almost grazing incidence θ=90° [see Fig. 2(a)]. This is the typical behavior for dielectric material and the Goos-Hänchen shift is expected to be large at the Brewster angle. On the other hand, for λ=248nm, the real part of the permittivity of metallic component is positive, and hence possesses the dielectric behavior. Therefore, the composite material for f=0.7 is dielectric too [ε_e(T=300K)=0.342+0.149i, ε_e(T=600K)=0.425+0.146i, and ε_e(T=1200K)=0.563+0.148i], and the reflectivity for f=0.7 [see Fig. 2(b)] takes similar behavior as the case for f=0.1. We further find that with increasing the temperature, the Brewster angle shifts to large value and the reflectivity spectrum becomes broad. Actually, such behavior is quite obvious for f=0.7, in which the dielectric component prevails [see Fig. 2(b)]. On the contrary, for λ=413nm and f=0.7, the composite material is metallic [ε_e(T=300K)=-1.62+1.74i, ε_e(T=600K)=-1.57+1.75i, ε_e(T=1200K)=-1.48+1.77i]. Therefore, one predicts that the reflectivity achieves a minimal value, which is far from zero.

Fig. 2. The reflectivity of the composite is plotted versus the incident angle θ for different temperatures. (a) f=0.1 and λ=295nm; (b) f=0.7 and λ=248nm and (c) f=0.7 and λ=413nm. The inserts of (a) and (b) show the reflectivity near the Brewster angle.

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Fig. 3. The GHS (Dp) of the material as a function of incident angle θ for different temperature: (a) f=0.1 and λ=295nm, (b) f=0.7 and λ=248nm, and (c) f=0.7 and λ=413nm.

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Next, we study the Goos-Hänchen shift for p-polarized light D_p as a function of the incident angle for various temperatures. As we know, the composite material exhibits the dielectric behavior for the given incident wavelengths and volume fractions in Fig. 3(a) and Fig. 3(b). As a result, D_p shows a characteristic, isolated (negative) resonance near the Brewster angle [32]. In addition, we observe positive GHS for the incident angle slightly larger than the Brewster angle. This is due to the usual Goos-Hänchen effect for total internal reflection, now modified because of the absorption in the composites [9]. However, for λ=413nm and f=0.7, as the composite material is metallic, D_p should exhibit a large negative value at grazing incidence, as expected [see Fig. 3(c)]. As far as the temperature effect is taken into account, we find that the temperature plays an important role in determining the GHS near and above the Brewster angle for dielectric composites. On the other hand, monotonic variation of negative GH shifts can be observed as a function of the temperature at the grazing incident angle for metallic composites, which is quite similar as the one for pure metal [6]. As a consequence, in what follows, we only concentrate on the temperature effect on the GHS for dielectric composites.

Fig. 4. The GHS (Dp) plotted versus temperature under different incident angle: (a) f=0.1 and λ=295nm, the curve for θ=89° is 10-fold magnified; (b) f=0.3 and λ=295nm and (c) f=0.7 and λ=248nm.

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Figure 4 shows GHS D_p as a function of T for dielectric composites. It is clearly seen that D_p can become more negative (or positive) at higher temperature for θ=50° in Fig. 4(a) and θ=30° in Fig. 4(b) [or for θ=55° in Fig. 4(c)]. Such temperature-dependence is quite similar as that for bulk metal [6]. However, for dielectric composites, it is possible to predict that D_p is less negative at higher temperature [see Fig. 4(b) for θ=70°]. To one’s interest, one predict the nonmonotonic temperature-dependence in both positive and negative GHS region. This should be in contrast to the case for pure metal or metallic composites. Moreover, with increasing the temperature, GHS can crossover from positive to negative values (or negative to positive values), and hence one may realize the reversal of GHS by the suitable adjustment of the temperature.

4. Numerical Simulations

To demonstrate the validity of the above analysis, we first perform numerical simulations based on the momentum method [11, 25, 26]. We consider an incident beam of the Gaussian shape with the incident angle θ [11],

E_{in} (x, y) ∣_{x = 0} = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} A (k_{y}) \exp ({ik}_{y} y) {dk}_{y} .

Here A(k_y)=w_y exp[-(w ² _y/2)(k_y-k _y0)²] is the angular spectrum of the Gaussian beam, where w_y=w ₀ secθ (w ₀ is the width of the beam at the waist) and k_y=k ₀ sinθ. Then, the reflected field is given by,

E_{r} (x, y) ∣_{x = 0} = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} R (k_{y}) \exp ({ik}_{y} y) {dk}_{y} .

where R is given in Eq. (7). To obtain the GHS, one must solve Eq. (11) to find the location where the reflected field achieves the maximum.

Numerical simulations with the momentum method are shown in Fig. 5. It is evident that both monotonic and non-monotonic dependences on T of the GHS are again predicted by the numerical simulations, and the numerical simulations are in qualitative agreement with the above theoretical analysis based on stationary phase method for small width w ₀. Note that there exists the discrepancy between the numerical and theoretical results due to the distortion of the reflected Gaussian beam [35, 36]. It is expected that the wider the incident beam is, the closer to the theoretical results the simulation data are. Moreover, a better measure for the GHS is the centroid of the beam intensity distribution [37, 38].

Fig. 5. The comparison for the results given by the Artmann’s formula (solid curve) and the Gaussian-shaped beams with w ₀=5λ for f=0.1,λ=295nm [(a) and (b)]; f=0.7,λ=248nm [(c) and (d)].

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Fig. 6. COMSOL Multiphysics simulation of the GH shifts for two Gaussian-shaped beams: (a) λ=413nm and θ=50° (ε_e=-1.62+1.74i); (b) λ=248nm, and θ=54° (ε_e=0.563+0.148i)

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To give a comprehensive understanding, we take one step forward to simulate the GHS of the reflected light using the COMSOL Multiphysics. In the simulations, we consider an incident beam with a spatial Gaussian profile again, and the width of the beam is 5λ. The results are shown in Fig. 6 for f=0.7 and T=1200K. For λ=413nm, the composite is metallic and the simulated shift based on the COMSOL Multiphysics is negative. This is in accordance with our theoretical results [see Fig. 3(c)]. On the contrary, for λ=248nm, the composite is still dielectric and the simulated shift is positive, which is agreement with that shown in Fig. 3(b). Therefore, our theoretical results are indeed reliable.

5. Conclusion

To summarize, we have outlined the stationary-phase method to the analysis of Goos-Hänchen shift of the reflected beam from the metal/dielectric composite material at elevated temperatures. It is generally found that the dependence of GHS on the temperature becomes significant near the Brewster angle for the dielectric composites and at the almost grazing incidence for the metallic composites. In detail, through the suitable adjustment of the temperature, one may achieve large negative (or positive) lateral shift, and even GHS takes nonmonotonic variation with increasing temperature. To one’s interest, through suitable adjustment of the temperature, one may realize the reversal of the GHS, and achieve large values of GHS. In addition, our theoretical results are compared with the numerical simulations, and reasonable agreement is found. Our investigation may be helpful for us to measure the relevant physical parameters and realize the optical temperature sensing based on Goos-Hänchen shift.

Acknowledgments

This work was supported by the National Basic Research Program under Grant No. 2004CB719801, the Natural Science of Jiangsu Province under Grant No. BK2007046, and the Key Project in Science and Technology Innovation Cultivation Program of Soochow University.

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Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites

Abstract

1. Introduction

2. Theoretical analysis

3. Theoretical Results

4. Numerical Simulations

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (11)

Optics Express