1. Introduction

Teraherz time-domain spectroscopy (THz-TDS) is an efficient tool for characterizing materials with broadband coherent terahertz radiation [1]. Typically, a femtosecond optical laser is a main component in THz-TDS. A nonlinear interaction of optical pulses with a terahertz emitter results in a burst of subpicosecond pulses spanning a range between a few hundred gigahertz and a few terahertz. The emitted pulses induce a local change in the electric field at the detector. Laser-gated detection at the detector can resolve the amplitude and phase of terahertz pulses with a high signal-to-noise ratio (SNR). In transmission measurements, the generated terahertz pulses are modified in amplitude and phase by interacting with a sample. This registered change can be used to extract the broadband complex optical constants or related quantities of the sample.

Typically, THz-TDS is used to characterize a sample that has a thickness comparable to or larger than the operating wavelength (1 THz = 300 μm). With some augmentation, the applicability of THz-TDS can be extended to thin-film sensing, where the sample thickness is much smaller than the wavelength. Example thin films include epitaxial semiconductors, dielectric insulators, and graphene, whose terahertz properties are technologically relevant in the development of high-speed circuits [2]. Biosensing is another area that THz-TDS becomes useful. Typically, a target substance or ligand establishes a thin film on top of immobilized sensing molecules. As biomolecules have characteristic responses at terahertz frequencies, the presence of the target thin layer can be detected with THz-TDS [3]. While specialized arrangements of THz-TDS are often more suited to such thin-film measurements, there are cases where characterization of the thin-film sample must be performed with the typical transmission configuration [4]. In these cases it is important to understand the measurement limitations.

For free-space transmission-mode THz-TDS, each material has an optimal thickness that yields the lowest uncertainty in the optical constants [5]. The dynamic range of the system clearly delimits the maximum thickness of a sample that still results in valid optical parameters [6]. For a thin sample, the measurement accuracy is significantly reduced, because measurement uncertainties become relatively strong compared with a change in the amplitude and phase induced by the sample. A limitation in thin-film characterization exists where the film is so thin that the extracted parameters are no longer reliable [7]. Likewise, detecting the presence of a thin film reaches a limit when the sample is no longer distinguishable from the background material [7]. Recently, a simple criterion for determining the minimal film thickness has been proposed with its application limited to freestanding films [8].

This article includes an evaluation of three different procedures for thin-film measurement using conventional THz-TDS. Because the system uncertainties become very influential for thin-film sensing, the measurement procedure must be carefully chosen to avoid any deceptive result. The article further provides a mathematical analysis on a limitation of free-space THz-TDS for thin-film detection. The analysis covers common thin-film configurations, including substrate-backed films and freestanding films. By considering major physical factors of the film-wave interaction, the analysis defines the critical film thickness as a function of the system SNR and the sample refractive index. The derivable closed-form condition is useful as a rule of thumb in estimating the reliability of a THz-TDS system. It also suggests when other advanced measurement techniques are necessary to sense particular film samples.

2. Thin-film measurement procedures

Several measurement procedures have been implemented for measuring dielectric samples with THz-TDS. Generally, these procedures can, to some extent, alleviate random and systematic errors in the measurement. The sources of random error include laser intensity fluctuation, optical and electronic noise, jitter in the delay stage, sample relocation, etc., whilst the sources of systematic error include registration error, mechanical drift, etc. [9]. For thin-film sensing, these errors become relatively strong. In the worst case, the systematic error that progresses linearly over time might be misinterpreted as the presence of a sample. It is therefore necessary to consider these measurement procedures cautiously. In this section, we qualitatively analyze the results of applying three common measurement procedures when a THz-TDS system is run without any sample present. Despite the lack of sample, the analysis further assumes (i) a linear drift in the measured response due to the systematic error and (iii) a systematic error that is much stronger than the random error.

It can be seen that the linear drift is accounted for properly only in the last measurement procedure (case 3). As mentioned earlier, the systematic drift considered here is the worst-case scenario. It is possible that in the real measurement the drift is random, which is unlikely to cause any misleading effect. In this case, any measurement procedure is valid. However, as the nature of signal drifting is not known a priori, it is advisable to treat the problem as a worst case. In the next section the limitation of thin-film sensing is analyzed with a presumption of Case 3 measurement procedure, i.e., the error from all the sources of uncertainty are fully incorporated into the confidence intervals.

 

Fig. 1 Typical measurement procedures. (a) Case 1: measuring the same target consecutively and then averaging. (b) Case 2: measuring the reference and sample alternately and then normalizing each adjacent pair before averaging. (c) Case 3: measuring the reference and sample alternately and then averaging each group. The dotted arrows denote the time progress, during which a number of measurements are conducted. The letters ‘R’ and ‘S’ represent reference and sample measurements, respectively. The amplitude values are for an explanatory purpose only. The shaded areas denote the confidence intervals, which are proportional to the standard deviations.

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3. Limitation for thin-film detection

In the context of an uncertainty analysis, a confidence interval localizes or bounds an expected value, i.e., the value that is free from random error. In other words, the error-free value could be anywhere within this interval. Therefore, in the sample detection process, if the confidence intervals of the averaged sample and reference measurements are well separated, it can be inferred that a sample is unambiguously detected. Contrarily, if the two intervals overlap, it is possible that the two measurements share an identical expected value. Hence, in this case, the presence of a sample cannot be judged. Figure 2 schematically illustrates both of the situations. Mathematically, a confidence interval for any variable x is defined as [11]

 

Fig. 2 Confidence intervals in the detection scheme. (a) The confidence intervals of the sample and reference measurements are well separated. The presence of a sample is confirmed. (b) The two confidence intervals partially overlap. The presence of a sample is undecided.

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3.1. Consideration of optical constants

From Eq. (1) and the discussion in the previous paragraph, it can be established that the film sample with a complex refractive index n_f (ω) − jκ_f (ω) can be distinguished from the reference material with a known index n₀ − jκ₀ by terahertz waves if either of these two criteria is satisfied:

The variances, $s_{\{n,\kappa \},l}^2$, could be discarded, provided that the error in thickness measurement is relatively easy to deal with or becomes irrelevant if the detection is directly based on the time-domain or frequency-domain data. Furthermore, $s_{\{n,\kappa \},E_{\text{sam}}}^2$ and $s_{\{n,\kappa \},E_{\text{ref}}}^2$ can be combined if the numbers of sample and reference measurements are equal. Hence, from the given assumptions, Eq. (2) can be simplified to

The criteria given in Eq. (3) do not contain an explicit link to the system uncertainty. In this case, it is possible to substitute s_{n,κ}(ω) with a standard deviation in the measured transmission spectrum as done earlier [8]. Nevertheless, the resulting criteria are approximated without considering the effect of the transmission coefficients at interfaces and the reflections inside a sample. For a precise evaluation, an exact model relating the measured spectrum to the complex refractive index will be considered. Since the refractive index has a linear relation with the phase, the phase spectrum together with its confidence interval will be considered in Section 3.2. The analysis is supplemented by considering the limitation of film detection arising from the amplitude part, as described in Section 3.3.

3.2. Consideration of phase component

Figure 3 illustrates possible geometrical configurations for thin-film sensing. The three cases cover most common forms of thin films, including immobilized biomolecules [12], epitaxial semiconductors [13], polymers films, etc. From Fig. 3(a), the complex spectra for the reference and sample measurements can be expressed as [14], respectively,

 

Fig. 3 Different geometries for thin-film measurement. (a) Sample on top of the substrate, typically obtained from spin coating, chemical vapor deposition, or molecular binding; n̂₀ = 1. (b) Sample embedded in the substrate; n̂₀ = n̂_s. (c) Freestanding film; n̂₀ = n̂_s = 1. In the first two cases, the substrate is considered thick enough to allow time-gating of the internal reflections originating therein.

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In analogy to Eq. (3a), the thin film can be detected if the total phase change or the phase difference between the sample and reference measurements is larger than a geometrical combination of the corresponding confidence intervals, or

To determine the critical thickness, the standard deviation s_arg(Eref) can be obtained from a set of repeated reference measurements via the Monte Carlo method after the phase of each measurement is properly unwrapped and extrapolated [14]. Since the alternating measurement procedure, Case 3 in Section 2, is employed for thin-film measurement, these reference measurements must be left out every other scan to mimic the actual condition. Increasing the number of measurements N_ref does not necessarily reduce the minimum sample thickness. In the situation where the long-term laser instability outweighs other random errors, the standard deviation s_arg(Eref) will be accentuated [9].

3.3. Consideration of amplitude component

As described in Section 3.1, the criterion based on the refractive index, or equivalently the phase component, is sufficient to determine the presence of a thin film. For the sake of completeness, the amplitude component is analyzed in this section. The result is a supplementary condition that defines the minimal film thickness, restricted by the uncertainty in the amplitude spectra.

The presence of a thin film can be confirmed via a change in the amplitude spectrum only if

Both of the conditions in Eqs. (13) and (18) can be used to determine the minimal film thickness that can be detected by a system. The two conditions clearly provide different minimal thicknesses, and the applicability depends on the component of interest, i.e., the amplitude or phase. However, as discussed earlier, a thinner dielectric film can be unambiguously detected by observing a phase change in the transmission spectrum. As for thin-film characterization, a difference in the sample amplitude or phase with respect to the reference should be at least one order of magnitude larger than the confidence interval, i.e., the film thickness for characterization should be at least ten times the minimum thickness.

4. Experiment

The proposed criteria for thin-film sensing are experimentally validated. A series of nominally identical photoresist films with different thicknesses are measured with transmission-mode THz-TDS by using the alternating reference-sample measurement procedure. The results are compared against the critical film thickness, determined from the system uncertainty.

4.1. Sample preparation, system, and measurement

The photoresist used for making the films is Shipley MICROPOSIT S1813, obtained through Rohm and Haas Electronic Materials LLC. The photoresist is spin-coated in single or multiple layers on top of a 626.6 μm thick silicon (Si) substrate with an n-type resistivity of 10 Ω-cm. Thin layers can be spin-coated in a single pass by adjusting the spin speed and duration, between 3000–4000 revolutions per minute (rpm) and 30–60 seconds duration, according to the manufacturer’s specification. For films with a thickness above 2 μm, multiple spins (up to 10) are performed, and between each spin the sample is baked on a hot plate at 115°C for 60 seconds. The size of each coated-silicon sample is 12 mm×24 mm, all of which is initially covered by photoresist. Then, half area of the sample is completely cleaned of photoresist to create a bare Si area to be used as a reference, as in the case shown in Fig. 3(a). This cleaning procedure uses an acetone-soaked lens tissue gently dragged in one direction across the reference area. The process is repeated several times with new tissues until the reference area of the wafer is completely clean. Four different thickness photoresist films, 0.83, 0.92, 1.10, and 2.70 μm, are precisely measured via both ellipsometry and mechanical profilometry. An extra thick photoresist film of 12.5 μm is fabricated for extracting the photoresist properties in the terahertz region. The properties of the photoresist and silicon in use are given in Appendix A.2.

A transmission THz-TDS system, shown in Fig. 4 [16, 17], is employed to characterize the photoresist properties and to study the limitation in thin-film sensing. The parabolic mirrors are arranged in 8-F confocal geometry, which provide excellent beam coupling between the transmitter and receiver. The Gaussian beam of terahertz pulses is focused to a frequency independent beam waist of 3.5 mm (1/e amplitude diameter) for characterizing small samples located midway between the parabolic mirrors M2 and M3, whose effective focal length are both 50 mm. To maximally avoid film thickness variations and beam clipping artifacts, the measured sample and reference areas on the wafer are separated by only 10–15 mm, leaving a minimum of 3 mm between the edge of the terahertz beam and the edge of the photoresist sample.

 

Fig. 4 Schematic diagram of modified THz-TDS setup with an 8-F confocal geometry. The notation M1,..., M4 denote the off-axis parabolic mirrors. The terahertz beam with a frequency-independent beam waist is obtained between mirrors M2 and M3.

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The transmitter and receiver are illuminated and gated with a beam of 30 fs pulses, having an average power of 10 mW, a wavelength of 800 nm, and a repetition rate of 88 MHz. This 8-F setup is able to produce broad usable bandwidth from 0.2 to 4.5 THz. As usual, the terahertz waveform is measured by means of lock-in amplification with a time-constant of 100 ms and a point-to-point measurement delay of 300 ms. A low-noise, battery-powered, transimpedance amplifier is used ahead of the lock-in to convert the weak current signal into a stronger voltage signal. The lock-in reference is established by optical chopping, applied directly to the terahertz beam.

In all cases, the reference and sample are measured alternately in a dry air chamber. The alternating measurement procedure ensures the reliability of data for thin-film sensing, as discussed in Section 2. Both reference and sample are mounted on a remote-controlled mechanical stage so that the dry air environment is not disturbed between measurements. Each time-domain measurement, of either the sample or reference, requires about five minutes. Before further processing, the obtained temporal signals are windowed and zero-padded to remove the reflections originating inside the silicon substrate.

4.2. Validation of critical film thickness

Four reference scan datasets, where the interlacing sample scans are ignored, are utilized to determine the critical thicknesses l_c,p and l_c,m according to Eqs. (13) and (18). Each reference dataset with N_ref of 20 yields the standard deviations for the phase spectrum, s_arg(Eref) and for the amplitude spectrum s_ln|Eref|. In order to demonstrate the robustness of the developed models, the estimation of the critical thicknesses relies on approximated and frequency-independent optical constants n̂_f and n̂_s for the photoresist and silicon, as opposed to the exact frequency-dependent values given in Appendix A.2. Specifically, the refractive indices for the film and the silicon are fixed at n_f = 1.6 and n_s = 3.4, respectively; the silicon is treated as lossless, and the extinction coefficient for the photoresist is fixed at κ_f = 0.05. The coverage factor k_P is set to 1 for a standard measurement uncertainty.

Figure 5 illustrates two sets of the critical thicknesses, one related to the phase component and the other for the amplitude component. In Fig. 5(a) the critical thicknesses of this photoresist obtained from the phase component are reasonably constant over the usable frequency range and over different groups of scans. Beyond 4.5 THz and below 0.5 THz, the critical thicknesses ascend sharply, corresponding to a rapid decrease in the SNR of the current system. It can be deduced that this particular THz-TDS system can be used to detect thin polymer films down to about 2 μm by observing a phase change between the averaged reference and sample measurements.

 

Fig. 5 Critical film thicknesses for the THz-TDS system under consideration. (a) Critical thicknesses, l_c,p, derived from the phase component of four different datasets by using Eq. (13), and (b) Critical thicknesses, l_c,m, derived from the amplitude component by using Eq. (18). The indices annotate the number designator of the dataset in use.

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On the other hand, the critical thicknesses shown in Fig. 5(b) obtained from the amplitude component vary greatly from one set of scans to the next, and vary greatly with the frequency. Interestingly, there seems to be no direct link in the trend between the amplitude- and phase-based critical thicknesses, albeit derived from the same dataset. In some datasets over a limited frequency range, the critical thickness is smaller than what is indicated by the phase component. Nevertheless, due to a strong and random variation in the critical thickness l_c,m, the amplitude component is not suitable for thin-film sensing applications.

A validation of the phase-based critical thicknesses in Fig. 5(a) is conducted by using four full sets of alternating sample and reference measurements for the four photoresist films with different thicknesses. Each film is measured to obtain 20 reference and 20 sample scans. In Fig. 6, for each film, the phase difference between the averaged sample and reference measurements is compared with the combined confidence interval. In order for the film to be detected, the condition in Eq. (9) must be satisfied, i.e., the phase difference must be larger than the combined confidence interval. It is clear from Fig. 6 that for the film thickness of 0.83 μm, the phase difference is far lower than the confidence interval. As the thickness increases to 0.92 μm and 1.10 μm, the phase difference is larger, but still below the confidence interval. For the film thickness of 2.70 μm, reasonably larger than the critical thickness of 2 μm, the phase difference overcomes the confidence interval over the usable frequency band. Thus, this film is detectable. This experiment confirms the critical thickness of 2 μm determined from the measurement uncertainty of the THz-TDS system under investigation.

 

Fig. 6 Phase difference compared with confidence interval. The phase difference is obtained from the averaged sample and reference phase spectra, |arg(E_sam) − arg(E_ref)|, and the confidence interval is from a geometrical combination of their corresponding confidence intervals, $k_{P\sqrt{s_{\text{arg}(E_{\text{sam}})}^2/N_{\text{sam}}+s_{\text{arg}(E_{\text{ref}})}^2/N_{\text{ref}}}}$, where k_P = 1. The numbers indicate different film thicknesses.

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4.3. Other common thin films

The film thickness conditions given in Eqs. (13) and (18) have been experimentally validated in Section 4.2. Based on the presented experimental results, this section extends the investigation to other common thin films to gain more insight into the effect of the material properties on the minimal film thickness. The calculation assumes the standard deviations of the phase and amplitude spectra s_arg(Eref) and s_ln|Eref| that are obtained from a reference dataset containing 20 scans or N_ref = 20. Estimated minimum detectable thicknesses of different films at 1 THz are given in Table 1. Note again that the amplitude-derived critical thicknesses, l_c,m, can vary greatly owing to a strong variation in s_ln|Eref| from one dataset to the other, and therefore should be taken as rough estimations. All the estimated thicknesses are tied with the system described in Section 4.1, and are provided for illustrative purposes only.

Table 1. Minimum detectable film thicknesses for some common materials. The phase-based and amplitude-based minimum thicknesses l_c,p and l_c,m are determined by using Eqs. (13) and (18), respectively. For a substrate-backed film, the reference is assumed to be a bare substrate, as illustrated in Fig. 3(a). For a freestanding film, the reference is assumed to be free space, as in Fig. 3(c). For all the calculations, s_arg(Eref) = 3.2 × 10⁻² rad, s_ln|Eref| = 3.8 × 10⁻³, N_ref = 20, ω = 2π × 10¹² rad, and k_P = 1.

View Table

Interestingly, from Table 1, it can be seen that the freestanding photoresist film lowers the phase-derived minimal thickness by half compared with the same type of film on the silicon substrate, i.e., the system is more sensitive to the freestanding film. For 2πn_f · l/λ ≪ 1, the Fabry-Pérot effect from these air-film-air interfaces results in additional phase delay to enhance the phase retardation from wave propagation inside the film, whereas the air-film-silicon interfaces yield phase advancement in the Fabry-Pérot term. Compared with the photoresist, a smaller refractive index in the PDMS film leads to an increase in the phase-derived minimum thickness, and a larger absorption leads to a decrease in the amplitude-derived thickness. In the case of a doped epitaxial GaN layer, its relatively high refractive index and extinction coefficient relax the thickness constraints to only a few hundred nanometers. It can be inferred that THz-TDS detection ability is enhanced if the material under test is conducting. For characterization purposes, it is suggested that the minimal thickness should be ten times higher than these limits.

5. Conclusion

In this article, different measurement procedures commonly used for sample characterization with THz-TDS have been assessed for their suitability in thin-film sensing. It is recommended that the alternating sample and reference measurement procedure be employed to avoid misleading results. Furthermore, the critical sample thickness has been derived on the basis of the uncertainty analysis that takes into account all the possible sources of error in the system. Provided that the standard deviation of reference terahertz signals from a particular transmission-mode system is known a priori, the proposed model can be used to estimate the minimal film thickness that can be detected by the system. It is revealed that the phase component is more reliable than the amplitude counterpart in detecting the sample. For polymer films with a refractive index around 1.6, the system in use can detect films with a thickness down to about 2 μm. Although the experimental validation is specific to a particular polymer, the theory is general and applicable to a wide range of films in different configurations, including epitaxial semiconductors and biomolecules. The concept of critical thickness can be readily extended to the applications of thin-film characterization.

Appendix A: Linear approximation of Fabry-Pérot term

In Sections 3.2 and 3.3, the phase and amplitude of the Fabry-Pérot function are approximated by linear functions. This section provides corresponding proofs.

A.1. Phase of the Fabry-Pérot term

The argument of the Fabry-Pérot function can be expressed as

(19)$$\text{arg}({\text{FP}}_f)=\text{arg}{\left[1-{\rho }_{\text{total}}\cdot \text{exp}(-jkl)\right]}^{-1},$$

where ρ_total = ρ_faρ_fs and k = 2n̂_fω/c. Evaluating the function at l = 0 for the zeroth-order approximation yields

(20)$$\text{arg}{({\text{FP}}_f)}_0=-\text{arg}(1-{\rho }_{\text{total}})=-\text{arg}({\tau }_{\text{total}}),$$

where τ_total = τ_afτ_fs/τ_as. By assuming that n̂_f and n̂_s has a negligible imaginary component so that ρ_total ≈ ℛ(ρ_total), Eq. (19) can be rewritten as

(21)$$\begin{array}{lll}\text{arg}({\text{FP}}_f)\hfill & =\hfill & \text{arctan}\left[ℐ({\text{FP}}_f)/ℛ({\text{FP}}_f)\right]\hfill \\ \hfill & \approx \hfill & \text{arctan}\left[\frac{ℛ({\rho }_{\text{total}})\text{sin}kl}{ℛ\left({\rho }_{\text{total}}\right)\text{cos}kl-1}\right].\hfill \end{array}$$

Then, the first-order approximation can be obtained from

(22)$$\begin{array}{lll}\text{arg}{({\text{FP}}_f)}_1\hfill & \approx \hfill & l\cdot \frac{d}{dl}\text{arctan}{\left[\frac{ℛ({\rho }_{\text{total}})\text{sin}kl}{ℛ({\rho }_{\text{total}})\text{cos}kl-1}\right]}_{l=0}\hfill \\ \hfill & \approx \hfill & 2n_f\frac{\omega l}{c}\cdot \frac{ℛ({\rho }_{\text{total}})}{ℛ({\rho }_{\text{total}}-1)}.\hfill \end{array}$$

Combining the two orders of approximation results in a Taylor expansion of the Fabry-Pérot phase function around the origin, or

(23)$$\text{arg}({\text{FP}}_f)\approx -\text{arg}({\tau }_{\text{total}})+2n_f\frac{\omega l}{c}\cdot \frac{ℛ({\rho }_{\text{total}})}{ℛ\left({\rho }_{\text{total}}\right)-1}.$$

This approximation is valid for n_fωl/c = 2πn_f· l/λ ≪ 1.

A.2. Amplitude of the Fabry-Pérot term

The amplitude of the Fabry-Pérot part can be described as

(24)$$\text{ln}\left|{\text{FP}}_f\right|=\text{ln}{\left|1-{\rho }_{\text{total}}\cdot \text{exp}(-jkl)\right|}^{-1}.$$

Taking zeroth order approximation of Eq. (24) at l = 0 yields

(25)$$\text{ln}{\left|{\text{FP}}_f\right|}_0=\text{ln}{\left|1-{\rho }_{\text{total}}\right|}^{-1}=-\text{ln}\left|{\tau }_{\text{total}}\right|.$$

Provided that the extinction coefficient in n̂_f is negligible, Eq. (24) can be expanded to

(26)$$\text{ln}\left|{\text{FP}}_f\right|\approx -\frac{1}{2}\text{ln}\left[{\left(1-ℛ({\rho }_{\text{total}})\cdot \text{cos}kl\right)}^2+{\left(ℛ({\rho }_{\text{total}})\cdot \text{sin}kl\right)}^2\right].$$

Equation (26) can be approximated to the first order as

(27)$$\begin{array}{lll}\text{ln}{\left|{\text{FP}}_f\right|}_1\hfill & \approx \hfill & l\frac{d}{dl}{\left\{-\frac{1}{2}\text{ln}\left[{\left(1-ℛ({\rho }_{\text{total}})\cdot \text{cos}kl\right)}^2+{\left(ℛ({\rho }_{\text{total}})\cdot \text{sin}kl\right)}^2\right]\right\}}_{l=0}\hfill \\ \hfill & \approx \hfill & 0.\hfill \end{array}$$

Hence, a Taylor expansion for the amplitude component of the Fabry-Pérot term is given as

(28)$$\text{ln}\left|{\text{FP}}_f\right|\approx -\text{ln}\left|{\tau }_{\text{total}}\right|.$$

Again, this approximation is valid for n_fωl/c ≪ 1.

Appendix B: Terahertz properties of the photoresist and silicon

The 12.5 μm-thick S1813 photoresist film on the 626.6 μm-thick silicon substrate is spin-coated and measured as described in Section 4. The silicon measurement is normalized with the free-space reference, whilst the photoresist-on-silicon measurement is normalized with the silicon reference. The optical parameters are extracted from the averaged reference and sample scans by fitting the model to the measured amplitude and phase spectra via the gradient descent method [19]. The extracted refractive index n and extinction coefficient κ for the photoresist and silicon are given in Fig. 7.

 

Fig. 7 Optical parameters of the photoresist and silicon. (a) Refractive index, n, and (b) extinction coefficient, κ. The shaded areas indicate unreliable parts of the data. At 1 THz, the complex refractive index for the photoresist equals 1.6 − 0.08 j, and the refractive index for the silicon is 3.44 with negligible loss.

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Acknowledgments

WW acknowledges financial support from the Australian Research Council ( DP1095151). WC, JFO, and WZ acknowledge financial support from the US National Science Foundation (Grant No. ECCS-1232081). The authors gratefully acknowledge valuable comments from Daniel Grischkowsky, School of Electrical & Computer Engineering and the Ultrafast Terahertz Optoelectronics Laboratory, Oklahoma State University, and technical assistance of Charan M. Shah, Functional Materials and Microsystems Research Group, RMIT University, Melbourne, Victoria, Australia.

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