## Abstract

Phase measurement of continuous-wave terahertz (CW-THz) radiation is a potential tool for direct distance and imaging measurement of optically rough objects due to its high robustness to optical rough surfaces. However, the 2π phase ambiguity in the phase measurement of single-frequency CW-THz radiation limits the dynamic range of the measured distance to the order of the wavelength used. In this article, phase-slope measurement of tunable CW-THz radiation with a THz frequency comb was effectively used to extend the dynamic range up to 1.834 m while maintaining an error of a few tens µm in the distance measurement of an optically rough object. Furthermore, a combination of phase-slope measurement of tunable CW-THz radiation and phase measurement of single-frequency CW-THz radiation enhanced the distance error to a few µm within the dynamic range of 1.834 m without any influence from the 2π phase ambiguity. The proposed method will be a powerful tool for the construction and maintenance of large-scale structures covered with optically rough surfaces.

© 2014 Optical Society of America

## 1. Introduction

One of the unique characteristics of terahertz (THz) radiation (frequency = 0.1–10 THz; wavelength = 30-3000 µm) is the insensitivity to optically rough surfaces with sub-millimeter to micrometer roughness because its wavelength is larger than the surface roughness, and hence scattering of the THz radiation is largely reduced in comparison with optical waves. Therefore, distance and imaging measurement of optically rough objects is one of the promising applications of THz radiation. In particular, recent progress made in the construction of large-scale structures on scales of a few meters to several tens of meters, such as buildings, bridges, airplanes, ships, and parabolic antennae, has increased the demand for direct distance and/or imaging measurement of optically rough surfaces, such as metal, plastic, rubber, and painted materials.

THz distance and imaging measurement techniques are classified into the time-of-flight (TOF) method with pulsed THz radiation [1, 2], the interferometric method with single-frequency continuous-wave THz (CW-THz) radiation [3], and the frequency-modulated-continuous-wave (FMCW) method [4]. In the TOF method, use of a temporally localized sub-picosecond THz pulse allows distance measurement in a wide dynamic range, and moderate distance resolution can be achieved by a pump–probe measurement of a sub-picosecond THz pulse [1]. Furthermore, a combination of this method and the asynchronous optical sampling technique [5–7] makes it possible to achieve real-time ranging of moving objects located at unknown distances without the need for time coincidence between the THz and probe pulses [2]. However, the signal-to-noise ratio (SNR) of the detected signal is relatively low due to the low average power of the pulsed THz radiation. On the other hand, in the case of the interferometric method, phase measurement using interference fringes provides higher distance resolution than that of the TOF method [3]. Furthermore, use of relatively strong CW-THz radiation contributes to a high SNR in the measurement. However, use of a moving arm for interference hinders the real-time phase measurement of moving objects. Also, precise spatial overlapping of two invisible THz beams for interference is not an easy task in real applications. Most importantly, the dynamic range of the distance measurement is limited to less than half the wavelength (*λ/2)* of the radiation because the phase measurement of single-frequency CW-THz radiation causes a 2π phase ambiguity, with the result that the practical applications of this technique are severely limited. In fiber-based swept-source THz radar, the limitations of such a 2π phase ambiguity have been reduced, and the difficulty of spatially overlapping two invisible THz beams has been alleviated [8]. However, the distance resolution is limited to 10 mm or greater due to the source bandwidth and the frequency dependent fiber attenuation. Although the sweep capable CW-THz system has been used for THz spectroscopy [9], the phase sensitive detection with this system enables THz distance measurement. For example, the maximum measurable distance can be largely enhanced by measuring the phase difference between two adjacent frequency points. However, it is difficult to achieve the distance resolution comparable to that achieved by the phase measurement of a single-frequency CW-THz radiation. The FMCW method has been used in the millimeter-wave region for automobile radar and recently has been transferred to THz frequencies [4]. In this method, the target distance can be determined in real time by measuring the beat frequency between the transmitted FMCW signal and the returned one. Although this full-electronic approach is easier to operate and enables the longer distance measurement, the achievable distance resolution is limited to a few mm due to the range of the frequency sweep in the FMCW source.

If the dynamic range of THz distance measurement could be extended to greater than *λ/2* while maintaining the same distance resolution as the phase measurement in CW-THz interferometry, the range of applications would be considerably widened. Recently, THz frequency combs [10] have attracted attention for CW-THz frequency measurement [11–14], CW-THz signal generation [15–17], and broadband spectroscopy [18–20] due to their high precision, broadband spectrum, and real-time capability. Furthermore, electro-optical heterodyne detection enables the real-time phase measurement of single-frequency CW-THz radiation at 600 GHz [21, 22] and 2.5 THz [23] without interference and hence a moving arm. Using this technique, a distance resolution on the order of sub-micrometer to micrometer was achieved in THz imaging of optically rough objects. However, it is still not possible to determine the absolute distance beyond its inherent ambiguity range of *λ/2*. If multiple phase measurements are repeated at different frequencies by using tunable CW-THz radiation, and the resulting phase values are unwrapped with respect to frequency as a phase slope, the measurable range will be greatly increased in the same manner as the wavelength scanning interferometry [24]. Furthermore, if the phase-slope measurement for ranging at distances greater than *λ/2* could be combined with phase measurement for ranging at distances below *λ/2* without the 2π phase ambiguity, the distance resolution of the phase measurement and the distance dynamic range of the phase-slope measurement would be achieved at the same time. In the work described in this article, we combined phase-slope measurement with tunable CW-THz radiation and phase measurement with single-frequency CW-THz radiation. To perform real-time phase measurement of the tunable CW-THz radiation without the need for an interference configuration, we used photoconductive heterodyne mixing between the CW-THz radiation and a photocarrier THz (PC-THz) comb in a photoconductive antenna (PCA) detector [11, 12]. Furthermore, we applied it to the distance measurement of an optically rough object with a large dynamic range of distance.

## 2. Principle

#### 2.1 Phase measurement at multiple frequencies

Consider the case where femtosecond mode-locked laser light (mode-locked frequency =
*f _{rep}*) is incident on a gap in a PCA. Figure 1 illustrates the spectral behaviors in the THz and radio-frequency
(RF) regions. The mode-locked pulse train emitted from the femtosecond laser constructs an
optical frequency comb in the frequency domain, whose mode spacing is equal to

*f*. When the PCA is illuminated by such a pulse train, a PC-THz comb is induced in the PCA due to demodulation of the optical comb in the PCA [10, 11]. Next, consider what happens when measured CW-THz radiation (frequency =

_{rep}*ν*) is incident on the PCA detector. The photoconductive mixing process in the PCA generates a group of beat signals between the CW-THz wave and the PC-THz comb in the RF region. Consider a beat signal at the lowest frequency ( =

_{THz}*f*), namely, the

_{b}*f*beat signal. Since this

_{b}*f*beat signal is generated by mixing the CW-THz wave (frequency =

_{b}*ν*) with the m-th mode of the PC-THz comb (frequency =

_{THz}*mf*) nearest in frequency to the CW-THz wave,

_{rep}*f*is given by

_{b}*ν*has been determined precisely from Eq. (1) by measuring

_{THz}*f*and

_{rep}*f*in RF region and calculating

_{b}*m*from them [11, 12]. On the other hand, if

*ν*and

_{THz}*f*are sufficiently stable and

_{rep}*f*is kept constant, precise phase measurement of CW-THz radiation will be possible in the RF region by using a phase meter, such as a lock-in amplifier.

_{b}When we extend this method to multiple phase measurements at different frequencies of the tunable CW-THz radiation, we have to pay attention to the *f _{b}* value. Although

*f*can be any number from 0 through

_{b}*f*/2 depending on

_{rep}*ν*, the phase measurement involves an error at

_{THz}*f*= 0 and

_{b}*f*=

_{b}*f*/2 due to zero frequency or overlapping of

_{rep}*f*and

_{b}*f*-

_{rep}*f*. This problem is avoidable if

_{b}*f*is always constant regardless of

_{b}*ν*To this end,

_{THz.}*ν*of the tunable CW-THz radiation is tuned stepwise at intervals of

_{THz}*f*. Since the resulting

_{rep}*f*value is always constant within the operating frequency range in the lock-in amplifier, the phase values can be measured correctly without the influence of

_{b}*f*= 0 or

_{b}*f*/2.

_{rep}#### 2.2 Distance measurement based on phase slope

A target distance *D* is calculated from a wavelength *λ _{THz}* (corresponding frequency =

*ν*) and total phase

_{THz}*ϕ*for single-frequency CW-THz radiation as follows

_{THz}*N*and

_{THz}*ε*are respectively the integer part and excess fraction in the total phase

_{THz}*ϕ*, and

_{THz}*c*is the speed of light. Since a phase measurement at a single frequency can only provide

*ε*in Eq. (2) due to the 2π phase ambiguity, it is difficult to determine the absolute distance at distances greater than

_{THz}*λ/2*. To extend the distance dynamic range to greater than

*λ/2*,

*N*has to be obtained by using another method. To obtain

_{THz}*N*while avoiding the 2π phase ambiguity,

_{THz}*D*has to be determined at a precision better than

*λ*, or

_{THz}/4*c/(4ν*.

_{THz})To this end, we consider the total phases *ϕ _{THzi}* at multiple frequencies

*ν*for a fixed distance

_{THzi}*D*as follows

*ν*and their corresponding phases

_{THzi}*ϕ*should be linear. Moreover, the slope between them, called the phase slope

_{THzi}*Δϕ*, should be proportional to the distance

_{THz}/Δν_{THz}*D*. Therefore, if the frequency of the CW-THz radiation can be tuned by

*Δν*, the phase will be changed by

_{THz}*Δϕ*depending on the distance

_{THz}*D*as follows:

*N*and

_{THz}*ε*are obtained from Eq. (4) as follows

_{THz}*INT (•)*is the integer number nearest to • and

*MOD (•)*is the modulo of

*•*. The phase-slope method determines

*N*+

_{THz}*ε*from Eqs. (5) to (7), corresponding to the total phase

_{THz}*ϕ*in Eq. (2), whereas the single-frequency phase method provides

_{THz}*ε*, corresponding to the wrapped phase within the range of 2π. When the total phase

_{THz}*ϕ*is much larger than the wrapped phase and the phase errors in both methods are comparable to each other, the distance precision of the phase-slope method will be worse than that of the single-frequency phase method. However, if the phase-slope measurement can determine

_{THz}*D*at a precision better than

*λ*, or

_{THz}/4*c/(4ν*, the integer part

_{THz})*N*will be determined from Eq. (6) without any influence from the 2π phase ambiguity. Therefore, if the phase-slope measurement of tunable CW-THz radiation is performed for determining

_{THz}*N*and then the phase measurement of single-frequency CW-THz radiation is performed for determining

_{THz}*ε*, a high distance resolution and a wide dynamic range will be achieved at the same time.

_{THz}## 3. Experimental setup

Figure 2 shows the experimental setup. Laser light
from a Kerr-lens mode-locked Ti:sapphire laser (FEMTOLASERS Femtosource Scientific Pro, with
pulse duration = 10 fs and central wavelength = 780 nm), whose mode-locked frequency
*f _{rep}* was stabilized at 81.8 MHz by laser control [11], was focused into the antenna gap of two bowtie-shaped,
low-temperature-grown GaAs PCAs (PCA1 and PCA2), respectively. This resulted in the generation
of PC-THz combs in PCA1 and PCA2, which were composed of a fundamental component and a series of
harmonic components of

*f*[11]. This PC-THz comb acted as local oscillators having multiple frequencies to beat down the CW-THz radiation to the RF region.

_{rep}Tunable CW-THz radiation was generated by an active frequency multiplier chain (Millitech AMC-10-R0000, with multiplication factor = 6), referred to as an AFMC source. The AFMC source multiplied the output frequency of a microwave frequency synthesizer (Agilent E8257D, with frequency = 12.5–18.33 GHz, and linewidth < 0.1 Hz) by six and then radiated the resulting electromagnetic wave into free space via a horn antenna. Since the frequency synthesizer was synchronized to a rubidium frequency standard (Stanford Research Systems FS725, with frequency = 10 MHz, accuracy = 5 × 10^{−11}, and stability = 2 × 10^{−11}), the AFMC source acted as an accurate, stable, tunable, narrow-linewidth, CW-THz source (tuning range = 75–110 GHz, linewidth < 0.6 Hz, average output power = 2.5 mW). Part of the collimated THz beam was reflected by a first Si beam-splitter (BS) and fed into PCA1 as a reference. The THz beam passing through the first and second Si BSs was reflected at a target and then redirected into PCA2 for measurement. The target was placed at the point of zero path difference between the reference beam and the measurement one. We defined this point as a target position of 0 mm.

The *f _{b}* beat signals in PCA1 and PCA2 (frequency = 50 kHz) were used as a reference signal and a measured signal for the phase measurement with an audio-frequency (AF) lock-in amplifier (NF Corporation LI 5640, with a frequency range of 100 kHz, time constant = 100 ms). To keep

*f*constant during frequency tuning of the CW-THz radiation, the CW-THz radiation was tuned stepwise at a frequency interval of

_{b}*f*( = 81.8 MHz) within a frequency range from 75 GHz to 110 GHz.

_{rep}## 4. Results

#### 4.1 Phase-slope measurement of tunable CW-THz radiation

We first investigated the change in the phase slope with respect to the target position
when a target was placed at different positions using a stepping-motor-driven translation stage
(positioning precision < 2 µm). To remove the influence of the surface roughness in
the distance measurement, we used an aluminum mirror as the target. We confirmed beforehand
that there were little difference of the detected signal intensity between the aluminum mirror
and an optical rough surface such as a metal plate, rubber sheet, polyethylene plate, and so on
(not shown), indicating that the signal loss on the optical rough surface is negligible. Figure 3(a) shows the relations between frequency and phase
with respect to target positions of 15, 30, 45, 60, and 75 mm, respectively. The acquisition
time of a single phase-slope was 41.9 sec, which was limited by the time constant of the
lock-in amplifier ( = 100 ms) and the number of data plots included in the phase slope ( =
419). Here, a phase unwrapping process was applied to each data set, resulting in linear
relationships between them. A linear approximation of the experimental data to the straight
line *y = αx* data was performed in order to obtain the phase slope of
each target position. The residual error of the calculated phase slope to the measured phase
values was within ± 10.8 degree within a frequency range from 75 GHz to 110 GHz.
Finally, we determined the target distance, *D*, at five target positions from
the phase slope using Eq. (4).

Next, we evaluated the error of the distance measurement with the phase-slope method by comparing the determined distance with the target positions. Figure 3(b) shows the relation between the target position and the target distance *D* measured with the phase-slope method. A good linear relationship was confirmed between the target position and *D*. The discrepancy between them at each target position is shown in Fig. 3(c). Since the positioning precision of the target (< 2 µm) was much smaller than these discrepancies, the data in Fig. 3(c) reflects the precision of the distance measurement in the phase-slope method. The mean and standard deviation of the distance error were −16 µm and ± 24 µm, respectively. We consider that the negative errors are caused by insufficient parallelism between the optical path of the THz beam and the moving direction of the translation stage.

#### 4.2 Connection of phase-slope measurement of tunable CW-THz radiation to phase measurement of single-frequency CW-THz radiation

As demonstrated in the previous section, the target distance could be determined with an error of a few tens µm using the phase-slope method. Since this precision is much smaller than *λ _{THz}/4* of CW-THz radiation ( = 750 µm for 100-GHz radiation, for example), this method enables us to determine

*N*in Eq. (2) without any influence from the 2π phase ambiguity. Therefore, if we determine the excess fraction

_{THz}*ε*in Eq. (2) by extracting the phase value at a certain frequency from the result of the phase-slope measurement, the target distance will be determined with a precision of

_{THz}*λ/100*to

*λ/1000*while maintaining the wide dynamic range of the distance measurement achieved by the phase-slope method.

We first evaluated the distance resolution for the phase measurement of single-frequency
CW-THz radiation. To this end, we repeated the phase measurement for a target fixed at a
certain position at five different *ν _{THz}*, namely, of 78.002,
82.995, 87.988, 92.980, and 102.97 GHz. Figure 4 shows
the result of phase fluctuation, defined as the standard deviation of five repeated
measurements. For five different frequencies, the phase fluctuated within a range of 1 degree.
This phase fluctuation is much larger than the phase stability ( = 0.01 degree/°C) of
the lock-in amplifier used here. The distance resolution based on the phase measurement of the
single-frequency CW-THz radiation is determined by this phase fluctuation. The right axis in
Fig. 4 shows the distance resolution at five different

*ν*, calculated from the corresponding

_{THz}*λ*and the phase fluctuation. This result clearly indicated that the distance resolution was less than 10 µm.

_{THz}Next, we connected the phase-slope measurement of the tunable CW-THz radiation to phase
measurement of single-frequency CW-THz radiation. We first obtained
*N _{THz}* at 101.25 GHz (corresponding wavelength = 2.9629 mm) from
Eq. (6). Then, we extracted

*ε*at 101.25 GHz from the phase measurement. Finally, we determined the target distance,

_{THz}*D*, by substituting

_{fin}*ν*,

_{THz}*N*, and

_{THz}*ε*into Eq. (2). Table 1 summarizes the results of

_{THz}*N*,

_{THz}*ε*,

_{THz}*D*, and discrepancy between target position and

_{fin}*D*at target positions of 15, 30, and 45 mm. Comparison of the results between Fig. 3(c) and Table 1 clearly indicated that the combination of the phase-slope measurement with the single-frequency phase measurement largely reduced the distance error in comparison with the phase-slope measurement alone. The mean and standard deviation of the distance error in the combined method were −0.67 ± 2.9 µm.

_{fin}#### 4.3 Profilometry of optically rough surface including a large step

To demonstrate the advantage of the wide dynamic range of the distance measurement in the
proposed method, we performed line imaging of optically rough surfaces including a large step.
The sample used here was a two-stepped, aluminum block with a step height of 8.37 mm, as shown
in Fig. 5(a), and was placed at unknown distance. For
line imaging, a THz beam for measurement was focused onto a sample with an off-axis parabolic
mirror (focal length = 50 mm), reflected at the sample surface, and redirected onto PCA2. Then,
the sample was moved at intervals of 2 mm to scan the measurement position across the two-step
surface while measuring the target distance based on the combined method described above. Figure 5(b) shows the result of line imaging of the sample,
correctly revealing the shape profile of the two-step surface. *D _{fin}*
of the first surface was determined to be 306.463 mm ± 102 µm [(mean) ±
(standard deviation)] within the sample position range from 0 to 16 mm. On the other hand,

*D*of the second surface was determined to be 298.026 mm ± 7 µm within the sample position range from 22 to 48 mm. The difference of the absolute distances between the two surfaces was 8.437 mm, which is in god agreement with the actual depth of 8.37 mm measured by a vernier caliper. Here, it is important to note that the surface profile of several to a few hundreds micrometers, the step profile of several millimeters, and the absolute distance of several hundred millimeters for the optically rough object were determined at the same time by the combined method. This demonstration clearly indicates the advantage of the wide dynamic range of the combined method over other conventional methods. The

_{fin}*D*profile in Fig. 5(b) indicated the temporary increase near the sample positions of 16 mm and then the decreased slope at the sample positions from 16 to 22 mm, which were different from the real shape of the sample in Fig. 5(a). We consider that a mixture of two THz beams returning from different surfaces causes the error in the phase measurement because the focused THz beam was irradiated onto both surfaces due to the beam diameter of several millimeters. Use of another higher-frequency CW-THz source, such as a tunable THz quantum cascade laser [25, 26], will improve the spatial resolution of THz imaging as well as the distance resolution.

_{fin}## 5. Discussions

As described in the previous section, we correctly determined the target distance located at a distance of several hundred millimeters. We here discuss the maximum measurable distance in the phase-slope method. In this method, it is difficult to perform the phase unwrapping calculation correctly between two adjacent plots in the phase slope when one increment of the frequency tuning induces a phase change greater than 2π. In the present demonstration, since stepwise tuning of the CW-THz frequency was repeated at an interval of *f _{rep}* ( = 81.8 MHz),

*D*is obtained from Eq. (4) as follows

_{max}*L*is the cavity length of our Ti:sapphire laser. It is important to note that this

*D*value greatly exceeds the dynamic range of the distance measurements based on the phase measurement of single-frequency CW-THz radiation alone ( = 3 mm in case of 100-GHz CW-THz radiation). Also, the

_{max}*D*value in Eq. (8) is significantly consistent with the dynamic range of distance in THz TOF method [1].

_{max}The frequency tuning step of 81.8 MHz was selected to keep *f _{b}* constant within the operating frequency range of our AF lock-in amplifier ( = 100 kHz). However, if a RF lock-in amplifier, operating at frequencies from 0 through

*f*/2, is effectively used while avoiding the situation where

_{rep}*f*= 0 and

_{b}*f*=

_{b}*f*/2, the frequency tuning step can further decrease. For example, if the frequency tuning step can be decreased to 1 MHz,

_{rep}*D*will achieve over 100 m.

_{max}On the other hand, the minimum measurable distance *D _{min}* is limited by the phase noise in the phase-slope measurement and the phase measurement. The residual phase error of ± 10.8 degree limits the

*D*in the phase-slope measurement to a few tens µm as shown in Fig. 3(c). This

_{min}*D*value is sufficient to transfer the distance determined by the phase-slope measurement to the phase measurement. As a result, the

_{min}*D*can be reduced to the distance resolution in the phase measurement (less than 10 µm, see Fig. 4) rather than the phase-slope measurement. Use of higher-frequency CW-THz source will further improve the

_{min}*D*to the order of sub-micrometer [22].

_{min}Finally, we discuss the difference of the phase error between the phase-slope measurement and the phase measurement. The residual phase error in the phase-slope measurement ( = ± 10.8 degree) was 10-times larger than the error in the phase measurement of a single-frequency CW-THz radiation (< 1 degree, see Fig. 4). In the experiments demonstrated above, internal reflections of CW-THz radiation in the optical setup cause unnecessary interference at specific THz frequencies because the narrow-linewidth CW-THz radiation has the considerably long coherence length. This interference effect disturbed the linearity of the phase slope, resulting in increase of the residual phase error in the phase-slope measurement. In the case of the phase measurement, we selected five different frequencies ( = 78.002, 82.995, 87.988, 92.980, and 102.97 GHz) to avoid unnecessary interference caused by internal reflections in the optical setup.

## 6. Conclusions

To simultaneously achieve high resolution and wide dynamic range in THz distance measurement, we proposed a combination of phase-slope measurement of tunable CW-THz radiation and phase measurement of single-frequency CW-THz radiation. Photoconductive mixing of CW-THz radiation with a THz comb effectively transferred the phase information in the THz region to the RF region without loss of precision; this leads to high-precision measurement of phase-slope and phase using an AF lock-in amplifier. The phase-slope measurement enables us to not only achieve a distance error of −16 ± 24 µm but also enhance the distance dynamic range to 1.834 m. The achieved dynamic range is three orders of magnitude larger than that in the phase measurement of single-frequency CW-THz radiation [21–23]. This dynamic range will be further enhanced if the frequency-tuning step of the CW-THz radiation is reduced. Moreover, the combination of this method with the phase measurement of the single-frequency CW-THz radiation makes it possible to improve the distance error to −0.67 ± 2.9 µm while maintaining the distance dynamic range of 1.834 m. The proposed method will be a powerful tool for the construction and maintenance of large-scale structures with optically rough surfaces.

## Acknowledgment

This work was supported by Collaborative Research Based on Industrial Demand from Japan Science and Technology Agency, Grants-in-Aid for Scientific Research Nos. 23656265 and 26246031 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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