Origin of pseudo-dispersion in non-dispersive media by terahertz time-domain spectroscopy

Farah Vandrevala; Erik Einarsson; Erik Einarsson

doi:10.1364/OE.27.033537

1. Introduction

The contactless and non-destructive nature of terahertz time-domain spectroscopy (THz–TDS) is making it a popular choice for characterizing the electrical properties of materials ranging from two-dimensional crystals [1,2] to biological samples [3,4]. Although transmission geometry is more commonly used, reflection geometry is needed for the study of highly reflective or strongly absorbing materials. In such measurements, a sample is typically placed atop a substrate, such as fused quartz or high-resistivity silicon (HR-Si), and its characteristics are determined by studying the reflection from the substrate–sample interface [5]. The process of parameter extraction involves forming a transfer function with a measurement from a blank substrate, which is used as a reference, and another measurement with the sample placed on the substrate. Consequently, accurate characterization of the substrate is a necessary first step of the sample parameter extraction process.

In this paper, we report the appearance of dispersion in the substrate when it is sitting atop a metal stage or when the substrate-supported sample is a bulk metal. When using the data collected in reflection geometry for a substrate suspended over air (air–substrate–air configuration), the extracted refractive index appears to be constant over the entire frequency range. However, when the measurement is repeated for the same substrate using a metal stage for support (air–substrate–metal configuration), the substrate refractive index extracted from the obtained data shows a dispersion-like effect over the same frequency range. We demonstrate that this behavior can be explained using a transmission line model to describe the propagation and reflection of the THz light. We show that when the substrate is supported and measured with respect to metal, it presents a different input impedance to the incident THz field, which when plotted as a function of frequency mirrors the pseudo-dispersion effect observed for the extracted refractive index.

2. Measurement method

To obtain accurate sample properties using THz–TDS, proper characterization of the supporting substrate is critical, which includes correctly ascertaining its physical thickness $d$. This is because an imprecise value of the substrate thickness can cause errors to cascade into sample property calculations related to the detected phase [6]. Most approaches numerically optimize the product of the substrate thickness and refractive index, but ideally the thickness should be determined independently. We recently presented a fractional reflection technique to do exactly this [7], which avoids complicated numerical optimization routines. Using this accurately determined value of the physical thickness, we can then determine the optical properties of the substrate, such as its complex refractive index, with high confidence.

Our THz–TDS system (Advantest TAS7500TS) has an aluminum metal stage with a hole in the center, which enables the system to be operated in transmission, upside reflection, or underside reflection measurement configurations. For the measurements in this paper, we use our system in the upside reflection geometry wherein the THz emitter and detector are above the substrate, which lies flat on the metal stage, and a THz pulse is incident on the substrate at an angle of $\theta _{i}= 10$°. If the substrate (medium 1) is placed over the hole, the medium above (medium 0) as well as below (medium 2) the substrate is air. This is illustrated in Fig. 1(a), which shows the corresponding ray diagram for a THz beam incident from above. If the substrate is placed on the metal stage, the second interface is substrate–metal. This is illustrated by the corresponding ray diagram in Fig. 1(b). To find the substrate refractive index, we use a self-referenced reflection method similar to the one described in [8]. The measured time delay (or phase difference) between the main and echo pulses contains information about the substrate refractive index as shown by

(1)$$n = \sqrt{{\biggl(}\dfrac{c}{2\omega d}{\bigl(}[\phi_\mathrm{echo}-\phi_\mathrm{main}]\pm \psi{\bigr)}{\biggr)}^{2}+\sin^{2}\theta_{i}},$$

where $\psi =\pi$ if $n_{1}>n_2$, and $\psi =0$ otherwise. Here, $n_{1}$ and $n_{2}$ corresponds to the refractive index of medium 1 and medium 2, respectively.

Fig. 1. Experimental setup and corresponding transmission line models.

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3. Results and discussion

When measuring the substrate in reflection geometry such that the THz field is first incident at an air–substrate interface, we note that the extracted refractive index of the substrate appears to vary depending on the second interface, i.e., whether it is measured with respect to air or with respect to metal. Using Eq. (1) to calculate the refractive index of HR-Si and fused quartz based on measurements for the suspended (Fig. 1(a), with respect to air) and metal-supported (Fig. 1(b), with respect to metal) cases, we obtain different values, as shown in Fig. 2. The refractive index obtained in the metal-supported case appears to vary with frequency, suggesting that these materials are dispersive, although it is well known that they are not [9–13]. Modeling this system as a transmission line can handily explain this erroneous result.

Fig. 2. Comparison of refractive index for HR-Si (top) and quartz (bottom) when extracted with respect to the second interface being air or metal.

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The experimental setups shown in Figs. 1(a) and 1(b) can be represented as equivalent transmission line models shown in Figs. 1(c) and 1(d), respectively. Doing so allows us to calculate the complex impedance of the material on each side of the interface using the equations:

(2)$$\tilde{\Gamma}_\mathrm{01} = \dfrac{\tilde{Z}_{1}-Z_{0}}{\tilde{Z}_{1}+Z_{0}},\qquad \tilde{\Gamma}_\mathrm{10} = \dfrac{Z_\mathrm{0}-\tilde{Z}_\mathrm{1}}{Z_\mathrm{0}+\tilde{Z}_\mathrm{1}}.$$

where $\tilde {\Gamma }_{01}$ is the reflection coefficient at each interface, and is equivalent to the corresponding Fresnel reflection coefficient at that interface [14].

For the case of air as the medium on both sides of the substrate, the equivalent transmission line is shown in Fig. 1(c). Equating the Fresnel reflection coefficient with $\tilde {\Gamma }_{01}$ leads to

(3)$$\dfrac{\tilde{Z}_{1}}{Z_{0}} = \dfrac{1}{\cos\theta_{i}}\dfrac{\cos\theta_{t}}{\tilde{n}_\mathrm{1}} .$$

This yields the complex impedance values shown in Figs. 3(a) and 3(b). The corresponding refractive index values, shown in Figs. 3(c) and 3(d), match the values that have been reported in literature for HR-Si and quartz [9–13].

Fig. 3. (a, b) Complex impedance and (c, d) the corresponding complex refractive index extracted for HR-Si, fused quartz, and air.

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When the substrate is placed on the metal stage, it is equivalent to the case of a transmission line terminated by a short-circuit, as shown in Fig. 1(d). Since our substrates are low-loss dielectrics, the characteristic impedance of the transmission line can be expressed using only the real part, $Z_\mathrm {c} = (L_\mathrm {eq}/C_\mathrm {eq})^{1/2}$, where $L_\mathrm {eq}$ and $C_\mathrm {eq}$ are the equivalent frequency-independent inductance and capacitance of the line [15]. In this case, the input impedance of the line is defined as

(4)$$\tilde{Z}_\mathrm{in} = jZ_\mathrm{c}\tan(\beta d_\mathrm{eff}) \quad\textrm{where}\quad \beta d_\mathrm{eff} = \dfrac{\omega}{c}n d\cos\theta_{t} = \omega\dfrac{d_\mathrm{eff}}{v_{p}}.$$

Here, the effective substrate thickness $d_\mathrm {eff}=d\cos \theta _{t}$ can be considered as the length of the transmission line, and the phase velocity $v_{p} = c/n=(L_\mathrm {eq}C_\mathrm {eq})^{-1/2}$ as the propagation velocity on the transmission line [15].

For any complex impedance $\tilde {Z}=R\pm jX$, the real part $R$ is resistance and the imaginary part $X$ is reactance. Since $\tilde {Z}_\mathrm {in}$ in Eq. (4) is purely imaginary, only the reactance portion remains, which is comprised of inductive and/or capacitive components. To understand the behavior of the transmission line at different frequencies, we consider the Mclaurin series expansion of the tangent function for any arbitrary value of $\beta d_\mathrm {eff}$ to be

(5)$$\tan(\beta d_\mathrm{eff}) = \beta d_\mathrm{eff} + \dfrac{1}{3}(\beta d_\mathrm{eff})^{3} + \dfrac{2}{15}(\beta d_\mathrm{eff})^{5} + \cdots$$

At low frequencies, $\beta d_\mathrm {eff} \ll 1$, or equivalently $2\pi d_\mathrm {eff}\ll \lambda$. Hence $\tan (\beta d_\mathrm {eff})\approx \beta d_\mathrm {eff},$ and

(6)$$\tilde{Z}_\mathrm{in} \approx jZ_{c}\beta d_\mathrm{eff} = j\sqrt{\dfrac{L_\mathrm{eq}}{C_\mathrm{eq}}}\omega d_\mathrm{eff}\sqrt{L_\mathrm{eq}C_\mathrm{eq}} = j\omega L_\mathrm{eq} d_\mathrm{eff}.$$

Thus, for wavelengths much longer than the effective substrate thickness, the transmission line behaves like an equivalent inductance to the incident electric field, as shown in Fig. 4(a).

Fig. 4. Equivalent transmission line model showing input impedance for (a) $2\pi d_\mathrm {eff}\ll \lambda$ and (b) $\lambda \lesssim 2\pi d_\mathrm {eff}\lesssim \lambda$.

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For frequencies slightly higher than the above case, or for wavelengths on the order of the effective substrate thickness [15],

(7)$$\tan(\beta d_\mathrm{eff}) \approx \beta d_\mathrm{eff} + \dfrac{1}{3}(\beta d_\mathrm{eff})^{3} = jZ_{c}\left(\beta d_\mathrm{eff} + \dfrac{\beta^{3}d_\mathrm{eff}^{3}}{3}\right) = j\omega L_\mathrm{eq}d_\mathrm{eff}\left(1+\dfrac{\omega^{2}L_\mathrm{eq}C_\mathrm{eq}d_\mathrm{eff}^{2}}{3}\right).$$

In terms of input admittance, this becomes

(8)$$\tilde{Y}_\mathrm{in} = \dfrac{1}{\tilde{Z}_{in}} \approx \dfrac{1}{j\omega L_\mathrm{eq}d_\mathrm{eff}} + j\dfrac{1}{3}\omega C_\mathrm{eq}d_\mathrm{eff}\,.$$

Thus, in the frequency range where the wavelength is comparable to the effective substrate thickness, the transmission line terminated in a short-circuit appears as an equivalent inductance connected in parallel to an equivalent capacitance, as shown in Fig. 4(b). In other words, a short-circuit transmission line can be reduced to the two cases depicted in Fig. 4 depending on the wavelength relative to the length of the line, which in this case is the effective substrate thickness.

Figure 5(a) shows how the input admittance changes with frequency for an arbitrary ratio of $L_\mathrm {eq}/C_\mathrm {eq}$. Since refractive index would be directly proportional to admittance, as implied by Eq. (3), the observed variation in the refractive index, plotted in Fig. 5(b) for the case of HR-Si and fused quartz substrates measured atop a metal stage, show the same trend.

Fig. 5. (a) Input admittance for arbitrary ratio of $L_\mathrm {eq}/C_\mathrm {eq}$, and (b) refractive index for HR-Si and fused quartz measured atop a metal stage.

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4. Conclusion

Using a transmission line model, we explain the pseudo-dispersion effect observed in a substrate when it is supported by a bulk conductor. This effect occurs due to the substrate having a frequency-dependent reactance, which makes the effective substrate impedance different for the many wavelengths comprising the incident THz pulse. To verify that the transmission line approach is valid, we also show that the complex impedance obtained from the transmission line model can be used to extract the refractive index value when the substrate is suspended over air. This value matches that obtained using the conventional Fresnel reflection approach, exemplifying that the two methods are equivalent. These findings demonstrate how a system being measured can be more than the sum of its materials, and this must be taken into consideration when measuring metal-supported samples. This could have implications for general characterization and metrology, as well as quality control or production line situations in which reflection geometries are the preferred configuration of the THz–TDS system.

Funding

Air Force Office of Scientific Research (FA9550-16-1-0188).

Disclosures

The authors declare no conflicts of interest.

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Origin of pseudo-dispersion in non-dispersive media by terahertz time-domain spectroscopy

Abstract

1. Introduction

2. Measurement method

3. Results and discussion

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (8)

Optics Express