1. Introduction

Surface plasmon (SP), a collective excitation at the interface between a conductor and a dielectric, have provided us various attractive aspects from the viewpoint of both fundamental sciences and advanced applications [1]. SP is a transverse magnetic (TM) mode, traveling along the interface with the field amplitude decaying exponentially in perpendicular to the interface. The electric field amplitude of SP at the boundary is strongly enhanced at the resonance frequency due to free conduction carriers, leading to further enhancement of light-matter interactions. Traditionally, SP has been investigated as a probe of the equilibrium electron density in surface science [2]. Recently, surface-enhanced Raman spectroscopy with SP was proposed for enhancing tiny signals from a single molecule so that they are detectable [3,4]. Furthermore, the small index change of the surrounding medium causes a sensitive frequency change on SP resonance, providing high-sensitive SP sensors for the sensing of diluted biomolecules [5–9].

SP has a remarkable character in the dispersion relation: it has a larger wave vector than that of light in a vacuum, prohibiting SP excitation in the normal light incidence due to the selection rules of the wave vector. Thus, it is necessary to use an external SP coupler for the excitation, such as prism-coupled method, small particles, surfaces with random roughness and periodic structures such as photonic crystal and grating [1, 10–13]. Essentially, the resonance frequency of SP is determined by the dielectric constant of the active material (conductors) but is strongly modulated by the external coupler. Such an active material with an external coupler gives rise to various unique optical properties, but coincidentally makes any explanation of the phenomenon much more difficult [10, 11, 14–16].

One of the most basic and simple experiments for the demonstration of modulation is a conventional prism coupling method in the Otto’s configuration (prism/air/active material) [10, 11, 15, 16]. This is called as attenuated total reflection (ATR) technique where the evanescent wave is generated by the internal total reflection in the prism and coupled with the SP. In this configuration, the coupling strength between the evanescent wave and SP can be controlled by changing the distance between the prism and the active material. A frequency shift in the SP resonance has been observed with changing the distance. Otto has explained the frequency shift using the additional damping mechanism, which comes from the presence of the prism [10, 11]. The ratio of the frequency shift Δω to the intrinsic SP resonance frequency ω _SP, |Δω/ω _SP|, is of the order of 0.001 [11]. The interpretation has been acceptable within the surface polariton picture whenever the damping constant of the active material is small enough in comparison with the plasma frequency. A simple expansion of this approximation is not applicable to a more general situation, e.g., in a system in which the surface polariton picture is invalid, or in a system with more complicated structures. It is crucial to develop an alternative interpretation scheme for a comprehensive understanding of the modulation of the resonance frequency in the ATR technique.

To understand the frequency shift of the SP resonance, a time-domain ATR technique with terahertz coherent pulses (THz TD-ATR) is powerful and promising [17]. This technique has two advantages for investigating SP resonance. One is that systematic measurements can be performed with the control of the prism-active material distance in the Otto configuration. The penetration length of the evanescent wave is almost the same as its wavelength, determining the typical length scale of the prism-active material distance. Since the wavelength of 1 THz pulse corresponds to 300 μm, which is much longer than that in visible light, the distance can be exactly controlled with the conventional translation stage. The other advantage is that time-domain spectroscopy gives precise information on the electric field. Remarkable developments of an ultrashort pulse laser technique bring rapid developments of coherent THz pulse generation and detection with the optical sampling methods, allowing the precise detection of phase as well as the intensity of reflected light [18–20]. Recently, the drastic phase change on surface plasmon resonance has become interesting from a research viewpoint for the purposes of novel high-sensitive sensing [6–9, 14]. Even interferometric phase measurement in a visible frequency region offers several orders of higher sensitivity than that in traditional intensity measurements [5]. Both the amplitude and phase of a reflected electric field should allow us to reveal the underlying nature of SP resonance.

In our previous work, we proposed THz TD-ATR spectroscopy, which can access the optical constants of various samples [17]. In this paper, we investigate the SP resonance in a doped semiconductor systematically with a THz TD-ATR technique. We experimentally demonstrate the alternative interpretation that a reflected electric field in an ATR configuration is given by the interference between the electromagnetic wave reflected at the prism surface and that reemitted from excited SP, which includes the Otto framework.

2. THz time-domain attenuated total reflection technique (THz TD-ATR)

Figure 1(a) shows the schematic experimental setup of the THz TD-ATR technique. Optical pulses from a Ti: Sapphire regenerative amplifier (Spectra Physics, Inc.) with a repetition rate of 1 kHz, an average power of 700 mW, a center of frequency of 800 nm, and a pulse duration of 150 fs (full-width at half-maximum-intensity) are split into two beams. One of these beams is used to generate coherent broadband THz pulses with optical rectification in (110) oriented-ZnTe crystal with a thickness of 2.5 mm. To prevent ZnTe crystal for THz pulses generation from damaging and to increase the efficiency of THz pulse generation, spot diameter of pump beam is expanded to ≈30 mm with the lenses. It induces the maximum incident fluence of a pump beam (≈0.1 mJ/cm²) on the ZnTe. In order to control the polarizations (s- and p-polarizations) of the incident THz pulses on the prisms, the <-110> crystallographic orientation of the ZnTe crystal is set at angles of 90° and 60° to the polarization of the pump beam [19,20]. The wire grid polarizer is used to adjust the polarization. The generated THz pulses are focused onto the (110) oriented-ZnTe crystal with a thickness of 1.0 mm with three off-axis parabolic mirrors. The other beam is used as the sampling beam of the THz pulses and is focused onto the same spot of the ZnTe crystal for detection through the small hole of a parabolic mirror, and the birefringence of the sampling beams modulated by the electric field of THz pulses is measured using a quarter wave plate, a Wolston prism, and two balanced Si detectors. To detect a small change in the sampling beams, we synchronously chop the pump beam at 500 Hz, which blocks every other pump beam. The sampling beam is then sampled in a box car integrator that inverts every electric signal. A computer then collects the electric signals and separates them to extract pump-influenced signals and non-influenced ones. By varying the time delay between the two pulse trains, the electric field amplitude of the THz pulses can be detected as a function of time with sub-picosenond resolution.

The experimental setup shown in Fig. 1(a) is based on conventional THz transmission geometry [18]. We realize total reflection geometry by inserting prisms with peculiar shapes. The prism is basically the Dove prism shown in Fig. 1(b) [21], in which total reflection occurs with the same optical axis of incident and outgoing beams. We also develop a prism as shown in Fig. 1(c), in which the incident electromagnetic wave has three total reflections. The chromatic aberration may be reduced since its incidence is perpendicular to the side face of the prism [21]. To obtain evanescent waves with different wave vectors, we used two different prisms made of material with different refractive indices (n ₁=${\epsilon }_1^{1/2}$) in the THz frequency region. One is a plastic prism shaped as shown in Fig. 1(c), and the other is a MgO prism shaped as shown in Fig. 1(b). The characteristic parameters of these prisms and the evanescent waves are summarized in Table 1. The wave vectors parallel to the interface, k _x(=${\epsilon }_1^{1/2}$ ω sinθ ₁/c), are 35 cm^-1 (plastic prism) and 82 cm^-1 (MgO prism) at 1 THz. The condition of internal total reflection (ε ₁ sinθ ₁>l) is maintained so that, in the absence of active material, an evanescent wave is generated whose electric field decreases exponentially away from the prism surface. The penetration lengths of the evanescent wave in the air, λ(=($k_{\mathrm{x}}^2$ -(ω/c)²)^-1/2), are then 116.5 μm (plastic prism) and 14.1 μm (MgO prism).

 

Fig. 1. (a) Schematic diagram of the experimental setup. WG: wire grid polarizer, PBS: polarized beam splitter, λ/4: quarter wave plate. (b), (c) Coupling prism used to produce an evanescent wave on the internal total reflection surface. The n-type (100) oriented-InAs crystal (7 mm×7 mm×0.5 mm) is separated from the prism by air with thickness d. The dimensions are (b) h₁=8 mm, w₁=20.7 mm, and α=104.8° for the MgO prism and (c) h₂=15.6 mm, m=9 mm, and β=135° for the plastic prism.

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Table 1. Parameters of the Prisms and the Generated Evanescent Wave^a

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We use an undoped (100)-oriented InAs crystal as the active material, which tends to be slightly of the n-type. The carrier density obtained by four contacts resistance measurement is 1.5(±0.3)×10¹⁶ cm^-3 and the corresponding plasma frequency is in THz frequency region. The InAs is held on the electrically moved stage at room temperature to control the distance d. We inject a visible white light beam into the air gap layer from prisms. Since both prisms have high transmissivity in the visible frequency region, the parallelism between the surfaces of the prism and InAs can be aligned by monitoring the spatial patterns of interference, and distance d can be calibrated from the interference spectrum within 1 μm accuracy. This white light is cut during the THz-TD-ATR measurements to avoid additional carrier injection.

3. Experimental results

Figure 2 shows the temporal profile of an electric field in ATR geometry with p-polarized incidence at different distances between the plastic prism and InAs crystal. The upper waveform E _Me(d=∞, t) is measured at a sufficiently large distance, which can be considered as the total-reflected pulse on the prism surface without InAs. This waveform in a long-delay time scale is shown in the inset, but we focus on the waveforms around the time delay of 0 ps to identify the characteristic features. The amplitude of the electric field was reduced, and its peak position, as denoted by red circles, was delayed by decreasing distance d.

 

Fig. 2. Time development of the electric field of p-polarized THz pulses obtained with changing the distance d between the plastic prism and InAs. The reference signal of the THz pulse is obtained when the distance is large enough to enable the assumption that d equals infinity. The inset shows a reference signal in a long-delay time scale. The red dots indicate a time delay in which the electric field has a maximum value as an eye guide.

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For further evaluation of the prism-InAs distance dependence of THz pulses, we evaluated complex reflective coefficients. Fourier transformation of the temporal profile yields both the amplitude and phase of the electric field E _Me(d, ω) in the frequency domain. We obtain the attenuated total reflectivity (ATR) and phase shift (Δϕ) from the experimental results:

where the normalization of reference data ensures that the spectra are independent of the generated THz pulse shape as well as of instrumental functions. Figure 3 shows (a) ATR and (b) Δϕ at different d, which are derived by using the temporal profiles in Fig. 2. Two prominent features are evident in Fig. 3. One feature is the frequency of ATR dips. In Fig. 3(a) for a large distance (d=187 μm), an ATR dip appears at 1.3 THz. However, its position shifts toward a higher-frequency region with d decreasing, as indicated by arrows. In Fig. 3(b), the other feature is the anomalous phase jump behavior. When the minimum value of ATR approaches zero, Δϕ shows a jump at the frequency at which ATR has a minimum value. At a distance larger than 53.2 μm, Δϕ monotonously decreases toward zero with the frequency. However, the direction of the phase jump is drastically changed at two specific distances. Δϕ increases toward 2π at d=31.9 μm and again decreases toward zero at d=11.9 μm. These features are specific to the p-polarized incidence. We also perform THz TD-ATR spectroscopy with s-polarized incidence. The experimental results, as shown in Fig. 4, indicate no dip in ATR and no steep phase jump.

 

Fig. 3. (a) Attenuated total reflectivity (ATR) and (b) phase shift (Δϕ) measured with different distances d by using the plastic prism (solid line). Incident THz pulses are p-polarized. The arrows in (a) indicate the frequency where ATR has minimum values. Calculated curves (dashed line) are also shown.

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Fig. 4. (a) Attenuated total reflectivity (ATR) and (b) phase shift (Δϕ) measured with different distances d by using the plastic prism (solid line). Incident THz pulses are s-polarized. Calculated curves (dashed line) are also shown.

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Figures 5(a) and 5(b) show the ATR and Δϕ spectra obtained by the THz TD-ATR with MgO Dove prism. The ATR spectra are similar to the experimental results obtained with the plastic prism, but the typical scale of the prism-InAs distance is approximately one-order smaller. The distance dependence of the ATR and Δϕ spectra is different from two prominent features in the case of the plastic prism. In Fig. 5(a), the minimum value of ATR lies near 1.6 THz at d=4.4 μm, and its position shifts toward the lower frequency region as d decreasing, as indicated by arrows. In Fig. 5(b), a phase jump occurs at the frequency at which ATR has a minimum value, as in the case of the plastic prism. However, there is only one direction change of the phase jump; the phase monotonously decreases toward zero when d is more than 2.1 μm, and it increases toward 2π when d equals 1.6 μm. The number of drastic direction changes with the MgO prism is different from that observed in the plastic prism.

 

Fig. 5. (a) Attenuated total reflectivity (ATR) and (b) phase shift (Δϕ) measured with different distances d by using the MgO prism (solid line). The incident THz pulses are p-polarized. The arrows in (a) indicate the frequency at which ATR has a minimum value. The calculated curves (dashed line) are also shown.

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4. Two-interface model

The experimental results in Figs. 3, 4, and 5 indicate that the spectra of ATR and Δϕ strongly depend on the wave vector of the evanescent wave, the refractive index of the prism, the incident angle, the polarization of the THz pulses, and the distance between the prism and the active material. In principle, ATR and Δϕ can be easily calculated with the standard theory for electromagnetic propagation through layered media [11]. We assume a plane monochromatic wave incident upon the two-interface system, as shown in Fig. 6. The reflection coefficient r ₁₂₃ is expressed as follows:

with

where k _zi is the z-component of the wave vector in the ith layer perpendicular to the surface of each medium, θ ₁ is the angle of incidence, and $r_{\text{ij}}^{p\left(s\right)}$ is the Fresnel coefficient of the p- (or s-) polarized electromagnetic wave at the interface between the ith layer and the jth layer. Considering that the observed spectra E _Me(d, ω)/E _Me(d=∞, ω) correspond to r ₁₂₃(d, ω)/r ₁₂, the dielectric constant ε ₃(ω) can be derived from the experimental data [17]. Figure 7(a) shows the real and imaginary part of ε ₃(ω) derived by using the experimental data in Fig. 3. These dispersion curves of ε ₃ are well described with the Drude model, represented by

where ε_b =16.3 is the background high-frequency dielectric constant [22], ω _p=(Ne ²/ε _b ε ₀ m ^*)^1/2 is the plasma frequency, N and m ^* are the carrier density and effective mass, respectively, and γ is the damping constant. The best-fitted parameters are ω _p/2π=1.91 THz (corresponding carrier density N=1.92×10¹⁶ cm^-3) and γ/2π=0.58 THz. Figure 7(b) shows the reflectivity measured with a p-polarized 45° incidence in conventional reflection geometry and that calculated with the derived ω _p and γ. The agreement of these individual experimental results with the calculations in Figs. 7(a) and (b) indicates that the derived ε ₃ is relevant. The evaluated plasma frequency is almost reproduced by that obtained with the electric measurement. In Figs. 3, 4, and 5, the theoretical curves derived by using Eqs. (3)–(4) with the obtained parameters are added as denoted by dashed curves. The prominent features in ATR and Δϕ spectra are well reproduced by the two-interface model. The frequencies of an ATR minimum are similar to the surface plasma frequency ω _p ${\epsilon }_{\mathrm{b}}^{1/2}$/(2π(ε _b+1)^1/2)=1.85 THz [10]. In addition, in Fig. 4, ATR dips and phase jumps cannot be observed with the s-polarization incidence. Furthermore, the inset in Fig. 7(a) shows that the condition for the existence of SP at the interface between air and active material (Re[ε ₃]<0) and Im[ε ₃]>0) is satisfied [23]. These results indicate that ATR dips and phase jumps originate from the SP resonance.

 

Fig. 6. Detailed layout of the three-layered media used for the calculation of r ₁₂₃. ε ₁, ε ₂, and ε ₃ indicate the dielectric constant in a prism, air, and InAs, respectively. Here, ${\epsilon }_2^{1/2}$=1 is assumed. θ ₁ is the angle of incidence, and θ _c is the critical angle of internal total reflection.

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Fig. 7. (a) Real and imaginary part of the dielectric constant ε ₃ (blue and red circles, respectively,) measured by the THz TD-ATR technique and calculated by the Drude model (blue and red dashed lines). The inset shows the expanded derived ε ₃ around 1.7 THz. The dotted line indicates zero of the real part of ε ₃ for an eye guide. (b) 45°-incidence reflectivity measured (solid line) with p-polarized THz pulses and that calculated (dashed line) with the Drude model.

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5. Destructive interference effect on the surface plasmon resonance

As shown in the previous section, the experimental results agree well with calculations derived by using the two-interface model. The ATR dips and phase jumps are attributed to the SP resonance. However, the physical meaning of the anomalous behaviors in the ATR dips and phase jumps are not clarified by the model calculation. The imaginary part of a dielectric constant at the frequency at which the dips of an ATR have a minimum is almost the same as the real part (|Re[ε(ω)]|≈Im[ε(ω)]), which is beyond the assumption for the explanation of the SP resonance introduced by Otto [10,11].

In order to obtain an alternative physical interpretation for the SP resonance, we divide the right-hand side of Eq. (3) into two components:

with

The first term in the right-hand side of Eq. (5) is the electric field reflected at the prism surface, and the second term is that reemitted from the SP excited at z=0 in the prism when the electric field amplitude of the incident electromagnetic wave is unity. The second term shows that the B(d, ω), which is the electric field amplitude at z=d in air, decays with the exponential component exp(-η _z2 d) and goes out to the prism with transmission coefficient t ₂₁. The superposition of these two components results in the observed reflected electromagnetic wave.

We elucidate the zero-ATR condition, in which two components in Eq. (5) are cancelled out by destructive interference. This can be derived by equalizing B(d, ω) = t₂₁^-1 r ₂₃exp(-η _z2 d) obtained by equalizing Eq. (5) to zero with Eq. (6), as follows:

Equation (7) includes two unknown variables of the frequency ω and the prism-active material distance d. This equation can be solved by separating the variables and dividing them into the phase and the amplitude components for r ₂₃:

where we use the total reflection condition of |r ₁₂|=1. The phase-matched condition of Eq. (8) provides the solution of ω ₀, and substituting the solution ω ₀ into Eq. (9) yields the solution of d ₀. An additional condition of r ₂₃ larger than unity at ω ₀ is required to obtain the real number d ₀.

Equations (8) and (9) can be easily solved with the graphical approach. Figures 8(a) and 8(b) show the phase and amplitude spectra of r ₂₃ with γ/2π=0.58 THz for the plastic prism and the MgO prism, respectively. The phase of r ₂₃, arctan(r ₂₃), is zero at ω=0, but it increases with frequency and reaches the maximum value. Arctan(r ₁₂)+π is shown by dashed lines, and the frequencies of intersecting points that correspond to the solutions of ω ₀, by blue circles. Substituting the derived ω ₀ into Eq. (9), we obtain two solutions, (ω ₀/2π=1.69 THz, d ₀=48.3 μm) and (ω ₀/2π=1.91 THz, d ₀=21.4 μm), in the case of the plastic prism, and one solution, (ω ₀/2π=1.48 THz, d ₀=1.8 μm), in the case of the MgO prism.

Although we succeeded in deriving the zero-ATR condition with our experimental parameters, it is difficult to determine experimentally the frequency of zero-ATR from the ATR data, since the minimum of ATR with nearly zero is attained in a wide range of d. This also prevents us from obtaining a precise identification of of d ₀ from the ATR data. Fortunately, as described later, the phase jump at the SP resonance offers the solution. The Δϕ spectra of the plastic prism indicate that distance d ₀ of the zero-ATR condition should be between 53.2 and 31.9 μm and between 31.9 and 11.9 μm. In the case of a MgO prism, d ₀ lies between 2.1 and 1.6 μm. These distances and the corresponding resonant frequencies are nearly consistent with the values derived theoretically. Distance d ₀ for a MgO prism is one order shorter than that for a plastic prism. This is reasonable because the value of |r ₂₃| is smaller and that of η_z2 is larger than those in a plastic prism, as shown in Figs. 8(a) and (b).

 

Fig. 8. Phase and amplitude of r ₂₃ calculated for (a) the plastic prism, (b) the MgO prism with γ/2π=0.58 THz, (c) the plastic prism, and (d) the MgO prism with γ/2π=0.058 THz. In (a) and (b), the dotted lines represent the frequency where the curves of arctan(r ₂₃) intersect with the lines of arctan(r ₁₂)+π represented by dashed lines for an eye guide. The blue and red circles for the eye guide show the points where the dotted lines intersect with the phase of r ₂₃ and with the amplitude of r ₂₃, respectively. The dashed lines in (c) and (d) are the same as those in (a) and (b), respectively.

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It is noteworthy that |r ₂₃| has a peak at the frequency away from ω ₀, as shown in Figs. 8(a) and (b). The condition of the denominator of r ₂₃ equalized to zero, ε ₃ k _z2+ε ₂k_z3=0 [23], provides the dispersion relation of a surface plasmon without a prism coupler. The peak position of |r ₂₃|, therefore, can be regarded as the surface plasmon resonance frequency ω _SP. However, as shown in Figs. 8(a) and (b), the amplitude of the frequency shift ratio Δω/ω _SP≈-0.25 for a plastic prism and Δω/ω _SP≈0.08 for a MgO prism, where Δω is the difference of ω ₀ from ω _SP, is so large that we can no longer apply the Otto approximation to explain the frequency shift in this situation [10,11].

For a comprehensive understanding of these features of resonant frequency, we consider the frequency shift ratio Δω/ω _SP for the case of a smaller damping constant, where Otto’s approximation may be applicable. Figures 8(c) and 8(d) show the amplitude and phase shift spectra of r ₂₃ for InAs with the damping constant reduced by one-tenth. In this case, the maximum |r ₂₃| is strongly enhanced by narrowing its bandwidth, and the phase spectrum of r ₂₃ has a steep change at the frequency at which the |r ₂₃| has its maximum. As shown in Figs. 8(c) and (d), the amplitude of Δω/ω _SP≈-0.01 for a plastic prism and ≈0.05 for a MgO prism are somewhat smaller than that of large damping case. In such a case, Otto’s interpretation can be applied at the shift of the ATR minimum.

Even in the case of considerably large damping, it can be shown that the dip structure in the ATR spectrum should be located at the ω _SP in the limit of d→∞. In d→∞, the second term of the SP reemission in Eq. (5) is approximately given by (1-r₁₂²)r ₂₃(ω)exp(-η _z2 d). The frequency dependence of r ₂₃ strongly dominates this term, considering that the exponential term monotonically increases with the frequency. In addition, the first and second terms in the right-hand side of Eq. (5) are almost out of phase, as shown in Figs. 8(a) and (b). Therefore, it can be concluded that the position of the ATR dip at the large distance d should be located at the ω _SP. In fact, the frequency of the ATR minimum 1.3 THz at d=187 μm in Fig. 3(a) and 1.6 THz at d=4.4 μm in 5(a) coincides with the frequency, in which |r ₂₃| has its maximum, 1.3 THz in Fig. 8(a) and 1.6 THz in Fig. 8(b). With approaching d ₀, the minimum ATR value approaches zero, and the frequency shifts extensively. This is because the phase condition for r ₂₃ represented by Eq. (8) manifests the frequency of the ATR minimum, as mentioned above.

Otto explained the small frequency shift of the SP resonance due to the phase condition by treating it as a tiny perturbation of the prism. This approximation is valid for the case of a small damping constant but does not explain the experimental results for a larger damping case. So the explanation that the frequency shift is dominated by the phase condition is more general because it does not require an assumption.

6. Phase jumps at the SP resonance

Next, let us consider a prominent feature of the anomalous phase jump behavior. The phase shift changes drastically at the frequency at which ATR has minimum values, and, particularly, the direction of the phase jump is inverted at distance d ₀, where the zero-ATR condition is realized.

 

Fig. 9. Parametric plot of complex reflective coefficients r ₁₂₃(d, ω)/r ₁₂ as a function of the frequency plotted for different d on a complex plane. (a) r ₁₂₃(d, ω)/r ₁₂ obtained from experimental results E _Me(d, ω)/E _Me(d=∞, ω) from 0.6 THz to 2.2 THz for d=53.2 μm (blue line), 31.9 μm (red line), and 11.9 μm (black line). The green arrow connects the origin with the value for d=31.9 μm at 1.35 THz. (b) r ₁₂₃(d, ω)/r ₁₂ calculated for the plastic prism in the frequency region from ω/2π=0 (opaque circle) to 3 THz (open circles), assuming ω_p /2π=1.91 THz and γ/2π=0.58 THz. d=53.2 μm (blue line), 31.9 μm (red line), and 11.9 μm (black line).

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In order to explain these features, we show the parametric plot of the complex reflective coefficients as a function of the frequency on the complex plane normalized by r ₁₂, r ₁₂₃(d,ω)/r ₁₂=E _Me(d, ω)/E _Me(d=∞, ω). In Fig. 9(a), three sets of experimental data show the steep phase jumps in Fig. 3(b), d=53.2, 31.9, 11.9 μm. The green arrow is an example of a complex reflective vector, where the square of its length and the polar angle correspond to ATR and Δϕ, respectively. Here, we focus on the near zero-ATR region in Fig. 9(a). All trajectories of r ₁₂₃/r ₁₂ go from the left-hand side to the right-hand side with increasing the frequency. If the trajectory goes through the upper side of the origin, Δϕ decreases with the frequency. In fact, as shown in Fig. 9, the experimental and calculated data at d=53.2 μm indicate the trajectory going through the upper side of the origin or turning clockwise around it. This means that the negative direction of the π-phase jump occurs at the frequency of the ATR dips at d=532 μm, as shown in Fig. 3(b). The curve of r ₁₂₃(d,ω)/r ₁₂ moves smoothly in the complex plane by changing d and passes through the origin at the zero-ATR distances of d ₀ and frequencies of ω ₀. When the trajectory goes through the zero-ATR distances of d ₀ (48.3 μm) and then through the lower side of the origin, Δϕ increases with increasing frequency. In Fig. 9, the experimental and calculated data at d=31.9 μm, corresponding to the distance between d=48.3 and 21.4 μm, indicate the trajectory turning anti-clockwise around the origin, causing a positive direction of the π-phase jump in Fig. 3(b). Again, when the trajectory goes through the zero-ATR distances of d ₀ (21.4 μm) and then through the upper side of the origin, Δϕ increases with increasing frequency; as a result, the data at d=31.9 μm shows the negative direction of the π-phase jump in Fig. 3(b). While a phase shift of 0 above the frequency at which ATR has a minimum is essentially identical with the phase shift of 2π in Fig. 9, the difference of the phase shift (0 and 2π) stands out by considering the continuous phase shift change along the trajectory.

As in the case of a plastic prism, the experimental data with the MgO prism shows the direction changes of the π-phase jump. The number of direction changes of the π-phase jump is different in different prism cases, which have different numbers of zero-ATR solutions. In this way, it is verified that the modulation of the zero-ATR condition by the small change of external parameters causes the direction changes of the π-phase jump.

In the visible region, the phase jump at the SP resonance has been investigated in ellipsometry measurements [24]. Recently, monochromatic interferometer methods can also access the phase jump at the SP resonance, which will be applied to develop ultra-sensitive sensors with single-atom thickness resolution. The basic experimental setup is a prism-coupled method, but an additional interferometer for phase detection in reflected light was fabricated. Even phase detection shows a dramatic gain in sensor characteristics, and its sensitivity to a refractive index change was estimated as 4×10^-8 in a model experiment with different gases [6–9]. In our work, the phase shift of an electric field in the time domain could be estimated without an additional interferometer. The detection of both amplitude and phase should bring higher sensitivity to the tiny index change of the surrounding medium, which will open new possibilities for bio- and chemical sensing in the THz frequency region [25].

7. Conclusion

In conclusion, we have systematically investigated the SP resonance in a (100) oriented-InAs surface with time-domain ATR spectroscopy in the THz frequency region. The position of the ATR minimum strongly depends on the wave vector of the evanescent wave, the refractive index of prism, the incident angle of the THz pulse, and the distance between the prism and the active material. In order to explain this experimental result, we have proposed an alternative interpretation, namely, that the observed electric field is given by the destructive interference between the electromagnetic wave reflected at the prism surface and that reemitted from the excited SP. The large SP resonance frequency shift can be attributed to the presence of the prism. This interpretation of the destructive interference has already been used in a qualitative explanation using an SP-enhanced near-field microscope and nano-particle SP sensors [14], but a quantitative explanation is complicated. We experimentally demonstrate a simple interpretation of the destructive interference on the SP resonance in a simple configuration with THz TD-ATR.

Furthermore, a steep phase shift at the resonance frequency is observed when ATR approaches zero. The π-phase jump is based on the π-phase difference between the electromagnetic wave reflected at the prism surface and that reemitted from the excited SP. The detection of both amplitude and phase should bring higher sensitivity to tiny index change of the medium originated from a bio- and chemical reaction in THz frequency region [6–9]. Nevertheless, a serious problem for the sensing applications appears when the damping constant γ becomes large because the enhancement of the electric field at the boundary is strongly suppressed. For the case of the active material with γ/2π=0.58 THz and the plastic prism case we used, the maximum enhancement factor |B(ω/2π=1.47 THz, d ₀=48.3 μm)| is 2.7. On the other hand, for the case of the active material with γ/2π=0.058 THz, |B(ω/2π=1.41 THz, d ₀=221.1 μm)| is 9.2 and is fairly enhanced. Therefore, active materials with a small damping constant, such as metamaterial and metal with photonic structures [26,27], are potential candidates for use in the development of an SP sensor in the THz frequency region.