Probing negative refractive index of metamaterials by terahertz time-domain spectroscopy

Jiaguang Han

doi:10.1364/OE.16.001354

1. Introduction

The concept of “Metamaterials” has become so popular and attracted a great deal of attention in recent years due to the realization that this phenomenon could lead to many interesting applications, as well as it will be guided to reach a fundamental understanding of the rich physics behind the complicated phenomenon. Metamaterials (or negative index materials, or left-handed materials) are media in which two important parameters, electrical permittivity ε and magnetic permeability μ, that determine their response to electromagnetic radiation are simultaneously negative. Since that, it is also called Double Negative (DNG) materials. The index of refraction n in metamaterials possesses negative value that leads to unusual reversed propagation phenomena of electromagnetic waves. Metamaterials have found a range of promising applications including in communications, electronics, optics, superlenses, near-field imaging and medicine [1–10]. Composite materials, such as SRRs and specially designed photonic crystals, have been demonstrated to realize negative refractive index that simultaneously has negative permittivity and permeability [11].

The initial work creates more interest to understand this novel effect. One of challenging problems is how to measure the negative refractive index of metamaterials directly. Hence, in this paper we introduce a novel method that allows us to extract the negative refractive index of metamaterials using terahertz time-domain spectroscopy (THz-TDS) measurement. Compared with the previous utilized method mostly based on Snell’s Law [11], this method offers a more direct and simple way to obtain the frequency-dependent real part and imaginary part of refractive index of metamaterials through THz-TDS measurement. It is also worthy of noting that metamaterials at THz frequencies have significant potential applications in optoelectronic devices such as compact cavities, adaptive lenses, tunable mirrors, isolators and converters. In particular, THz metamaterials will be expected to benefit THz near-field imaging and its biomedical applications [10, 12–14]. The realization of artificial electric and magnetic materials in THz regime burdens an important step filling the gap between microwave and optical frequencies, where THz technology is suggested the potential applications in semiconductor, tomographic imaging, label free genetic analysis, cellular level imaging, biological sensing, and so on [15]. The new metamaterial structures may act an important role in the development of THz technology, such as THz waveguide, source generator, modulator, switch, emitters, and others.

2. Theoretical description

Terahertz time-Domain Spectroscopy (THz-TDS) is a new spectroscopic technique and provides a powerful tool to measure the refractive index of a medium, as well as the power absorption and the real and imaginary parts of the complex dielectric function. The motivation comes from whether THz-TDS has capabilities to obtain the negative refractive index of metamaterials directly. To do this, firstly it is of help to turn to a description of THz-TDS method applied to characterization of materials in a general case. As shown in Fig. 1, THz-TDS normally requires two measurements: one reference waveform E_R(t) measured with the reference of known dielectric properties, and a second measurement E_S(t) through the sample. Here we consider a sample of thickness d, referred as medium 2, placed between two media 1 and 3, where the corresponding reference is denoted as medium 0 with the same thickness as the sample. The terahertz radiation goes through the sample from medium 1 to medium 3, while the reference measurement is as same as the sample case, shown by Fig. 1.

Fig. 1. Sketch of THz-TDS measurement for reference and sample.

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The transmitted electric field of THz pulses through the sample and the reference are recorded in time domain and then the corresponding frequency spectra are obtained by numerical Fourier-transform. The complex transmission spectra of the reference E _R(ω) and the sample E _S (ω) are determined by the product of the input spectrum E_i(ω) and the total transmission function, given by [16–19]:

E_{R} (ω) = E_{i} (ω) \frac{t_{10} t_{03} \exp (i k_{R} d)}{1 + r_{10} r_{03} \exp (2 i k_{R} d)},

E_{S} (ω) = E_{i} (ω) \frac{t_{12} t_{23} \exp (i k_{S} d)}{1 + r_{12} r_{23} \exp (2 i k_{S} d)},

where t₁₀, t₀₃, t₁₂, t₂₃ and r₁₀, r₀₃, r₁₂, r₂₃ are the Fresnel transmission and reflection coefficients of the THz pulses propagating through medium 1-reference, reference-medium 3, medium 1-sample, and sample-medium 3 interfaces, respectively, and d is the sample thickness. k_R(ω)=ωn_R(ω)/c and k_S(ω)=ωn_S(ω)/c are the propagation wavevectors through reference and sample, respectively, in which n_R(ω) and n_S(ω) are refractive index of the reference and the sample. If defines the angle of incidence as θ_i and the angle of refraction as θ_t, the Fresnel coefficients for s-polarization are [18]:

t_{ab}^{s} = \frac{2 n_{a} \cos θ_{i}}{n_{a} \cos θ_{i} + n_{b} \cos θ_{t}}, r_{ab}^{s} = \frac{n_{a} \cos θ_{i} - n_{b} \cos θ_{t}}{n_{a} \cos θ_{i} + n_{b} \cos θ_{t}},

while for p-polarization,

t_{ab}^{p} = \frac{2 n_{a} \cos θ_{i}}{n_{b} \cos θ_{i} + n_{a} \cos θ_{t}}, r_{ab}^{p} = \frac{n_{b} \cos θ_{i} - n_{a} \cos θ_{i}}{n_{b} \cos θ_{i} + n_{a} \cos θ_{y}} .

Because of the relatively clean separation in time between the main transmitted pulse and the first internal reflection, it becomes possible to keep only the first directly transmitted terahertz pulse and hence the initial data analysis was performed on the main pulse only [17, 19]. For this simpler case, as well as considering the mostly employed normal incidence profile, the Eqs. (1a) and (1b) can be simplified as:

E_{R} (ω) = E_{i} (ω) t_{10} t_{03} \exp (i k_{R} d),

E_{S} (ω) = E_{i} (ω) t_{12} t_{23} \exp (i k_{S} d),

with Fresnel coefficients for both s-polarization and p-polarization:

t_{ab} = \frac{2 n_{a}}{n_{a} + n_{b}} .

The complex transmission T(ω) of the sample is obtained by dividing the recorded sample signal E _S (ω) by recorded reference signal E _R(ω):

T (ω) = \frac{E_{S} (ω)}{E_{R} (ω)}

It can be seen obviously that the phase shift can be calculated to determine the refractive index of the sample. Furthermore, if the measured sample is the lossy substance, which means it will loses its energy due to various loss mechanisms, the refractive index n_S(ω) becomes the complex refractive index of form: ñ_S(ω)=n_S(ω)+iκ_S(ω). Thus, the complex transmission T(ω) is rewritten as:

T (ω) = \frac{E_{S} (ω)}{E_{R} (ω)}

where the power absorption α(ω)=2ωκ(ω)/c [17]. It is important to notice that the Fresnel coefficients t₁₂ and t₂₃ in Eq. (7) now become complex number related to both real part n_S(ω) and imaginary part κ_S(ω) of complex refractive index ñ_S(ω). As a result, the analytic expressions for n_S(ω) and κ_S(ω) can not be found and then the complex refractive index ñ_S(ω) of the sample need be estimated using the iterative approximation. However, for most low-loss (low-absorption) cases, κ_S(ω)≪n_S(ω). Thus analytic expression for the complex refractive index of the sample can be easily deduced from Eq. (7). Comparing the ratio and phase change between the sample and reference, the real part n_S(ω) and imaginary part κ_S(ω) of refractive index of studied sample can be obtained simultaneously. The frequency-dependent effective complex dielectric response of the studied sample is further determined by the recorded data of power absorption and refractive index through the relationship:

ε (ω) = ε_{r} (ω) + i ε_{i} (ω) = {(n_{S} (ω) + i κ (ω))}^{2},

with:

ε_{r} = n_{S}^{2} - {(\frac{α c}{2 ω})}^{2},

ε_{i} = \frac{α n_{S} c}{ω} .

Optical properties of materials are typically characterized by the frequency dependence of either n_S(ω) and κ_S(ω) or ε_r(ω) and ε_i(ω). Here we mainly take a low-loss or lossless case into account.

Fig. 2. Illustration of reflection and refraction of a p-polarization plane wave incidence upon the interface of DPS and DNG medium.

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The above procedure is most common in practice, as well as the principle to measure refractive index of materials using THz-TDS. Based on this, we will concentrate on how to obtain the negative refractive index of metamaterials by THz-TDS. Due to the special characteristics of metamaterials, one must exercise more care with the definitions of the electromagnetic properties in such a DNG medium. If the medium is of negative refractive index, then the refracted angle, according to Snell’s law, should also be on the same side of the interface normal as the incident angle is, in which the wave and Poynting vectors point in opposite directions, as shown in the Fig. 2 [20]. Here the p-polarization is taken into account as an instance. The wave vector is directed in a direction to the interface, but the Poynting vector away from the interface. For the extraction procedure, we would obtain the reflection and transmission coefficients first of all. The polarized plane wave (time dependence exp(-jωt) being assumed) falls on a boundary between the double positive (DPS) medium 1 of permittivity ε₁>0 and permeability μ₁>0, and DNG medium 2 of negative permittivity ε₂<0 and permeability μ₂<0, in which we do not take losses into account and treat ε and μ as real numbers. The wave is split into two ways: a transmitted wave proceeding into the medium 2 and a reflected one back into medium 1. Real negative refractive index of medium 1 and medium 2 are defined as: $n_{1} = \sqrt{ε_{1} μ_{1}}$ and $n_{2} = - \sqrt{ε_{2} μ_{2}}$ , respectively. Taking the plane of incidence as the xz-plane and denoting θ_i, θ_r and θ_t as the angle of incidence, reflection and refraction, as shown by Fig. 2, we have the electric and magnetic fields of the incidence:

{\vec{E}}_{i} = {\vec{E}}_{i 0} \exp [j ({\vec{k}}_{i} \cdot \vec{r} - ω t)]

{\vec{H}}_{i} = {\vec{H}}_{i 0} \exp [j ({\vec{k}}_{i} \cdot \vec{r} - ω t)]

with $k_{1} = \frac{ω}{c} n_{1} = \frac{ω}{c} \sqrt{ε_{1} μ_{1}}$

Similarly the reflected fields have:

{\vec{E}}_{r} = {\vec{E}}_{r 0} \exp [j ({\vec{k}}_{r} \cdot \vec{r} - ω t)]

{\vec{H}}_{r} = {\vec{H}}_{r 0} \exp [j ({\vec{k}}_{r} \cdot \vec{r} - ω t)]

The transmitted fields have:

{\vec{E}}_{t} = {\vec{E}}_{t 0} \exp [j ({\vec{k}}_{t} \cdot \vec{r} - ω t)]

{\vec{H}}_{t} = {\vec{H}}_{t 0} \exp [j ({\vec{k}}_{t} \cdot \vec{r} - ω t)]

with $k_{2} = \frac{ω}{c} ∣ n_{2} ∣ = - \frac{ω}{c} n_{2} = \frac{ω}{c} \sqrt{ε_{2} μ_{2}}$ .

Let the boundary surface be at z=0 and the tangential components of E⃗ and H⃗ are continuous [18]. We derived from Eq. (10) to Eq. (15):

E_{i 0} \cos θ_{i} - E_{r 0} \cos θ_{i} = E_{t 0} \cos θ_{t},

\sqrt{\frac{ε_{1}}{μ_{1}}} E_{i 0} + \sqrt{\frac{ε_{1}}{μ_{1}}} E_{r 0} = \sqrt{\frac{ε_{2}}{μ_{2}}} E_{t 0} .

And then the reflection and transmission coefficients r ₁₂ and t ₁₂ are determined from Eq. (16a) and (16b) as:

r_{12} = \frac{E_{r 0}}{E_{i 0}} = \frac{\sqrt{\frac{ε_{2}}{μ_{2}}} \cos θ_{i} - \sqrt{\frac{ε_{1}}{μ_{1}}} \cos θ_{t}}{\sqrt{\frac{ε_{2}}{μ_{2}}} \cos θ_{i} + \sqrt{\frac{ε_{1}}{μ_{1}}} \cos θ_{t}}, t_{12} = \frac{E_{t 0}}{E_{i 0}} = \frac{2 \cos θ_{i} \sqrt{\frac{ε_{1}}{μ_{1}}}}{\sqrt{\frac{ε_{2}}{μ_{2}}} \cos θ_{i} + \sqrt{\frac{ε_{1}}{μ_{1}}} \cos θ_{t}} .

For normal incidence

r_{12} = \frac{E_{r 0}}{E_{i 0}} = \frac{\sqrt{\frac{ε_{2}}{μ_{2}}} - \sqrt{\frac{ε_{1}}{μ_{1}}}}{\sqrt{\frac{ε_{2}}{μ_{2}}} + \sqrt{\frac{ε_{1}}{μ_{1}}}}, t_{12} = \frac{E_{t 0}}{E_{i 0}} = \frac{2 \sqrt{\frac{ε_{1}}{μ_{1}}}}{\sqrt{\frac{ε_{2}}{μ_{2}}} + \sqrt{\frac{ε_{1}}{μ_{1}}}} .

Similar deductions are available for the s-polarization configuration. At normal incidence, both s-polarization and p-polarization are of the same reflection and transmission coefficients expressions as denoted by Eq. (18). From the above derivations, we have seen that the basic equations relating to the propagation of the wave in a DNG metamaterial medium differ from those relating to propagation in a normal medium.

Turning back to Fig. 1 measurement sketch, we would rewrite the extraction procedure supposing that the measured sample is DNG medium ( $n_{S} = - \sqrt{ε_{2} μ_{2}} < 0$ without loss) under normal incidence. By analogy with Eq. (4), the complex transmission spectra of the reference E _R(ω) and the sample E _S(ω) are obtained as following:

E_{R} (ω) = E_{i} (ω) t_{10} t_{03} \exp (i k_{R} d) = E_{i} (ω) t_{10} t_{03} \exp (i \frac{ω}{c} n_{R} d),

E_{S} (ω) = E_{i} (ω) t_{12} t_{23} \exp (- i k_{S} d) = E_{i} (ω) t_{12} t_{23} \exp (- i \frac{ω}{c} ∣ n_{S} ∣ d)

with:

t_{10} = \frac{2 n_{1}}{n_{1} + n_{R}}, t_{03} = \frac{2 n_{R}}{n_{3} + n_{R}}, t_{12} = \frac{2 \sqrt{\frac{ε_{1}}{μ_{1}}}}{\sqrt{\frac{ε_{1}}{μ_{1}}} + \sqrt{\frac{ε_{2}}{μ_{2}}}}, t_{23} = \frac{2 \sqrt{\frac{ε_{2}}{μ_{2}}}}{\sqrt{\frac{ε_{2}}{μ_{2}}} + \sqrt{\frac{ε_{3}}{μ_{3}}}} .

Since the wave vector k_S in DNG sample medium is of opposite direction compared with DPS medium, the negative sign is required in the term exp(-ik_sd) of Eq. (20), which is quite dissimilar as that in Eq. (4b). In practice, we can generally choose the medium 1 and 3 as nonmagnetic substances for μ ₁=μ ₃=1.0, and then:

t_{12} = \frac{2 n_{1}}{\sqrt{\frac{ε_{2}}{μ_{2}}} + n_{1}}, t_{23} = \frac{2 \sqrt{\frac{ε_{2}}{μ_{2}}}}{\sqrt{\frac{ε_{2}}{μ_{2}}} + n_{3}} .

The complex transmission T(ω) is then given as:

T (ω) = \frac{E_{S} (ω)}{E_{R} (ω)}

Although the Eq. (23) looks identical with the Eq. (6), it should be emphasized that n_S(ω) is indeed negative and Fresnel transmission coefficients are defined under a more general expression. For lossy sample medium, we have the complex refractive index of metamaterial as: ñ_S(ω)=n_S(ω)+iκ_S(ω) with real part n_S(ω)<0, and then

T (ω) = \frac{E_{S} (ω)}{E_{R} (ω)}

with α(ω)=2ωκ(ω)/c. For most low absorption cases [21], according to the phase change $Δ ϕ (ω) = \arg (\frac{E_{S} (ω)}{E_{R} (ω)})$ , we then can obtain the real part of negative index of metamaterials as:

n_{S} (ω) = n_{R} (ω) + \frac{Δ ϕ \cdot c}{ω d} .

The imaginary part κ(ω) can be got from the ratio of electric fields through the sample and reference,

κ (ω) = - \frac{c}{ω d} \ln (\frac{t_{10} t_{03}}{t_{12} t_{23}} \cdot ∣ T (ω) ∣) .

3. Simulations

So far we have presented the extraction procedure how to get the negative index of metamaterials based on THz-TDS measurement. In order to validate this extraction procedure, here we test it on several typical cases based on frequency domain finite element method (FEM) simulation, where all of simulations match well the practical experimental THz-TDS measurement cases without loss. The extraction procedure successfully gives a good agreement between the assigned refractive index value and the extracted one.

Let us consider a fairly general case that metamaterial sample is placed between two uniform normal medium as displayed by Fig. 3. In the simulation, the sample medium 2 is assigned to possess isotropic negative permittivity ε=-16.0 and permeability μ=-1.0, while both of medium 1 and medium 3 have positive permittivity ε=3.0 and permeability μ=1.0. The reference is assumed to be air with refractive index 1.0. As shown by Fig. 3, the THz wave is launched by the line source illumination at 1.5 THz placed at the air, then is passed through the medium1, sample 2 (or reference 0), medium 3, and finally is probed at air. Figure 3 represents the electric field intensity distributions of such a sandwich THz-TDS measurement profile matching the practical experimental measurement situation. During the course of our simulation, it is of interest to note that the animated electric field magnitude as a function of phase in the sample region appears to be moving backwards. Considering the phase change of measured sample, we found that the phase is increased from the air-1 interface to the 1–2 interface, then is reduced between 1–2 interface and 2–3 interface, and after that is enhanced again; while for reference measurement, the phase keeps increase during total propagation process. In terms of phase change between the sample and the reference measurements, we can extract the refractive index of simulated sample with value -4.08 utilizing the above proposed extraction procedure, which matches the assigned value -4.0 for the sample material in the simulation. The deviation may be due to the simplification from Eq. (1) to Eq. (4), where total reflection contribution has not been involved in. However, for a real THz-TDS measurement, the reflection can be distinguished experimentally from the time-domain spectra, and hence the extraction procedure offers quite accuracy to obtain the refractive index, which has been verified by many previous experimental reports [16,17].

Fig. 3. Demonstration of theoretical simulated distributions of electric field intensity for THz TDS measurements on metamaterial sample (upper) and corresponding reference (lower), respectively.

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It is perceivable that one can discuss various complex configurations of wave interaction with metamaterials besides the above presented one. As a typical instance, we describe a more simple case, called free-standing sample measurement, where THz wave is excited through line source at air, passed through the sample and received at air, as shown by Fig. 4(a). Further, we assume the negative refractive index of sample is a function of frequency and when the incident wave changes from 1.0 to 1.7 THz, the refractive index varies from -6 to 2, in which we keep the assigned permeability μ=-1.0 below 1.6 THz and μ=1.0 above 1.6 THz fixed, and let assigned permittivity ε vary with increasing frequency, as shown by the triangles in Fig. 4(b). Running the FEM simulations and employing the proposed extraction procedure, we can obtain the index of refraction of measured free-standing sample dependence on the frequency, as shown by open circles. The good agreement between assigned values and extracted values quite demonstrates that THz-TDS offer a good method to measure the negative index directly based on our presented extraction procedure.

As an interesting and promising application, metamaterials offer the ability to act as the phase compensation or phase conjugation [20]. Here we provide an illustrative example to highlight such an effect. As shown by Fig. 5 (upper), when the wave propagates through DPS medium 1 of length d and index n₁, the phase will be enhanced ωn ₁ d/c; while the wave passes through the DNG slap sample of the same length d and negative index n_S, the phase is deduced ω|n_s|d/c. In such a case, the total phase change between the front interface A and the back interface B of this two-layer structure is ωn ₁ d/c-ω|n_S|d/c. It implies that if we chose n₁=|n_S|, then the total phase change through this two-layer structure becomes zero, in which the DNG sample slab acts a phase compensator and DPS medium1 and the DNG sample construct a DPS-DNG pair. Figure 5 show the FEM simulation results for electric field distributions, where the 1.5 THz wave is launched from air upon the sample, as well as upon the reference. Here the reference medium is assumed as medium 1. During the simulation, if we detect the phase at interface A and B, we found that they have the same phase value, and it means that the phase change between interface A and B keeps zero. Comparing the phase change that wave passed through sample and reference cases, the total phase change $Δ ϕ (ω) = \arg (\frac{E_{S} (ω)}{E_{R} (ω)})$ equals to -2ωn ₁ d/c. Carrying out the extraction procedure, we get the refractive index of measured sample of value -3.13, which is quite consistent with the assigned value -3.16 for the simulated material. As shown by Fig. 5 (upper), the backward wave can be seen obviously when the wave passes through the DNG sample. The matched DPS-DNG pair could realize the phase compensation and lead a variety of potential application in many optical systems.

Fig.4. (a). Sketch of THz-TDS measurement on the free-standing sample. (b) Comparison of extracted refractive index (circles) and assigned value (triangles) dependence on frequency.

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Fig. 5. Theoretical demonstration of DPS-DNG pair. Electric field intensity distributions simulated for the sample case (upper) and reference case (lower). The animated electric field magnitude as a function of phase of the simulated DPS-DNG pair is seen clearly from the video. (AVI video, figure_05.avi, 1.43 MBytes). [Media 1]

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4. Conclusion

In conclusion, we have presented a detailed procedure to extract the negative refractive index of low-loss metamaterials based on THz-TDS measurement. Detecting the phase change of THz wave passed through the metamaterial sample, as well as the corresponding reference, we can obtain the refractive index of studied sample using the presented extraction procedure. The method provides a direct and simple way to probe the real part and imaginary part of refractive index of metamaterials. Its validity has been well demonstrated through our theoretical simulations on several typical cases. The work may constitute a novel way for THz metamaterials investigations, designs and applications.

Acknowledgments

The author acknowledges financial support from the MOE Academic Research Fund of Singapore and the Lee Kuan Yew Fund.

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Probing negative refractive index of metamaterials by terahertz time-domain spectroscopy

Abstract

1. Introduction

2. Theoretical description

3. Simulations

4. Conclusion

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (5)

Equations (45)

Optics Express