High resolution continuous wave terahertz spectroscopy on solid-state samples with coherent detection

De-Yin Kong; Xiao-Jun Wu; Bo Wang; Yang Gao; Jun Dai; Li Wang; Cun-Jun Ruan; Jun-Gang Miao

doi:10.1364/OE.26.017964

1. Introduction

Emerging in 1990s, terahertz (THz) science and technology have already experienced rapid development during these thirty years. THz spectroscopy [1–3] and imaging [4–7] are facing a critical stage from the experimental research to the real applications. The improvement of terahertz time-domain spectroscopy (THz-TDS) greatly promotes the development of terahertz spectrum technology. However, the resolution of THz-TDS is ~1 GHz, for some specific applications like gas-phase sensing in environment monitoring, human breath, and drugs and explosives detections, high resolution THz continuous wave frequency-domain spectroscopy (THz-FDS) is also highly demanded [8–10]. Besides, THz-TDS has some imperfections such as high cost and low frequency resolution. In the laboratory, most of the measurements are conducted without the influence of atmospheric water vapors. However, in most practical applications, it is not so convenient to purge the systems with dry nitrogen gases or pump the system in chambers to rule out the interferences. Therefore, it is very important but challengeable to obtain accurate information in the THz frequencies from the measured samples or objects in atmospheric environment. Previous THz-FDS can only measure the intensity of the transmitted or reflected THz signals with the disadvantage of lacking the phase information. As a consequence, Kramers–Kronig (K-K) transform method has to been employed to extract the permittivity of the measured materials, which requires the full spectrum measurements from $- \infty$ to $+ \infty$ to obtain accurate values. Fortunately, the recent emerged THz-FDS with coherent detection based on photomixing technique can overcome this weakness [11]. Due to the coherent detection, the phase information hidden in the original data can be directly extracted. However, in real measurements, due to low signal-to-noise radio (SNR) and the interference of water vapors, it is still very challengeable to extract accurate permittivity of materials. Although THz-FDS has already been developed into a compact and fully fiber-coupled system, which may benefit for the real applications, there is still a big gap between laboratory research and real applications, if the data analysis problems were not fully addressed.

For real applications, no matter in THz-TDS or THz-FDS, a reference signal is always required to extract the refractive index and the absorption coefficient of samples. For THz-FDS with coherent detection, the signals of water vapor can be used as a reference so that the THz optical constants can be calculated. The original data obtained in the coherent detection THz-FDS is as a cosine function of frequency. In [12], A. Roggenbuck et al. have already extracted the refractive index and absorption coefficient of lactose monohydrate pellets. However, they only considered the extrema and zeros of the original data and ignore the rest data points. They did not offer the approach of errors eliminating during the calculation of the extrema order. In order to address this problem, further manipulation like phase unwrap method is required. In the aforementioned Finding extrema method, too many data points except the extrema and zeros are ignored which decrease the frequency resolution. However, in Hilbert transform analysis method, all the data are used, which offers an extremely high frequency resolution. For example, in [13], D. W. Vogt et al. employed this method to investigate the high Q whispering gallery mode bubble resonators in which they push the frequency resolution to the limit of the laser linewidth of ~4 MHz. However, they did not try to restrain the Fabry–Pérot (FP) effect but it interferes the determination of the resonance frequencies as well as its intensity.

In our work, we systematically investigate the data analysis methods related to coherent detection THz-FDS. We comparably study the aforementioned Finding extrema method and Hilbert transform analysis and also give the data analysis details. We demonstrate that Hilbert analysis is very appropriate for the coherent detection THz-FDS, because the system noise from FP effect can be successfully eliminated when the frequency-domain signals are transformed into time domain, and partial signals are set to be zero. To the best of our knowledge, this method has never been applied to the data analysis in this coherent detection THz-FDS. Our work benefits for mastery of the systems and paves the way for fast and accurate data analysis, which promotes the industrial and real applications.

2. Experimental setup

Figure 1 exhibits the THz-FDS system used in our experiments. The system is driven by two distributed feedback laser (DFB) lasers (TOPTICA Photonics, TeraScan 780) with different frequencies of 783 nm and 785 nm, respectively. The lasers are coupled by optical fibers into a laser combiner to form a beat signal which is divided into two beams with the same average power of ~36 mW and similar phase. One of the two beat signals is employed to pump the low-temperature grown GaAs photoconductive antenna and the THz wave is generated based on photomixing technique. The emitted continuous wave (CW) THz spectrum covers from 0.05 to 2.0 THz with a varied output power. At 0.1 THz, the maximum average power is ~2 $μW$ at 0.1 THz which decreases to 0.3 $μW$ at 0.5 THz. The generated THz beam is focused by a super hemisphere silicon lens with a focal length of ~30 mm. The focused THz beam is then collimated by a 2-inch $90^{\circ}$ off-axis parabolic mirror. For the sample measurements, the THz beam is focused by the second 2-inch $90^{\circ}$ off-axis parabolic mirror. The transmitted signal is collimated and focused by a symmetric 2-f system and then collected by the receiver attached with another silicon super hemisphere lens. All the parabolic mirrors for the THz propagation are 4-inch focal length. The detected THz signal drives the photo-carriers pumped by the other laser beat to form a photocurrent amplified by a lock-in amplifier. The photocurrent is recorded by the DLC Smart controller and send to the computer.

Fig. 1 (a) schematic diagram of the coherent detection THz-FDS, and (b) the photo of the four parabolic mirrors for THz propagation from the transmitter to the receiver.

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3. Data analysis of the THz-FDS

3.1 The coherent detection principle of the THz-FDS

Figure 2 shows the principle of the coherent detection of the THz-FDS system. In order to obtain the THz optical constants, we have to know the phase difference and the amplitude ratio between the sample and the reference. The phase difference at the two laser combiner outputs are always the same, hence we set the phase at the two positions to be zero. Although the laser fibers do not propagate THz waves, it contributes the phase shift of the system. The main signal that transfers the information starts at the combiner output 1 and ends at the receiver.

Fig. 2 The schematic diagram of the optical path used for THz optical constants extraction. $L_{T}$ and $L_{R}$ are the optical paths for the laser path in fibers while $L_{T H z}$ is the THz propagation distance in air. $S_{b i a s}$ is a bias voltage with a frequency of ~40 kHz. $S_{T H z - T}$ and $S_{T H z - i n p u t}$ are the signals from the transmitter and the receiver, respectively. The subscript $T$ and $R$ are used to distinguish the signals of these two mixers.

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Assuming the contribution of the transmitter and the receiver to the phase is constant at a certain frequency, which can be seen as zero during the study. Thus, the emitted THz wave from the transmitter can be describe as [14]:

E_{T H z} \propto \cos (2 π (f_{+} - f_{-}) t + Δ φ)

in which the

Δ φ

is:

Δ φ = 2 π \frac{n_{f i b e r} \cdot L \cdot f_{+}}{c} - 2 π \frac{n_{f i b e r} \cdot L \cdot f_{-}}{c} = 2 π \frac{n_{f i b e r} \cdot L \cdot (f_{+} - f_{-})}{c}

where

Δ φ

is the phase difference of two lasers at the transmitter or receiver, which originate from the different frequencies of the two DFB lasers and the optical paths

L = L_{T}

or

L = L_{R}

;

n_{f i b e r}

is the refractive index of the propagation medium;

c

is the light speed in vacuum;

f_{+}

and

f_{-}

are the frequencies of the two laser beams, and

f_{T H z} = f_{+} - f_{-}

, where

f_{T H z}

is the generated THz frequency. The frequency of the bias voltage added on the antenna is ~40 kHz which is very low compared to that of the emitted THz waves. Therefore, the THz signal is a double-sideband suppressed-carrier signal (DSB-SC). At the receiver, the THz wave is treated as a bias voltage. The original output photocurrent has an oscillation frequency of ~40 kHz whose amplitude is detected by the lock-in amplifier. The detected signal is expressed as:

I_{p h} = A_{S a m / R e f} \cos (Δ Φ_{S a m / R e f})

where the

I_{p h}

is the detected photocurrent,

Δ Φ_{S a m / R e f}

is the signal phase when measuring the sample or reference.

Δ Φ_{S a m / R e f}

is expressed as:

Δ Φ_{S a m / R e f} = [n_{f i b e r} (L_{T} - L_{R}) + n_{T H z} (L_{T H z} - d) + n_{S a m / R e f} d] \frac{2 π f_{T H z}}{c}

where

n_{T H z}

is the refractive index of air in the THz propagation path. The

A_{S a m / R e f}

is the signal amplitude expressed as:

A_{S a m / R e f} = A α_{S a m / R e f} α_{T H z}

where

A

,

α_{S a m / R e f}

and

α_{T H z}

represent for the attenuation of the system, the sample/reference, and air in the THz propagation path, respectively. Equation (3) obtained here is the original data acquired from the system and it agrees well with that in [12].

3.2 Methods of extracting the amplitude and phase from the original measured data of the samples

In order to extract the permittivity of the sample, the amplitude ratio and phase difference have to be calculated from the original data described above. However, we first have to obtain the amplitude and phase of the sample and the reference, respectively. There are many approaches to solve this problem, such as Fitting method, Finding extrema, and Hilbert transform analysis.

The fitting method is one of the most universal methods. Normally, it is used to fit the whole data points with a known function. However, in our case, the signal is difficult to be fitted due to the lack of complete function for an unknown material. Nevertheless, we can attempt to obtain the amplitude and phase of a single point by fitting several points before and after it. With this method, the resolution is equal to the sampling step. Equation (6) is the curve function used for fitting where $I_{p h}$ is the photocurrent, $A$ , $B$ is the unknown parameters and $f$ the frequency.

I_{p h} = A \cos (B f)

Another method is to find the extrema of the recorded signal. The values of the peak and trough of a cosine signal can be directly acquired, and their phases are known to be $2 m π$ and $2 m π + π$ , respectively. For each extremum, the order $m$ is calculated by the frequency. Frequency difference of the target and the adjacent peaks are used to calculate the approximate period $Δ f$ of the cosine signal. Then the $m$ and the phase of the maxima can be calculated [12]:

m = [\frac{f_{target}}{Δ f}] \pm k

Φ_{p e a k} = 2 m π, Φ_{t r o u g h} = 2 m π + π, Φ_{z e r o} = m π + \frac{π}{2}

where

[]

is the rounding symbol;

f_{target}

is the frequency of the target;

Φ_{p e a k}

,

Φ_{t r o u g h}

and

Φ_{z e r o}

are the phase of the peak, trough, and the zero point;

k

is the phase jump error when calculating the

m

, and

k

is a integer. If only the peaks were used, the resolution was the period of the cosine signal. If both of the peaks and troughs are employed, the resolution would be half of the period, which is still about ten times lower than those of the other two methods. The zeros can also be used to calculate the phase, but the amplitude cannot be obtained at zeros. To increasing the resolution, one need to change the path difference to decrease the period of the cosine signal.

The third one is Hilbert transform analysis. It is a commonly used method in decomposition of an amplitude-modulated and frequency-modulated (AM-FM) signal [15]. In signal processing, especially in automatic speaker identification (SID), Hilbert transformation is extensively used to get the signal envelope, thus it is also suitable for the FDS [16,17]. For a signal, it must be bandlimited for applying this method, or the method is equivalent to extra addition of a low-pass filter [18]. These require the amplitude of Eq. (3) to change slower than that of the cosine function. It is necessary to increase the optical path difference to increase the frequency of the cosine function, for the samples with very narrow absorption peaks. By Hilbert transform, we can get the imaginary part and the spectrum in complex form [13,18,19]:

{\tilde{I}}_{p h} = {I^{'}}_{p h} + i Η ({I^{'}}_{p h})

where

Η

means Hilbert transform. The real part of

{\tilde{I}}_{p h}

is the original data

{I^{'}}_{p h}

. Note that before Hilbert transform, the frequency interval of data needs to be interpolated to be uniform. During the calculation, the phase of

{\tilde{I}}_{p h}

need to be unwrapped. Phase in the low SNR part may be vague and difficult to unwrapped. There is no effective method yet to deal with this problem.

Figure 3 illustrates the procedure of the data analysis using Hilbert transform in THz-FDS. In this figure, we can see that after the Hilbert transform the data in frequency domain can be converted to time-domain by using inverse fast Fourier transform (IFFT). Then in time-domain, part of the oscillations from interfaces can be easily distinguishes which will be discussed in detail later. With this method, the system noise can be effectively suppressed.

Fig. 3 The procedure of the data analysis in THz-TDS and THz-FDS

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We summarize the properties of the aforementioned three methods in Table 1. The form of the independent variable used in Fitting method and Finding Extrema method is continuous, while it is discrete in Hilbert transform. The calculation speed for Fitting method is low, but that for both the Finding Extrema and Hilbert transform is high. Both of the Fitting method and the Finding Extrema do not have Gibbs phenomenon, but Hilbert Transform do have the phenomenon. In order to compare the time consumption of the three algorithms, we use the Matlab software platform to analyze the data, and record the time in the same computer. The Finding extrema method, the Hilbert transform method, and the fitting method take ~0.46 s, ~0.35 s, and ~50 min, respectively.

Table 1. Properties comparison of fitting, Finding extrema and Hilbert transform

View Table

Fitting is the most universal method to make full use of all the original data points but it is heavily time-consuming. Finding extrema is a normal and reliable method, but it neglects some information of the data not located at peaks and troughs. Hilbert transformation makes use of all the data points and it is convenient, fast and easy to acquire the amplitude and phase information from the original data. However, the phase difference of sample and reference has a $2 k π$ error for the ambiguity at the low frequencies [13].

3.3 Extracting the transmission coefficient and refractive index of the sample

For all the three method, the transmission coefficient of the sample can be obtained by dividing the amplitude of the sample with respect to that of the reference, as is shown below:

t_{S a m} = \frac{A_{S a m}}{A_{R e f}} = \frac{A α_{S a m} α_{T H z}}{A α_{R e f} α_{T H z}} = \frac{α_{S a m}}{α_{R e f}} = α_{S a m}, T_{S a m} = t_{S a m}^{2}

where

t_{S a m}

is the transmission coefficient, and

T_{S a m}

is the transmittance. The transmission coefficient of the reference has to be known. In our case, the sample thickness is 0.522 mm which is very thin, and the reference is air so that the transmission coefficient of the reference is close to 1.

In order to calculate the refractive index of the sample, the phase difference between the sample and the reference is required which is calculated as:

Δ Φ = Δ Φ_{S a m} - Δ Φ_{R e f} = (n_{S a m} - n_{R e f}) \frac{ω_{T H z} d}{c} = (n_{S a m} - 1) \frac{ω_{T H z} d}{c}

Thus, the refractive index of the sample is:

n_{S a m} = Δ Φ \frac{c}{ω_{T H z} d} + 1

The above method to acquire the refractive index is suitable for all the three methods. However, in Finding Extrema method, the process of signal phase become the manipulation of the order $m$ . We modify the equation in ideal condition given by the manual reference of TeraScan 780 [20], which is developed from the method in [12], and make it suitable for the condition of relatively low SNR. The optimized equation is:

[n_{S a m} (f) - 1] d = [\frac{f_{r e f, m_{r e f}}}{f_{s a m . m_{s a m}}} - 1] \cdot L_{r e f} - (m_{s a m} - m_{r e f} + n_{e r r o r}) \cdot \frac{c}{2 f_{s a m . m_{s a m}}}

where

f_{r e f, m_{r e f}}

and

f_{s a m . m_{s a m}}

are two closest frequency of reference at the order

m_{r e f}

and the sample at the order

m_{s a m}

, respectively;

n_{e r r o r}

is the error caused during the calculation of orders;

n_{S a m} (f)

represents for the refractive index of the sample while

f = f_{r e f, m_{r e f}} \approx f_{s a m . m_{s a m}}

;

L_{r e f}

is the average length at order

m_{r e f}

which is:

L_{r e f} = \frac{c}{2} \cdot \frac{N}{f_{r e f, +} - f_{r e f, -}}

where

N

is the number of periods;

f_{r e f, -}

and

f_{r e f, +}

is the frequency of

N / 2

points before and after the point at order

m_{r e f}

, respectively. To eliminate the

n_{e r r o r}

, a procedure like phase unwrapping is applied.

3.4 The Fabry–Pérot effect in the system

There are many interfaces in the THz-FDS system, such as the interfaces between the silicone lens and the air, the air and the sample, the fibers and the connectors and so on. These interfaces will cause oscillations in the amplitude and noise in phases due to the FP interference [21]. This kind of noise can be calculated by transfer-matrix method or simulated by commercial electromagnetic simulation software. However, it is difficult to acquire the accurate length of all the optical paths, especially inside the transmitter and the receiver.

It is common and convenient to reduce such a periodic oscillation by using low-pass filter. But the spectrum data are complex values which is unsuitable for normal low-pass filter designed for real signal. Because the refractive index depends on the frequency, the oscillation period is also variable. Therefore, it requires the filter with a frequency locking feature. In this work, for the Hilbert transform method, inverse fast Fourier transform (IFFT) is used to convert the frequency-domain data to calculated time-domain signals. As is always employed in THz-TDS which removes the oscillations by cutting the reflection signals, some parts of the calculated time-domain signal can be set zero. With this method, the oscillation in the amplitude and phase is partially reduced. This is a new method we introduced and it is useful for the data analysis in THz-FDS. However, the drawback of this method is that the resolution is decreased due to narrow time window of the time-domain signal. In the Finding extrema method and Fitting method, this smoothing analysis is not suitable because the frequency intervals are not equally distributed. Interpolation may make the setting zeros method applicable to the two aforementioned methods. The locally weighted scatterplot smoothing (LOWESS) in the commercial software Origin is used to reduce the oscillation in these two methods. The points of window in the LOWESS are 50 and 500, respectively.

4. The analysis results of Lactose Monohydrate

We first take the atmospheric water vapor (air) as a reference signal. Figure 4 shows the original data obtained in our measurement. For the method of Finding extrema, the resolution is the interval between two peaks ~0.210 GHz, while for the Hilbert transformation, the resolution is equal to the sampling step ~0.014 GHz.

Fig. 4 The original reference data of the atmospheric water vapor. The inset is the enlarged signal from 0.899 to 0.901 THz.

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We choose Lactose Monohydrate powder as the sample, because it has obvious absorption peaks at ~0.53 THz, ~1.20 THz, and ~1.37 THz. The powder is compressed to a cylindrical tablet with a diameter of 20 mm and the thickness of 0.522 mm. The pressure added to the tablet is ~26 MPa. The relative humidity in the measurement is 8.7 ± 0.9% and the temperature is 24.9 ± 0.5°C. Figure 5 shows the original data of the lactose monohydrate. From this figure, we can see that there are two distinct phonon absorption peaks. The inset plots the enlarged absorption resonance at 0.53 THz. Due to the huge absorption coefficient of the lactose, the resonance at 0.56 THz of water vapor molecule cannot be well resolved. The contribution from the water vapor can be eliminated when we conduct the data analysis. With the atmospheric water vapor, the absorption can also be well resolved which demonstrates that the THz-FDS can be used to characterize materials for real applications.

Fig. 5 The measured data of Lactose Monohydrate. The inset shows the enlarged absorption peak at 0.53 THz.

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Figure 6 shows the different envelopes of reference calculated by the three methods. The Hilbert transform method has the largest oscillations, the Finding extrema method takes the second place, and the Fitting method has the smallest. The maximum oscillation in Hilbert transform method may be caused by interpolation, while the Fitting method have a smoothing effect. Because we use data points ~1 period before and after the target point to fit which is like the adjacent averaging. From the Fig. 6(b), we can find that the Finding extrema method can’t acquire the small oscillations. Although the noise in Hilbert transform method is big, but the oscillations is obtained. Therefore, Hilbert transform method is more conducive to the analysis of the system situation and the origin of noise.

Fig. 6 (a) Envelope of the reference calculated by the three methods, and (b) their corresponding enlarged figure at ~0.48 THz.

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We first use the Finding extrema method to extract the transmission coefficient and the refractive index of the lactose monohydrate, as is shown in Fig. 7(a) and Fig. 7(b). From these figures, we can see that the three absorptions are located at 0.53 THz, 1.20 THz and 1.37 THz, respectively, which agrees very well with the reported results [12]. The average transmission coefficient curve equals to ~0.95. The average refractive index is ~1.7 and the resonance areas obey Lorentz oscillator model. The real and the imaginary parts of the calculated permittivity are shown in Fig. 7 (c) and Fig. 7 (d). The average real part of the permittivity is ~3.2 while the average value for the imaginary part is almost tending to zero.

Fig. 7 (a)-(d) Calculated transmittance coefficient, refractive index, real part and imaginary part of the permittivity of lactose monohydrate with different data analysis methods. The results from Finding extrema are offset for clarification.

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In Hilbert transform method, in order to eliminate the oscillation of the system noise, we convert the data from frequency domain to the time domain. The method that transforms between time domain and frequency domain is also used in Fourier Self-deconvolution (FSD) in the data analysis of Fourier Transform infrared spectroscopy (FTIR). FSD method is similar to our method, and some parts may also be applied to our system. However, the FSD is only suitable for the absorption spectrum, our method is used for original transmittance spectrum [22,23]. The time-domain waveforms of the reference and sample are illustrated in Fig. 8. We can see that there are ~12 oscillations in reference signal and two more in sample signal, which can be well resolved in the enlarged figures of Fig. 8(b). However, most of the oscillations are difficult to attributed, except them who may be from the interfaces between the silicon lens and the air located at ~7.9 ns. The time difference between this reflection signal to the main signal is ~5.6 ns and the calculated THz propagation length is ~1680mm, just twice of the THz propagation path ~840mm. The phase shift of the sample with respect to the reference can also be resolved in Fig. 8(c). The main signal can be distinguished with its intensity. When we calculate the transmission coefficient, refractive index, and the permittivity of the sample, we only use the main oscillation signal in Fig. 8(c) so that the system noise is easily restrained. The final results are shown in Fig. 7 with red color, respectively. All the resonances are obtained which agree very well with those from the Finding extrema method. Although we obtain almost the same results with these two methods, as shown in Fig. 7, the Hilbert transform benefits for later manipulations because its frequency interval is uniform. The Hilbert transform method has higher resolution and more information about the system. However, for the lactose monohydrate sample which has broad absorption lines, this advantage is not well performed. It will be very promising for other materials, high Q devices or gap-phase molecular with much narrower oscillation.

Fig. 8 (a) the calculated time-domain signal of reference and sample, (b) the enlarged figure form 1.5 ns to 9 ns, and (c) the enlarged main signal.

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The result of Fitting method is also shown in Fig. 7. The noise in the result is small compared with other two methods. The smoothed result agrees very well with those obtained from other two methods.

It is difficult to eliminate the interference caused by the two interfaces of the sample/air. The reflection signals are hidden in main signal, which cannot be cut off in the time domain. This problem also happens in THz-TDS which cannot distinguish the multiple reflections and transmissions in such thin sample. In Fig. 7, the transmission coefficient, refractive index and permittivity of the sample measured by THz-TDS are also exhibited in black dotted lines. Although the absorption features can be resolved, the frequency resolution is not good as that from the THz-FDS.

In both THz-TDS and THz-FDS data analysis, FP cavity transmission coefficient function is applied in extracting the sample permittivity. However, the interference from the thin sample is not well reduced. Consequently, there are still some oscillations which makes the sample signal located at 1.20 THz hard to be resolved, especially in the imaginary part of the permittivity in Fig. 7(d).

In comparison, as shown in Fig. 9, the frequency resolution of Finding extrema method is good enough to characterize the absorption features of the lactose monohydrate. However, for samples with an absorption peak width of ~200 MHz, the Finding extrema method would not be suitable to calculate the permittivity. Although the frequency resolution of Finding extrema can be scaled up to ~4 MHz, it requires not only decreasing the pump laser linewidth down to ~0.4 MHz but also increasing the optical path difference with variable delay stages or fiber length.

Fig. 9 Transmission spectrum for the frequency range of (a) 0.42-0.75 THz, and (b) 1.05-1.54 THz, and their corresponding refractive indexes (c-d) calculated by three methods after smoothing.

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As shown in Fig. 10, Setting zeros method has larger oscillations than the LOWESS, especially at high frequency. There are two possible reasons. First, the oscillations in the envelope obtained by the Hilbert transform is relatively large. Secondly, it may be due to its inherent features.

Fig. 10 (a) the transmission coefficients and (b) the refractive indexes smoothed by the Setting zeros method and LOWESS, respectively.

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5. Conclusion

In summary, we comparatively investigate three data analysis methods in coherent detection THz-FDS and find that Hilbert transform is the most appropriate method to extract the permittivity of the samples. Under the atmospheric water vapor, we successfully extract the permittivity of the lactose monohydrate. We introduce the time-domain data manipulation into the Hilbert transform so that the system noise due to FP effect in THz-FDS can be well restrained which is very helpful to material characterization. Our work is very helpful for signal processing in THz-FDS and promotes THz spectroscopy in real applications.

Funding

National Natural Science Foundation of China (Grant 61222110); High-Level Talent Introduction Project of Beihang University (Grant No. 29816248); Youth-Top-Talent Support Project of Beihang University (Grant No. KG12000801); “Zhuoyue” Program of Beihang University (Grant No. GZ216S1711); National Natural Science Foundation of China (Grant No. 61775233); National Basic Research Program of China (Grant No. 2014CB339800).

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	Fitting	Finding extrema	Hilbert transform
Resolution	Sampling step	Decided by path difference	Sampling step
Independent variable	Continuous	Continuous	Discrete
Calculation speed	Low	High	High
Gibbs phenomenon	Doesn’t have	Doesn’t have	Has

High resolution continuous wave terahertz spectroscopy on solid-state samples with coherent detection

Abstract

1. Introduction

2. Experimental setup

3. Data analysis of the THz-FDS

3.1 The coherent detection principle of the THz-FDS

3.2 Methods of extracting the amplitude and phase from the original measured data of the samples

3.3 Extracting the transmission coefficient and refractive index of the sample

3.4 The Fabry–Pérot effect in the system

4. The analysis results of Lactose Monohydrate

5. Conclusion

Funding

References and links

Cited By

Figures (10)

Tables (1)

Equations (14)

Optics Express