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In this study, inspired by the frequency-modulated continuous-wave (FMCW) method, an operation scheme of continuous-wave (CW) terahertz (THz) homodyne system is proposed and evaluated. For this purpose, we utilized the fast and stable wavelength tuning characteristics of a dual-mode laser (DML) as a beating source. Using the frequency-modulated THz waves generated by DML, a cost-effective and robust operation of CW THz system to be applicable to the measurements of thickness or refractive index of a sample is demonstrated. We believe that the proposed scheme shows a potential to the implementations of compact and fast CW THz measurement systems that can be useful in many THz applications.
© 2015 Optical Society of America
1. Introduction
The wavelength range lies between microwaves and infrared waves, so-called terahertz (THz) regime, has gained increasing interests for its potential in a wide variety of applications such as spectroscopy, imaging, and sensing [1–5]. Although recent achievements in the THz time-domain spectroscopy (THz-TDS) has demonstrated various possibilities in many applications such as spectroscopy, security imaging, and non-contacting diagnostics, the limitations such as the system cost, bulky size, and time-consuming measurement schemes still obstruct its widespread industrial utilizations [6,7]. As a consequence, an ever-increasing interest to the continuous-wave (CW) THz systems has been existed. For the CW THz systems, photomixing technique that are using the semiconductor lasers as a beating source has been widely-adapted as a potential candidate to the implementations of cost-effective and compact THz systems [6,7]. Combined with the homodyne-detection method, several implementations of compact CW THz spectrometers that can measure the complex refractive indices of samples over the broadband ranges have been reported [8–12]. In a contrast to the pulse-based TDS system, there can be 2π phase ambiguity in conventional implementation of CW THz homodyne systems. To remove such phase ambiguity, multi-frequency based approaches have been proposed [13,14]. These methods can be roughly summarized as the methods using multiple frequencies to utilize TDS-like quasi-pulse from CW fields. However, such methods require scanning of mechanical delay line which inevitably increases the size of system and the measurement time. To eliminate the mechanical delay line, alternative approaches based on a frequency-scanning method has been reported [15]. The frequency scanning method has proven a possibility of fast measurement with more compact system without scanning mechanical delay [15].
Recently, modulations of frequency [16,17] or phase [18,19] of beating light sources with externally added frequency tuning devices or phase modulators have been introduced. In all these methods, although the phase information conveyed onto the THz waves made it possible to eliminate the mechanical delay lines and verified the potential of frequency modulations in THz wave domain. However, except for the fact that these approaches require external modulators, the using of two separated sources to generate THz beating waves may have drawback that makes the system size to be too bulky [16,17] or requires additional optical fiber links and couplers that may increase the system complexity and may requires higher care to maintaining the stability of the system [18,19]. As far as the authors’ knowledge, there have been no implemented reports on the applications of the technique that modulates CW THz waves in an integrated beating source.
As for the tunable THz beating source, recently-developed dual-mode lasers (DMLs) have several advantages such as their compactness, polarization stability, and ease in wavelength tuning while preserving the coherence [20–22]. In its composition, a DML consists of two sections of distributed feedback laser diodes (DFB LDs) that share one optical cavity in common but have an interleaved phase section that suppresses the compound cavity mode [20–22]. For the thermal tuning of the lasing wavelengths to tune the THz beating frequency, overlaid μ-heaters are integrated on each DFB section as in Fig. 1. Since the μ-heaters are integrated very close to the active region of the DML laser, the required heat capacity to tune the wavelength can be reduced to accomplish a fast heat responses with a suppressed thermal crosstalk between laser sections [15,20–22].
Fig. 1 (a) Schematic diagram of 1.3μm DML with its constitutional parts; two λ/4 phase-shifted DFB laser sections, a phase section, and two integrated μ-heaters. (b) Optical spectrum of the DML used in this study when the μ-heaters are not biased.
The fast and stable tuning feature of DMLs can be further utilized than the simple frequency-scanning method as in [15]. By tuning the THz beating frequency of DML, the frequency modulation (FM) in the THz waves can be readily accessed. In general, using modulated signals can provide fluent and advantageous ways to many applications via easy-extractions of information with less sensitive to the noise. By adopting proper modulation techniques for CW THz system, plenty of favorable usages of THz waves can be found. However, excluding the field of THz communications [23], there have few literatures on the application of modulated signals in THz regime.
In this study, we propose a cost-effective and compact real-time CW THz system based on a FM technique. As a proof-of-concept for the CW THz system that utilizes modulated THz waves, we adopted a well-known and widely used scheme in the field of ladar systems: optical frequency modulation continuous-wave (FMCW) method [24]. From the experimental results, the proposed scheme shows its potential in realizing a low-cost and real-time THz system for the measurements of thickness or the refractive index. We believe that the proposed system can provide a preferable route to the portable CW THz measurement system.
2. Experimental setup and theoretical analysis
2.1. Experimental setup
A schematic configuration of FM-based coherent CW THz homodyne system for transmission type thickness measurement setup is shown in Fig. 2. In principle, the basic configuration is almost similar to the conventional coherent CW THz homodyne systems that are based on DML [22,23]: The optical beating signal from DML is amplified by external semiconductor optical amplifier (SOA) that is connected via a single mode optical fiber. After that, the beam is split by 50:50 fiber coupler to be provided to the transmitter (Tx) and receiver (Rx) modules, respectively. Each module consists of a log-spiral antenna integrated InGaAs photomixer and a hemispherical silicon (Si) lens. For the efficient homodyne detection, the polarizations of beating signals that are provided to Tx and Rx should be aligned to be parallel. For this objective, fiber-type polarization controllers (PCs) are inserted in each line.
Fig. 2 Schematic diagram of the proposed FM-based CW THz homodyne system. A sample of which thickness to be measured is inserted between Tx and Rx modules. Instead of using a mechanical delay line, the beating frequency from DML is repeatedly modulated by the bias pattern fed by a function generator.
The distinctive feature in this configuration to the conventional CW THz homodyne configurations is that the beating frequency of DML is modulated by the application of external waveform to the μ-heater bias. With this source modulation, the usually adopted mechanical delay line can be removed from the system configuration. For the THz detection, electrical chopping to the Tx via applying sinusoidal bias to antenna is used. Synchronized to this chopping signal, the Rx current is probed by lock-in amplifier (LIA) as in Fig. 2.
In principle, one or both of the μ-heaters in DML can be modulated by the periodic function(s) of voltage or current. In this study, for the simplicity, we have modulated the driving voltage for only one of the μ-heaters that is closer to the front facet of DML (heater 2). However, modulating both of the μ-heaters in either way of independent or correlated modes can also be possible for more complex modulation schemes.
In their basic operations, the μ-heaters of DML can be regarded as the electrical resistances of which typical current-voltage relation is linear [22]. Therefore, the frequency shift of a lasing beam from a laser section in DML shows a linear dependence to the resistive heating. Since the amount of resistive heating is in a quadratic dependence to the applied voltage, the beating frequency shows a quadratic relation to the heater voltage as shown in Fig. 3(a). Therefore, to obtain a linearized beating frequency profile, we have to consider an inverse relation for the applied voltage as in Fig. 3(b). Figure 3(b) shows the time-dependent waveform of saw-tooth modulation. In this waveform, to relieve the possible thermal fluctuation that may be caused by the abrupt changes in driving voltage, a short relief times with flat zero-bias voltage between pulses are introduced as in Fig. 3(b).
Fig. 3 A linearization of frequency sweeping profile for the driving voltage of μ-heater. (a) The dependence of the beating frequency of DML to the applied voltage. (b) Time-dependent voltage output from a function generator for 100GHz saw-tooth waveform modulation.
As a matter of fact, the measured beating frequencies in Fig. 3(a) are obtained under the static operating conditions of DML. Therefore, the actual dynamic beating frequencies during the modulation of DML may have different values from their static values. Since the tuning of beating frequency is based on a thermal method, such dynamic feature can be varied by the operation conditions such as the modulation frequency, tuning range, and the shape of waveform. Resolving such discrepancy between the static values and the dynamic values are not an easy task both in theoretical nor in experimental aspects that will require some sort of feedback loop (for example, refer to [24, 25]). Although such linearization will enhance the performance of the system, we will neglect such factor in this proof-of-conceptual study and will use rather simple form as described in Fig. 3.
2.2. Basic theory of FM-based CW THz homodyne system
Let us consider the basic principle of FM-based CW THz system. On its basic principle, our proposed method has many common features with optical FMCW (OFMCW) [24]. As a matter of fact, it can be roughly regarded as an application of OFMCW to the THz homodyne system. However, there are some differences in practical aspect of implementation: At first, instead of using a single light source as in the conventional OFMCW, we will use two coherent light beams that are beating. Although there is an approach of using multiple incoherent light sources in OFMCW [26], the use of two coherent light waves for the purpose of using their beating nature in our case is essential for the photomixing-based CW THz systems. In addition, in the conventional OFMCW, the phase difference that contains the distance information is obtained from high-speed photo detector (PD) by measuring the intensity of the interference between reference (source) and reflected waves. In our implementation, the Rx photomixer module in Fig. 2 takes the role of high-speed PD in OFMCW. In the Rx photomixer, the photocurrent that are generated in the Rx photomixer by the optically fed beating signal is gated by the voltage of received THz field is measured [6,11]. However, interestingly enough, the simplified mathematical model of our FM-based CW THz homodyne system can be summarized to almost the same form with that of conventional OFMCW. This will be shown in the followings.
The time-dependent light field E(t) emitted from a DML can be expressed by
where Ei (t) and ϕi (t) with i = 1, 2 are the vibration amplitude and the phase of two optical waves from DML, respectively. Although the field amplitude itself is a function of time under the condition of FM operation of laser diodes (LDs) that acts as a source of intensity noise [27], we will neglect such time-dependency and further assume that the amplitudes for the two light beams are the same as E1 (t) = E2 (t) = 2E0, for the analytical simplicity. Under a FM operation, the time-dependent phase ϕi (t) can be obtained from the integration of instantaneous angular frequency ωi (t) aswhere ϕi0 is constant initial phase.Let us consider we modulate one of the μ-heaters (heater 2) with a saw-tooth waveform as in Fig. 4. For the simplicity, let us assume that the other μ-heater (heater 1) is unmodulated to keep a constant frequency, ω1 (t) = ω1. The periodically modulated frequency ω2 (t) can be expressed as
where t’ is the shifted time that is defined as t’ = t – mT when T and m are the modulation period and a largest integer that makes mT to be a number less than or equal to t, respectively. By defining the faction of the flat relief time to the period T as s (0 ≤ s < 1), the offset frequency ω20 is defined to as the DC frequency during the relief time sT. In Eq. (3), and the angular frequency modulation rate α can be expressed with the angular frequency modulation depth Δω2 asUnder these assumptions, Eq. (1) can be expressed byAfter being separated by the optical splitter, the received fields at Rx after experiencing the path-length-dependent delay τη in each optical path respectively can be expressed aswhere the subscript η = S or R denotes the signal and reference paths, respectively.Fig. 4 Relations between the frequency profiles of the modulated waves passed through the two paths in Fig. 2. The cases are (a) τ0 < 0 and s < |τ0 |/T, (b) τ0 < 0 and s ≥ |τ0 |/T, (c) τ0 ≥ 0 and s ≥ τ0 /T, and (d) τ0 ≥ 0 and s < τ0 /T. The shaded regions stand for the time interval that corresponds to the signal pulse from the previous (p = −1, (a) and (b)), or from the next (p = 1, (c) and (d)) period. For all other time between (−1, 1) and outside of the shaded region, p = 0.
When a sample is placed between Tx and Rx as in Fig. 2, the additional delay introduced by the sample will change the value of signal delay τS. Since we are interested in the relative path difference between the two paths, Eq. (7) can be further simplified by assuming that τR = 0 and τS = τ0, without loss of generality. Depending on the sign and magnitude of τ0, the relative relation between the waves through the two paths can be categorized into four cases as in Fig. 4.
At the Tx photomixer under a DC bias, the ideal THz output power radiated from an antenna-integrated photomixer is proportional to the square of the incident optical power [3]. Therefore, the radiated THz field ETx from it will be proportional to the optical power and can be expressed from Eq. (7) as
where the integer p = −1, 0, or 1 denotes that the signal wave measured at a time t corresponds to the next, the same, or the previous period of the reference wave, respectively. To properly denote the corresponding angular modulation rate, a subscript p is used for αp, which value at a time t can be 0 or α as shown in Fig. 4. In Eq. (8), the AC components will only contribute to the THz radiation via the antenna coupling. Therefore, the actual field arriving at the Rx photomixer can be expressed aswhere ω0 and ϕ0 are the offset THz beating frequency and initial phase, respectively.At the Rx photomixer, the generated photocurrent will also be proportional to the incident power of the optical field. However, in this case, the THz field, in Eq. (9), that is coupled by the antenna will provide the bias voltage [3]. Therefore, at the Rx module, the measured photocurrent can be approximately expressed as
With some trigonometric identities, we can expand Eq. (10) aswhere τp = τ0 – pT, and ωα = ω0 + αsT.The time-dependent instantaneous frequencies in Eq. (11) can be obtained by differentiating the phase components in each term. For the first and second terms, the angular frequencies are approximately in the order of ω0 and 2ω0, respectively. Since the magnitude of ω0 is about few hundreds of GHz, such high frequency components are not efficiently measured by typical electronic devices. Therefore, they can only give an average intensity. Consequently, the measured photocurrent at the Rx module can approximately be expressed with the oscillating third term as
where M denotes the ratio of the oscillating term to the DC term.From Eq. (10), we can define the beating frequency in the measured current ωb as
In Eq. (13), during the time αp = 0, ωb will be either 0 or a value in the order of Δω2, which values again are not efficiently measured by electronic devices of moderate speed. Therefore, only during the time αp = α, the beating frequency gives a time delay-dependent response. In addition, during this period of αp = α, the phase term in Eq. (12) can give additional information that depends to the time delay τp. Usually, the quadratic term of αpτp2/2 is too small to be measured and is regarded to as a noise term [24]. Therefore, from the remained phase term of ωατp, a very small change of the phase variation can be extracted provided if we can chose a proper value of Δω2 that makes the total phase change is within the 2π limit, to avoid the phase ambiguity. In addition, when p ≠ 0, the beating frequency will be in the order of Δω2, of which value is too large to be properly measured. As a consequence, we will assume that p = 0 and hence τp = τ0 hereafter from here.Now, let us consider a sample of which refractive index and thickness are n and d, respectively is placed in the signal path in Fig. 2. In this case, the shifted time delay τs can be expressed as
If we denote the ωb with the sample as ωbs, and that for the case without the sample as ωb0, the difference Δωb = ωbs - ωb0, can be obtained from Eqs. (13) and (14) asSimilarly, the difference in the beating phase Δϕb can be expressed as3. Result and discussions
There are several limitations in a direct adaptation of the optical FMCW methods to the CW THz systems: the low optical-to-THz conversion efficiency and the non-uniform frequency response of the conventional photomixers. Although the affordable light power from our DML is over 18dBm after passing a semiconductor optical amplifier (SOA), the low conversion efficiency of the photomixers used in the experiments makes it inevitable to use a lock-in amplifier (LIA) for detecting very weak CW THz waves. In addition, the non-uniformity in the frequency response poses an upper-limit in the modulation depth. At the present state, we are aiming to overcome such limitations by developing more efficient photomixers or adopting photodiodes as the Tx component. However, in this study, we believe that an operational demonstration of the scheme in the CW THz system will be sufficient to show the potential of the utilizing FM in the field of CW THz systems.
For the frequency modulation, we applied a saw-tooth-like waveform in Fig. 3(b) on the μ-heater 2 of DML via dual channel function generator (Tektronics AFG 3052C). From the other synchronized channel of the function generator, a sinusoidal voltage signal of which frequency is more than 100 times higher than the modulation frequency is generated and applied to the Tx photomixer to electrically chopping the bias voltage of the THz antenna. The triggering output from bias voltage is used as the reference signal for the lock-in amplifier (LIA, Stanford Research System SR850) for the detection of the CW THz signal. Since the detected level of the oscillating component in THz signals is very smaller than its DC level, LIA’s DC filtered and AC amplified output is used for an efficient probing of the THz signal. The major operation parameters used in this experiment are shown in Table 1.
Table 1. Parameters used in the experiments
The spectra from DML for its static and modulation dynamic cases are depicted in Fig. 5. In Fig. 5(a) two four-wave mixing (FWM) peaks that indicate strong phase correlations of the two lasing peaks in DML are clearly shown. Before the modulations, the frequency sweep range was about 530-633 GHz. When the μ-heater 2 is modulated, a line broadening in the lasing peak for DFB 2 (right peak) is observed with a slight but observable line broadening in the unmodulated peak of DFB 1 (left peak). This means that the actual dynamic beating frequency will undergo some distortion from its static values and the actual modulation depth in its dynamic operation will be decreased. However, the observable FWM peaks in Fig. 5(b) show that, even under the modulation condition, the two laser beams are still strongly correlated to generate a strong modulated beating signal. Since the optical power levels for the main lasing peaks and the FWM peaks in Fig. 5(a) and 5(b) are not much changed, we can presume that the major origin of the increased background noise level in Fig. 5(b) is caused by the fast OSA scanning speed that is slightly increased to obtain the modulated spectrum.
Fig. 5 Measured optical spectra of DML for the cases of (a) static (μ-heater 1 is DC biased and μ-heater 2 is unbiased) and (b) μ-heater 2 is modulated with 20 Hz waveform.
Before measuring sample thickness, the basic operations of the system is investigated by measuring the path difference introduced by distance change in the free space. For this objective, we added a variable mechanical delay line to one of the optical path in the reference path in Fig. 2 and measured the beating frequency values that depend on the delay positions. The experimental results for the delay positions from 0 to 150ps with 10ps steps are presented in Fig. 6.
Fig. 6 (a) Delay position dependent beating frequencies obtained by peak counting method. The curve for number of counted peaks is depicted with rectangular markers and that of the calculated beating frequencies is depicted with circular markers. (b)-(c) Examples of the obtained beating signals for several delay positions of A, B, and C are shown in, respectively.
As a first approach to calculate the beat frequency, we applied a simple peak counting method. For this, we programmatically counted the total number of maxima and minima Npeaks, in a fixed duration of time within a modulation period. With this number and the locations of the first and the last extreme (maximum or minimum) points t1 and t2 respectively, the beating frequency can be approximately calculated as
The counted number of peaks and the corresponding beat frequencies are depicted in Fig. 6(a). As in Fig. 6(a), although the number of peaks for each delay position is a discrete number, the slight variation in the corresponding peak positions makes the beating frequency changes in an almost linear manner to the change of the delay position. However, the calculated beating frequency curve exhibits some observable wiggling points. The possible cause of such off-linear response can be various. First of all, any distortion or noise in the measured signal may affect the accuracy in locating the peaks. Much severely, any miscounting of the number of the peaks is, for example when missing one minimum point as in the point ‘B’ in Fig. 6(c), will induce a critical error to the measured result. As an alternative approach to extract the beat frequency, using the locations of zero-crossing points can also be an applicable choice. However, in a noisy signal that may have ripples near the zero-crossing points, such a method also gives an erroneous result.Instead of time domain analysis, frequency domain approach such as fast Fourier transformation (FFT) method can be a readily selectable choice. However, to obtain reliable spectral data from FFT, sufficiently long enough signals over many periods of time are desirable. Therefore, we have chosen the autocorrelation based extraction approach since it can be more proper method if the spectrum presents only a few peaks [28]. The obtained results from autocorrelation-based beating frequency extraction method are shown in Fig. 7. In contrast to other methods that uses the point value based frequency extractions, the autocorrelation method is based on an integration based convolutional approach. Therefore, as shown in Fig. 7(a), the beating frequencies extracted are less sensitive to the error or noise to provide much linear result for our case. In its practical implementation, the autocorrelation can be easily obtained with FFT algorithm [29].
Fig. 7 (a) Amount of frequency shift dependent on delay position obtained by autocorrelation method is depicted in a curve with triangular markers. For the comparison of the linearity, the beating frequency curve obtained in Fig. 6(a) is plotted in the same panel. (b) The autocorrelation functions for several delay positions. In these data, an inverse of the second autocorrelation peak corresponds to its beating frequency.
As a next step, we measured the thicknesses of polytetrafluoroethylene (PTFE or Teflon) disks that are prepared in a form of one-inch diameter and various thickness. Before the measurement, the thicknesses of each Teflon disk are measured with a micrometer for the evaluation of the measured values. The obtained results are shown in Fig. 8.
Fig. 8 (a) Autocorrelation results for Teflon disks. (b) Sample-dependent frequencies for the correlation peaks. In (b), the zero-th sample is free space as a reference. The data was obtained at the delay position of 100 ps.
Using the beat frequency measured without a sample as the reference value, sample 0 in Fig. 8(b), the relative differences in the beating frequencies for four disks are calculated from the autocorrelations. Under an assumption of the refractive index of the Teflon is n = 1.4, the thickness values are estimated and compared with the corresponding pre-measured ones in Table. 2. In this calculation, we assumed that the frequency-dependent variation in the value of the refractive index of Teflon in the THz range is negligible, which is a usually adopted approximation. If the refractive index of Teflon has value of other than the approximated one, the estimated thickness values will be greatly affected. Therefore, precise knowledge on the actual value of the refractive index is essential. In other aspect, if we take the pre-measured values of the thickness as the real and accurate thickness of the sample, we can inversely estimate the refractive indices of the Teflon disks. In this case, the approximation of the negligible dispersion in the sample for the used spectral regime is also required. The estimated refractive index values are depicted in the last column of Table 2. From these values, the average of n for the Teflon disk is n = 1.3906, which approximately coincides to the known value [30].
Table 2. Measured thickness of Teflon disk
Although we will not show it in here, the minimum resolvable thickness limit can be far more increased by utilizing the phase information. For such an application, an algorithm to check the measured phase is within the 2π range is required. Since the value of ωα in Eq. (15) is very large in the period, the magnitude and the measurable range of the phase variation should be very small. Therefore, only for the very limited cases of measuring extremely thin samples, phase based measurement can be applicable. In addition, large fraction of s for the flat region in Fig. 4 may be required to clearly identify the reference point to extracting the phase information.
4. Conclusion
Based on the fast tuning speed of a DML, an operation scheme of FM-based CW THz measurement system is proposed and evaluated. The main objective of this study is to examine the possibility of implementing a low-cost, portable and real time CW THz measurement system. In the course, the reliable operations of 20 Hz frequency modulation of DML with 0.1 THz modulation depth have been demonstrated. With an application of OFMCW method to the CW THz homodyne system, the feasibility of implementing a compact system to measure the thickness or refractive index of a sample is evaluated. In principle, the proposed scheme can also be constructed via using two individual single mode lasers. However, the advantageous compactness and stability of DML as a beating source, along with its suitability in generating frequency modulated beating signal in the sense of its μ-heater based fast-tuning speed and the stability in polarizations of two lasing beams during the modulation will construct a distinctive feature of our system. We hope that our proposed system will find many applications in various fields of THz applications such as thickness measurements, range finder, and other THz applications such as THz imaging system [31]. We believe that the beating source-based FM technique that is introduced in this study is the first report in the field of CW THz systems and hope that more various and advanced studies on utilizing modulated THz waves for many applications to be reported.
Acknowledgment
This work was partly supported by the IT R&D program of MOTIE/KEIT [10045238, Development of the portable scanner for THz imaging and spectroscopy], the Public Welfare & Safety Research Program through the National Research Foundation of Korea (NRF) Technology (NRF-2010-0020822), and Nano Material Technology Development Program through the NRF of Korea (NRF-2012M3A7B4035095).
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