A terahertz time-domain spectrometer for simultaneous transmission and reflection measurements at normal incidence

Fabian D. J. Brunner; Arno Schneider; Peter Günter

doi:10.1364/OE.17.020684

1. Introduction

The terahertz frequency range of the electromagnetic spectrum is typically defined to be between 0.1 and 10 THz. Today, terahertz waves are used for imaging and spectroscopy in many fields of science and technology (see [1] for a review). Terahertz time-domain spectroscopy (TTDS) is a phase-sensitive technique for the determination of the refractive index and the absorption coefficient at terahertz frequencies. TTDS experiments are most often performed in transmission setups, because transmission spectrometers are easier to implement and give more accurate results. However, strongly absorbing samples can only be measured in reflection.

The majority of the reflection spectra reported in the literature have been measured at a non-normal angle of incidence [2–5]. However, the geometry of normal incidence is favorable for both terahertz spectroscopy and imaging. In reflection spectroscopy, the refractive index and the absorption coefficient can be calculated from the reflected terahertz signals by an exact analytical formula, which is more simple for normal incidence. In terahertz time-of-flight tomography, the terahertz beam experiences a lateral offset if it is reflected from different layers of a sample at non-normal incidence, which limits the maximum accessible depth of the three-dimensional image [6]. Due to the lateral offset of the terahertz beam at non-normal incidence, also the positioning of the sample is more critical. For reflection at 45°, the tolerance of the distance between the sample and the terahertz emitter and detector is less than the diameter of the terahertz beam. A much larger tolerance is obtained by the use of normal incidence, which is crucial for remote sensing applications such as standoff detection of explosives. An experimental setup with normal incidence of the terahertz beam onto the sample can be implemented by introducing a beamsplitter into the terahertz beam path at the expense of reducing the detected terahertz signal in amplitude [7, 8].

In this article, we present a spectrometer in a transceiver configuration in which the terahertz pulses are generated and detected in a single electro-optic crystal. This geometry allows reflection measurements at normal incidence without the drawback of a reduced terahertz signal caused by a beamsplitter. The concept of a terahertz transceiver has been discussed in the literature for both photoconductive antennas and electro-optic crystals with the zinc blende structure [9–11]. However, the configuration using ZnTe presented in Ref. [11] has several limitations. On the one hand, the optimum crystal orientation for terahertz generation is different than for detection, leading to a significantly decreased overall efficiency of the transceiver compared to a setup with two separate crystals for generation and detection. More importantly, the experiment in Ref. [11] uses a quadratic detection scheme (crossed polarizers, zero optical bias), i.e., it is not the electric field E(t) that is detected, but rather its square E(t)². This is sufficient for time-of-flight imaging, but impractical for terahertz time-domain spectroscopy, where the undistorted waveform E(t) is essential. It is possible to correct the distortions of terahertz waveforms, but only with a non-zero optical bias [12]. However, the introduction of an optical bias in the setup of Ref. [11] would drastically increase the noise level. The transceiver configuration presented in this article does not suffer from any of these limitations. For the birefringent organic crystal 4-N,N-dimethylamino-4′-N′-methyl-stilbazolium tosylate (DAST), the optimum orientation is the same for terahertz generation and detection, and the detection scheme of terahertz-induced lensing does not require crossed polarizers and is intrinsically distortion-free. The terahertz generation does not influence the terahertz detection, if these two processes take place in one DAST crystal, because they are separated in time by 2 ns in our setup. To our knowledge, there is no excitation with such a long lifetime taking place in the DAST crystal. Possible thermal effects can also be excluded since the crystal is well transparent for the pump beam, and the power of the absorbed terahertz radiation is negligibly small. Thus, the dynamic range and bandwidth of our transceiver type spectrometer are the same as for the setup using two separate DAST crystals for terahertz generation and detection [13].

Apart from a more compact experimental setup and a more simple data analysis in reflection spectroscopy, the transceiver configuration has further intrinsic advantages. We demonstrate that our spectrometer can also be used for double pass transmission measurements without any modification of the experimental setup between the reflection and the transmission experiments. Thus, we can precisely measure the refractive index and the absorption coefficient of both transparent and strongly absorbing materials in a single experimental setup.

2. Experimental setup

2.1. Reflection measurements

The experimental setup is schematically shown in Fig. 1. The laser used for this work is an erbium-doped fiber laser delivering 70 fs pulses with an energy of 1.8 nJ at a repetition rate of 100MHz and a central wavelength of 1560 nm. The laser beam is split into a pump and a probe beam. Terahertz pulses are generated through optical rectification of the pump pulses in a 0.630mm thick DAST crystal exploiting its largest nonlinear optical susceptibility tensor element χ ⁽²⁾111=480 pm/V [13, 14]. The terahertz beam is reflected from a glass plate coated with indium tin oxide (ITO) [15]. The reflection of the pump beam from the ITO coated glass plate is blocked by a germanium crystal. The terahertz beam is focused onto the sample using an off-axis elliptical mirror. The terahertz beam reflected back from the surface of the sample is refocused onto the same DAST crystal by the elliptical mirror. The probe beam is transmitted through the ITO coated glass plate and propagates collinear with the terahertz beam though the DAST crystal. The electric field of the terahertz pulse is measured by electro-optic sampling in the DAST crystal [16]. Due to the small pulse energy of the laser, the relatively weak detected terahertz signal needs to be measured using lock-in amplification. This technique requires a modulation of the terahertz signal, which is achieved by mechanically chopping the terahertz beam. The experiments are conducted in a dry air atmosphere to avoid absorption of the terahertz wave due to ambient water vapor.

As a reference measurement, the terahertz signal reflected from a gold coated planar mirror is recorded. Taking into account that the refractive index of air is n≈1 and gold is a nearly perfect reflector at terahertz frequencies [17], the ratio of the sample and the reference spectra is given by

\frac{∣ E_{sample} (ν) ∣}{∣ E_{ref .} (ν) ∣} \exp (i [ϕ_{sample} (ν) - ϕ_{ref .} (ν)]) \equiv r (ν) \exp [i ϕ (ν)] = \frac{n (ν) + iκ (ν) - 1}{n (ν) + iκ (ν) + 1},

where ν is the terahertz frequency, r is the amplitude ratio, ϕ is the phase difference, and n and κ are the real and the imaginary part of the complex refractive index ñ, respectively. Equation (1) can be solved analytically, and we find the following expressions for the refractive index n(ν) and the power absorption coefficient α(ν)=κ(ν)4πν/c:

n (ν) = \frac{1 - r {(ν)}^{2}}{1 + r {(ν)}^{2} - 2 r (ν) \cos ϕ (ν)},

α (ν) = \frac{4 πν}{c} \frac{2 r (ν) \sin ϕ (ν)}{1 + r {(ν)}^{2} - 2 r (ν) \cos ϕ (ν)} .

The precise placement of the reference mirror with respect to the surface of the sample is crucial, since a minor displacement δx in longitudinal direction leads to a phase error δϕ=4πνδ x/c, which may result in a large error in the calculated refractive index and absorption coefficient. In order to minimize δx, we have designed a special sample holder. Both the sample and the reference mirror are pressed from behind to an aluminum frame which can be moved perpendicular to the terahertz beam using a motorized translation stage (see Fig. 1).

In the measurements of the optical properties of the ionic crystals CsI, KBr, and NaCl presented later in this article, we could correct the phase error δϕ as follows. If the displacement δx is introduced as an additional parameter into the theoretical functions which describe the dispersion of the crystals given by Eqs. (8)–(10), δx can be obtained along with other parameters of the functions by a fitting procedure (see Section 3.1 for details). The same procedure can also be used to correct the phase error for other material classes whose dispersion is described by different theoretical functions, e.g., by the Drude theory for the dispersion of free carriers in doped semiconductors [2].

2.2. Double pass transmission measurements

The experimental setup shown in Fig. 1 can also be used for double pass transmission measurements without any readjustment of the optical components. In these measurements, the terahertz pulse propagates through the sample, is then reflected back and propagates a second time through the sample, now in the opposite direction. There are two different configurations for such a measurement: Either the terahertz pulse is back-reflected from a spherical mirror with its focus at the sample position (see Fig. 1). Alternatively, the terahertz pulse reflected from the rear surface of the sample E ^{rear surface} _sample is recorded. For simplicity, we restrict the following discussion on optically thick samples, where the reflections of the terahertz pulses from the front and the rear surface of the sample are well separated in time. The data analysis for optically thin samples can easily be derived for the first configuration in analogy to the data analysis in standard transmission measurements of optically thin samples [18, 19].

Fig. 1. Experimental setup of the terahertz time-domain spectrometer for simultaneous transmission and reflection measurements: BS, beam splitter; ITO, glass plate coated with indium tin oxide; DAST, 4-N,N-dimethylamino-4′-N′-methyl-stilbazolium tosylate crystal used as terahertz transceiver; Ge, germanium crystal. Inset: The sample holder can be moved perpendicular to the terahertz beam into three different positions. In the first position, the terahertz pulse reflected from the sample or transmitted through the sample is measured. In the second and third positions, the terahertz signal reflected from the planar mirror or from the spherical mirror is used as a reference signal for reflection or for double pass transmission measurements, respectively.

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For the measurement of the reference signal in the first configuration, the sample is moved out of the terahertz beam (position 3 of the sample holder, see Fig. 1). The measurement and the reference measurement are denoted by E ^2-pass _sample and E ^2-pass _ref., respectively. For a weakly absorbing material (κ(ν)≪1), the imaginary part of the Fresnel transmission and reflection coefficients can be neglected. The refractive index n and the absorption coefficient α can then be calculated from the phases ϕ and the amplitudes |E| of the two measurements using the following relations:

n (ν) = \frac{c (ϕ_{sample}^{2 - pass} (ν) - ϕ_{ref .}^{2 - pass} (ν))}{4 πνl} + 1,

α (ν) = - \frac{1}{l} \ln (\frac{∣ E_{sample}^{2 - pass} ∣}{∣ E_{ref .}^{2 - pass} (ν) ∣} \frac{{(n + 1)}^{4}}{{(4 n)}^{2}}) .

The second configuration where the terahertz pulse is reflected from the rear surface of the sample is preferable for materials with a high refractive index, since the Fresnel reflection losses increase with the refractive index. For a material with a refractive index n larger than 4.2, the measured spectral amplitude |E ^{rear surface} _sample (ν)| is larger than |E ^2-pass _sample(ν)| and thus has a better signal-to-noise ratio. Many materials have a refractive index larger than 4.2 in the terahertz frequency range, e.g., LiNbO₃ (n_o≥6.7, n_e≥5.1), LiTaO₃ (n_o>6.4, n_e>6.3), or rutile [20,21]. For this configuration, the pulse reflected from the planar mirror E _ref. can be used as a reference. The refractive index n and the absorption coefficient α can be calculated using the following equations:

n (ν) = \frac{c (ϕ_{sample}^{rear surface} (ν) - ϕ_{ref .}^{} (ν) - π)}{4 πνl},

α (ν) = - \frac{1}{l} \ln (\frac{∣ E_{sample}^{rear surface} (ν) ∣}{∣ E_{ref .} (ν) ∣} \frac{{(n + 1)}^{3}}{4 n (n - 1)}) .

Note that the phase difference ϕ ^{rear surface} _sample (ν)-ϕ _ref.(ν) in Eq. (5a) is affected by the same phase error as in reflection spectroscopy. However, the phase error is much less critical for the determination of n and α in transmission spectroscopy. The displacement δx and the error in the determination of the thickness of the sample δl are typically of the same order of magnitude (1–30 µm). They lead to a systematic error of the refractive index given by:

δn = \frac{n}{l} \sqrt{{δl}^{2} + {δx}^{2} .}

To circumvent the phase problem in the second configuration, one can use the pulse reflected from the front surface of the sample E _sample as a reference. The refractive index n and the absorption coefficient α are then given by:

n (ν) = \frac{c (ϕ_{sample}^{rear surface} (ν) - ϕ_{sample}^{} (ν) - π)}{4 πνl},

α (ν) = - \frac{1}{l} \ln (\frac{∣ E_{sample}^{rear surface} (ν) ∣}{∣ E_{sample} (ν) ∣} \frac{{(n + 1)}^{2}}{4 n}) .

3. Measurements

3.1. Phonon polariton dispersion of CsI, KBr, and NaCl

The performance of the spectrometer in the reflection mode has been tested by measuring the phonon polariton dispersion of the salt crystals CsI, KBr, and NaCl. The samples were polished crystal windows purchased from Sigma-Aldrich. The real and imaginary parts of the dielectric function ε(ν)=ε′(ν)+iε″(ν) of the alkali halide crystals can be described in the harmonic approximation by [22]

ε' (ν) = ε_{\infty} \frac{(ν_{LO}^{2} - ν^{2}) (ν_{TO}^{2} - ν^{2}) + γ^{2} ν^{2}}{{(ν_{TO}^{2} - ν^{2})}^{2} + γ^{2} ν^{2}},

ε^{″} (ν) = ε_{\infty} \frac{(ν_{LO}^{2} - ν_{TO}^{2}) γν}{{(ν_{TO}^{2} - ν^{2})}^{2} + γ^{2} ν^{2}},

where ν _TO is the transversal optical phonon frequency, ν _LO is the longitudinal optical phonon frequency, γ is the damping parameter, and ε _∞ is the high frequency dielectric constant. The refractive index n(ν) and the absorption coefficient α(ν) can be obtained from ε′(ν) and ε′(ν) by

n (ν) = \sqrt{\frac{1}{2} [\sqrt{ε' {(ν)}^{2} + ε ″ {(ν)}^{2}} + ε' (ν)],}

α (ν) = \frac{4 πν}{c} \sqrt{\frac{1}{2} [\sqrt{ε' {(ν)}^{2} + ε ″ {(ν)}^{2}} - ε' (ν)]}

Theoretical functions for the amplitude ratio r(ν) and the phase difference ϕ(ν) are calculated from n(ν) and κ(ν)=α(ν)c/(4πν) using Eq. (1). The phase error is included into the theoretical phase difference by adding 4πν δx/c. The theoretical functions for r and ϕ are fitted to the measured data. The parameters of the functions determined for CsI and KBr are listed in Table 1. They are in reasonable agreement with the literature values which are also given in Table 1 for a comparison. Since the spectral amplitude measured with our system is relatively low for frequencies above ≈4.3 THz, we could not precisely determine the parameters for NaCl. However, the phase-error of the measurement could be corrected by a fitting procedure using the parameters reported in Ref. [22].

For a comparison, the displacement δx was measured independently, and a reasonable agreement with the value determined by the fitting procedure was found. The terahertz pulse reflected from the front surface of a silicon crystal mounted in the sample holder was shifted in time by 153 fs compared to the terahertz pulse reflected from the reference mirror, which corresponds to a displacement of δx=23.0 µm. Note that the very small absorption of silicon does not give rise to a significant phase change of the reflected terahertz wave. In a subsequent spectroscopy measurement of CsI, a displacement of δx=21.5 µm was determined using the fitting procedure. The difference of these two values of δx of 1.5 µm lies within the mechanical reproducibility of mounting a sample in the sample holder.

The measured refractive index and absorption coefficient of CsI and the theoretical functions calculated from Eqs. (8)–(10) using the parameters from Table 1 are plotted in Fig. 2. The measured spectra are in very good agreement with the theoretical functions and with the TTDS measurements previously presented in Ref. [23]. The features in the spectra at 3 THz shown in Fig. 2 can be explained by the increased error due to a phonon resonance in the DAST transceiver crystal [13]. All the measurement data in this article are presented for frequencies ν≥1.4 THz due to the strong absorption in the DAST crystal near the resonance frequency of 1.1 THz [13].

The measured data and the theoretical curves for KBr are shown in Fig. 3. The measured dispersion is in very good agreement with the theoretical one calculated from the Lorentz oscillator functions.

The results for NaCl are shown in Fig. 4. The theoretical curves plotted in Fig. 4 are calculated from Eqs. (8)–(10) using the parameters from Ref. [22]. There is also a good agreement between the measurement data and the theoretical functions in consideration of the fact that the errors in this measurement are large for frequencies above 4.3 THz.

Table 1. Parameters for the dielectric dispersion of the alkali halide crystals CsI, KBr, and NaCl in the harmonic approximation.^a

View Table

Fig. 2. Refractive index and absorption coefficient of CsI measured by terahertz time-domain spectroscopy in reflection at normal incidence. Dots: measured data; solid lines: best fit to the measured data using a Lorentz oscillator function (see Eqs. (8)–(10) and Table 1).

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Fig. 3. Refractive index and absorption coefficient of KBr measured by terahertz time-domain spectroscopy in reflection at normal incidence. Dots: measured data; solid lines: best fit to the measured data using a Lorentz oscillator function (see Eqs. (8)–(10) and Table 1).

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Fig. 4. Refractive index and absorption coefficient of NaCl measured by terahertz time-domain spectroscopy in reflection at normal incidence. Dots: measured data; solid lines: theoretical dispersion calculated from Eqs. (8)–(10) using the parameters from Ref. [22].

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3.2. Double pass transmission measurements on PTFE and Si

The effectiveness of the double pass transmission measurement technique has been demonstrated using polytetrafluoroethylene (PTFE, Teflon) and silicon whose terahertz spectra are well known from the literature [27, 28]. The PTFE sample was a 10mm thick plate. Due to the low refractive index of PTFE, the first measurement mode was used where the terahertz beam is back-reflected from an external spherical mirror (see Fig. 1). The measured refractive index and absorption coefficient of PTFE are plotted in Fig. 5.

Both measurement modes have been performed on a 2.137mm thick polished silicon crystal purchased from Sigma-Aldrich. In the second measurement mode, the terahertz beam is back-reflected from the rear surface of the sample. The results are shown in Fig. 6. The absorption coefficient of Si is lower than the detection limit in these experiments (α<0.2mm^-1). The features at 3 THz in the spectra of Si measured by the second mode can be explained by the large error due to a phonon resonance in the DAST transceiver crystal [13]. All the spectra are in good agreement with the ones published in the Refs. [27, 28].

Fig. 5. Refractive index and absorption coefficient of PTFE measured by terahertz time-domain spectroscopy in a double pass transmission configuration where the terahertz pulse is back-reflected from an external spherical mirror (see text for details).

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Fig. 6. Refractive index and absorption coefficient of silicon measured by terahertz time-domain spectroscopy in two different double pass transmission configurations (see text for details). Solid lines: the terahertz pulse is back-reflected from the rear surface of the sample. Dashed lines: the terahertz pulse is back-reflected from an external spherical mirror.

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4. Conclusions

A versatile terahertz time-domain spectrometer for simultaneous transmission and reflection measurements at normal incidence is presented. The use of normal incidence not only facilitates the data analysis in reflection spectroscopy, but it is also ideal for terahertz reflection tomography. Without any modifications of the experimental setup, we can measure the refractive indices and absorption coefficients of transparent samples in transmission or of absorbing samples in reflection. Thus, we can also determine the optical properties of one material in its transparency and its non-transparency range of the terahertz spectrum in a single experiment.

The refractive index and absorption spectra of the materials CsI, KBr, NaCl, PTFE, and Si obtained with this spectrometer are in good agreement with the literature data. The measured phonon polariton dispersion of the salt crystals CsI, KBr, and NaCl can be well described by a harmonic approximation. The parameters of Lorentz oscillator function have been determined for CsI and KBr.

The experimental setup presented here can also be used with different combinations of an electro-optic crystal and a femtosecond laser, e.g., 4-N,N-dimethylamino-4′-N′-methyl-stilbazolium 2,4,6-trimethylbenzenesulfonate (DSTMS) in combination with an erbium-doped fiber laser, 2-cyclooctylamino-5-nitropyridine (COANP) pumped with a Ti:sapphire laser, or 2-[3-(4-hydroxystyryl)-5,5-dimethylcyclohex-2-enylidene]malononitrile (OH1) combined with a laser operating at a wavelength near 1300 nm, in order to make use of the optimum phase-matching properties [29–31]. COANP and OH1 crystals were proven to be very efficient emitters of a gap-free terahertz spectrum ranging from 0.3 to 3.0 THz [30, 31].

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13. A. Schneider, M. Neis, M. Stillhart, B. Ruiz, R. U. A. Khan, and P. Günter, “Generation of terahertz pulses through optical rectification in organic DAST crystals: theory and experiment,” J. Opt. Soc. Am. B 23, 1822–1835 ( 2006). [CrossRef]

14. F. Pan, G. Knöpfle, Ch. Bosshard, S. Follonier, R. Spreiter, M. S. Wong, and P. Günter, “Electro-optic properties of the organic salt 4-N,N-dimethylamino-4′-N′-methyl-stilbazolium tosylate,” Appl. Phys. Lett. 69, 13–15 ( 1996). [CrossRef]

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Material	Ref.	ν _TO (THz)	ν _LO (THz)	γ (GHz)	ε _∞
CsI	this work	1.84±0.01	2.65±0.02	45±3	2.8±0.1
	[24]	1.84±0.01	2.45±0.01	69
	[25]	1.86	2.74		3.02
KBr	this work	3.44±0.01	4.65±0.03	120±6	2.6±0.1
	[26]	3.39	4.74	150	2.43
NaCl	[22]	4.92	7.86	270	2.31

A terahertz time-domain spectrometer for simultaneous transmission and reflection measurements at normal incidence

Abstract

1. Introduction

2. Experimental setup

2.1. Reflection measurements

2.2. Double pass transmission measurements

3. Measurements

3.1. Phonon polariton dispersion of CsI, KBr, and NaCl

3.2. Double pass transmission measurements on PTFE and Si

4. Conclusions

References and links

Cited By

Figures (6)

Tables (1)

Equations (14)

Optics Express