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We propose and demonstrate a robust terahertz self-heterodyne system using a phase noise compensation technique. Conventional terahertz self-heterodyne systems suffer from degraded phase noise performance due to phase noise of the laser sources. The proposed phase noise compensation technique uses an additional photodiode and a simple electric circuit to produce phase noise identical to that observed in the terahertz signal produced by the self-heterodyne system. The phase noise is subsequently subtracted from the terahertz signal produced by the self-heterodyne system using a lock-in amplifier. While the terahertz self-heterodyne system using a phase noise compensation technique offers improved phase noise performance, it also provides a reduced phase drift against ambient temperature variations. The terahertz self-heterodyne system using a phase noise compensation technique shows a phase noise of 0.67 degree in terms of a standard deviation value even without using overall delay balance control. It also shows a phase drift of as small as approximately 10 degrees in an open-to-air measurement condition without any strict temperature control.
© 2015 Optical Society of America
1. Introduction
Terahertz (THz) waves have interesting features including an outstanding spectral fingerprint, long penetration depth in dielectrics, and high rectilinear propagation, making them attractive for sensors, spectroscopy, and imaging applications. In particular, the spectral fingerprint characteristics of THz waves have been applied to various applications including nondestructive structural testing [1], biomedical diagnostics [2], security systems [3], and pharmaceutical analysis [4].
For the above applications, pulse type or continuous wave (CW) THz waves are used. The pulse type THz waves are used for time domain analysis and are especially advantageous in transmission and reflection imaging [5]. While pulse type THz sources offer a broadband spectrum (0.1 – 5 THz), they are relatively costly because usually expensive femtosecond lasers are used to generate the pulsed THz wave. On the other hand, CW THz sources used for frequency domain analysis offer advantages of high spectral resolution and frequency selectivity and are more cost-effective than the pulse type THz sources. In addition, continuous THz waves can also be applied to short-range, high-data-rate wireless communications [6].
There are many techniques for generating continuous THz waves including strategies using molecular far infrared (FIR) lasers, quantum cascade lasers (QCLs), backward wave oscillators (BWOs), electrical monolithic microwave integrated circuits (MMICs), and a photomixer. Among them, the use of a photomixer to generate CW THz waves is advantageous since it affords controllability of the signal frequency with a relatively wide tuning range. In this photomixing technique, CW THz waves are generated from beating of two laser sources having different wavelengths and thus the generated THz signal frequency can be relatively easily tuned by adjusting the wavelength difference. Using distributed-feedback lasers as laser sources, CW THz signals having the spectral linewidth as small as a few MHz were generated [7].
To characterize materials using THz waves, a self-heterodyne system using a photomixing technique can provide high phase sensitive measurements [8]. This system is suitable for characterizing materials with weak absorption since measurement of the phase variation offers a higher image contrast compared to measurement of the intensity variation in weak absorption materials. However, the performance of this technique is limited by the phase noise of the detected signal originating from the laser sources that are used. Inherently, the phase noise can be reduced by using laser sources with an ultra-narrow linewidth. However, the price of laser sources increases as the linewidth becomes narrower. Another solution to reduce the phase noise is to increase the time constant of the lock-in amplifier (LIA), which is used to obtain the amplitude and the phase information of THz waves. Increasing the time constant of the LIA is, in effect, averaging the amplitude and phase responses of the detected signal for a longer period. While increasing the time constant of the LIA reduces the phase noise of the detected signal, it also increases the data acquisition time, which can be a limiting factor for imaging applications requiring measurements of many points in a short period.
Overall delay balance control is a good alternative to reduce the phase noise without increasing the time constant of the LIA. In the overall delay balance control scheme, the total delays from the laser sources to the photomixer are precisely controlled using delay lines such that the difference in delays should be much smaller than the correlation length of the laser [9]. Matching the total delays using delay lines, however, is a cumbersome process since the conventional self-heterodyne system incorporates many optical components, including an EDFA, EO modulator, and fiber couplers, that can contribute to path length difference. The overall delay balancing process should be repeated if some components are added or removed since this will change the optical path lengths.
The self-heterodyne system is inherently vulnerable to phase drift induced by ambient temperature variations. This system uses optical fibers to link the laser sources to the THz transmitter and receiver and the temperature variations affect the length and the refractive index of the optical fibers [10]. As a result, when the optical path length is long and the data acquisition progresses for a long period, the temperature variations can induce phase drift, causing ambiguity of the phase response of the self-heterodyne system [11].
In this paper, we propose a robust self-heterodyne system with a phase noise compensation technique that can reduce the phase noise without using overall delay balance control. This technique also effectively reduces the phase drift induced by the ambient temperature variations.
2. Operational principle of the conventional self-heterodyne system
2.1 Phase noise in the conventional self-heterodyne system
Figure 1 shows the configuration of a self-heterodyne system based on a photomixing technique. The self-heterodyne system can utilize the variation of the phase of the continuous wave THz signal within a test sample to characterize its material properties, and thus its phase noise characteristics are very important for the overall system performance. The phase noise of the system, which mainly originates from the phase noise of laser sources, can be reduced by overall delay balance control [9].
Fig. 1 Configuration of a conventional self-heterodyne system utilizing a photomixing technique. FS: frequency shifter based on EO phase modulator. ωs: modulation frequency of the phase modulator. lm: length of the fibers indicated by arrows.
The outputs of the laser diodes (Laser 1 and 2) are split into two optical signals. One of the split optical signals from Laser 1 is blue-shifted by ωs1 and one of the split optical signals from Laser 2 is red-shifted by ωs2. The shift of the optical frequency is achieved by modulation of an optical phase modulator with a sawtooth wave. The photomixer has two inputs, an optical probe signal (Ep) in the form of a CW THz signal generated by an O/E converter and an optical local oscillation (LO) signal (ELO); they can be written as
where ωm and φnm are the optical frequency and the phase noise of each laser and ωs1 and ωs2 are the shifted optical frequencies, respectively. τi is the propagation delay and the delays (τ1, τ2, τ3, and τ4) are associated with the optical paths (P1, P2, P3, and P4). The optical paths P2 and P4 for the optical LO signal (ELO) correspond to the paths from each laser source to the optical input of the photomixer. The optical paths P1 and P3 for the optical probe signal (Ep) correspond to the paths from each laser source to the THz signal input of the photomixer including the air propagation path of the THz signal (i.e., from the O/E converter to the THz signal input of the photomixer), as shown in Fig. 1. ΔφTi is the variation of the phase in each path associated with ambient temperature variations. For the time being, ΔφTi is set to zero assuming that the ambient temperature is perfectly controlled. The effect of non-zero ΔφTi will be addressed later.A CW THz wave is generated by O/E conversion using a waveguide-integrated p-i-n photodiode. Due to the limited bandwidth of the p-i-n photodiode, a CW THz wave (ERF) having a frequency of ωTHz + ωs1 is generated and radiated through an integrated antenna. The CW THz wave is expressed as
where ωTHz is the optical frequency difference between the two lasers (ω1 and ω2) and ϕs is the phase shift produced by the test sample. The radiated CW THz wave is passed through a test sample and then directed to the THz signal input of the photomixer.In the photomixer, the optical LO signal (ELO) generates photo-carriers that change the conductivity of the photomixer [8]. Assuming that there is no parasitic amplitude modulation, the variation of the photomixer conductance can be described as
At the output of the photomixer a current produced by mixing CW THz wave (ERF) and optical LO signal (ELO) is generated and then converted to a voltage signal using a trans-impedance amplifier (TIA). The output of the TIA can be expressed as
where Δωs is the frequency difference between two shifted frequencies, (ωs1-ωs2) and As is the amplitude of the signal.Finally, the phase response of the system, which is extracted by the LIA, can be expressed as
The terms φnm[t – τi], which originate from the phase noise of the laser sources, lead to phase error that is critical in some applications utilizing the phase response. The phase error can be eliminated by using overall delay balance control, where the optical path lengths for the optical probe signal and LO signal are adjusted using optical delay lines to make τ1 = τ2 and τ3 = τ4. Under this condition of overall delay balance, the laser induced phase noise terms are canceled out and the phase response produced by the lock-in amplifier can be expressed asHowever, overall delay balance control using an optical delay line is complicated and cumbersome since all optical components within the optical signal paths have to be considered, and the process should be repeated whenever any components are added or removed.
2.2 Variation of the phase response in the conventional self-heterodyne system due to the ambient temperature variation
A practical conventional self-heterodyne system using overall delay balancing has finite phase noise due to imperfect overall delay balancing, although the phase noise is reduced if the system involves a small amount of mismatch of the optical signal path lengths. In addition, imperfect overall delay balancing produces variation of the phase response caused by the ambient temperature variation. The ambient temperature variation changes the characteristics of the optical components. Among them, changes of the characteristics of optical fibers including the refractive index and the length of the fibers have the most significant effects on the phase delay of the optical signal [10].
The output of the TIA for the conventional self-heterodyne system under imperfect overall balancing including the effect of the ambient temperature variation can be expressed as
where ΔφTi is the phase delay of the optical signal in optical path Pi induced by the temperature variation and is proportional to the temperature variation and the length of the optical fiber. The phase drift induced by the ambient temperature variation can be expressed aswhere c, ΔT, TCDk, and lk are the velocity of light, temperature variation, thermal coefficient of delay of the optical fiber, and length of the optical fibers described in Fig. 1, respectively. The thermal coefficient of delay is determined by the structure of the fiber [10]. In this analysis the structure of the optical fibers used in this system is assumed to be identical and thus all the thermal coefficients of delay, TCDk, are assumed to be identical.Finally, the phase variation can be simplified as
since ω1 and ω2 are much larger than ωTHz and thus the third term in the last line of Eq. (9) is negligible. As shown in Eq. (10), the phase drift induced by the ambient temperature variation is only related to the specific optical fibers in the system [11] and is proportional to the mismatch of the length of fibers. The above analysis results do not include the temperature dependence of optical components (e.g. frequency shifters and delay lines) within the optical signal path lengths that may aggravate the phase variation.The presence of the phase variation induced by the ambient temperature variation in a practical conventional self-heterodyne system under imperfect overall delay balancing requires calibration of the system whenever the ambient temperature changes.
3. Proposed self-heterodyne system with phase noise compensation
The phase noise of the self-heterodyne system induced by the phase noise of the laser sources can be reduced by using a phase noise compensation technique. The phase noise compensation can also effectively remove the phase variation associated with the ambient temperature variation, thereby making system calibration against ambient temperature change unnecessary. A schematic diagram of the self-heterodyne system using the phase noise compensation technique is shown in Fig. 2. The primary structural difference between the conventional and the proposed self-heterodyne system is the presence of a photodiode (PD) that produces the same phase noise and temperature dependent phase variation terms as the THz signal described in Eq. (8). This signal is used as a reference signal of the LIA so that the common phase noise and temperature dependent phase variation terms can be subtracted from the output of the TIA in the lock-in process.
Fig. 2 Configuration of a self-heterodyne system with a phase noise compensation technique. FS: frequency shifter based on EO phase modulator. ωs: modulation frequency of the phase modulator. lm: length of the fibers indicated by arrows. Note that l3 = l3a + l3b and l8 = l8a + l8b.
To simplify the analysis, initially the ambient temperature is assumed to be perfectly under control. Under the assumption, the optical LO, probe signal, and the output signal of the TIA are identical to Eq. (1), Eq. (2), and Eq. (5), respectively.
The input of the PD shown in Fig. 2 consists of two input optical signals and can be expressed as
Note that the propagation delays (τm) are associated with the optical path (Pm). Considering the limited bandwidth of the PD, the output signal of the PD is described aswhere Voffset is the DC offset voltage of the PD that can be removed by the DC block placed prior to the electrical mixer. The output of the electrical mixer that squares the output signal of the DC block is described asAs can be seen in Eq. (13), the output of the mixer has three frequency terms, DC, a high frequency term (ωs1 + ωs2), and a low frequency term (ωs1-ωs2). By electrical bandpass filtering, the DC and high frequency terms are eliminated, leaving the output of the bandpass filter described asIn the phase noise compensation technique, the signal described in Eq. (14) is used as the reference signal of the LIA.The role of the LIA can be simplified as mixing two input signals and subsequent lowpass filtering and can be expressed as
where Ainput and Bref are the input and the reference signal, respectively. When the output signal of the TIA given by Eq. (5) and the output of the bandpass filter given by Eq. (14) are used as the input and the reference signal of the LIA, respectively, the output of the LIA can be expressed asIf the optical path length described in Fig. 2 satisfies the conditions of l3a = l9 and l8a = l10, then the propagation delays become τ1 = τ5, τ2 = τ6, τ3 = τ7, and τ4 = τ8 and the laser induced phase noise terms described in Eq. (16) are canceled out. Consequently, the proposed system only has to consider the specific region related to l3a, l9, l8a, and l10 to reduce the phase noise, which is a more convenient solution than the overall delay balance control of the conventional self-heterodyne system.
As discussed in section 2.2, phase variation is induced by the ambient temperature variation in the proposed system. Under imperfect delay balancing, the output of the TIA is identical to Eq. (8) and the phase variation of the TIA’s output is identical to Eq. (10). Also, the reference signal for the LIA has phase variation due to the ambient temperature variation and this can be described as
From Eq. (17), the phase variation can also be expressed asSince ω1 and ω2 are much larger than ωTHz, the third term in the third line of Eq. (18) is much smaller than the first and second terms. The phase variation of the output of the bandpass filter is almost identical to that of the output signal of the TIA described in Eq. (10). Therefore, the phase variation caused by ambient temperature variation can be greatly reduced by the lock-in process despite the system being under imperfect delay balancing. Consequently, the proposed phase compensation technique not only reduces the phase noise induced by the laser sources but also is robust to the ambient temperature variation.
4. Experiment and results
A proof-of-concept experiment is performed using the setups shown in Fig. 1 and Fig. 2. The InGaAs/InP-based O/E converter and photomixer (Toptica) have an integrated bow-tie antenna. Two tunable laser sources (81640A, 81949A, Agilent) having a linewidth of approximately 100 kHz are used to generate a 1.5-μm light wave. Two phase modulators (COVEGA) driven by a sawtooth wave signal are used as frequency shifters. Two polarization-maintaining EDFAs (KEOPSYS) are used to increase the power of the optical signals. All the optical components are connected by polarization maintaining fibers. A high gain trans-impedance amplifier (DLPCA-200, FEMTO) and a lock-in amplifier (SRS830, Stanford Research Systems) are used for high dynamic range.
For comparison, the frequency and the phase responses of the LIA output for the conventional and the proposed self-heterodyne systems shown in Figs. 1 and 2, respectively, are measured. The wavelengths of the lasers are set to 1545 nm and 1547.39 nm, which correspond to approximately a 300 GHz frequency difference. The modulation frequencies of the phase modulators (ωs1 and ωs2) are set to 400 kHz and 450 kHz, respectively. All optical path lengths from the laser source to the O/E converter and the photomixer are roughly identical without a precise overall delay balance control (i.e. under imperfect overall delay balancing). The reference frequency and the time constant of the LIA are set to 50 kHz and 1 ms, respectively. A frequency counter is connected to the output of the TIA to obtain the frequency.
The frequency of the TIA’s output signal is 50 kHz since, as described in Eq. (5), it corresponds to the difference between the modulation frequencies (ωs2 - ωs1). However, as can be seen in Figs. 3(a) and (b), the frequency of the TIA output and the phase of the LIA output, respectively, fluctuate due to the phase noise in the laser sources and imperfect overall delay balancing. The unacceptably large fluctuations of the frequency of the TIA output and the phase of the LIA output can be reduced by the use of laser sources having a very narrow linewidth, perfect overall delay balance control using optical delay lines [9], or optical phase-lock-loops [12]. However, these measures make the system expensive, complex, and cumbersome.
Fig. 3 (a) Frequency response of the TIA output signal of the conventional self-heterodyne system. The gate time of the frequency counter is 1 ms. (b) Measured phase response of the LIA output signal of the conventional self-heterodyne system. The time constant of the LIA is 1 ms.
To demonstrate the advantages of the phase noise compensation technique proposed in this paper, the frequency and the phase responses of the system are measured under identical experiment conditions described above.
Figures 4(a) and 4(b) show the frequency of the TIA output and the phase of the LIA output, respectively. Despite the frequency fluctuations of the TIA output signal shown in Fig. 4(a), the phase response of the LIA output signal shown in Fig. 4(b) is reduced drastically. The reduction of the phase noise of the LIA output signal is due to the phase noise compensation technique, where the common phase noise in the TIA’s output and the BPF’s output is subtracted. The phase response has a standard deviation of 0.67 degree, which is superior to that of a previous report [8] achieved using overall delay balance control and an LIA time constant of 1 ms.
Fig. 4 (a) Frequency response of the TIA output signal of the proposed self-heterodyne system. The gate time of the frequency counter is 1 ms. (b) Measured phase response of the LIA output signal of the proposed self-heterodyne system. The time constant of the LIA is 1 ms.
The frequency response of the proposed self-heterodyne system is measured from 0.3 THz to 1.6 THz. To change the generated THz-wave frequency, the wavelength of Laser 1 is set to 1545 nm and that of Laser 2 is changed from 1547 nm to 1558 nm with a step of 0.02 nm. Figure 5 shows the SNR of the proposed system that is obtained using the procedure reported in [13]. The SNR-limited phase standard deviation at 300 GHz, which is estimated from the SNR data shown in Fig. 5, is 0.40 degree. The difference between the SNR-limited phase standard deviation (0.40 degree) and the measured phase standard deviation (0.67 degree) from Fig. 4(b) is attributed to the excess phase noise caused by imperfect matching of fiber lengths (l3a~l9 and l8a~l10), which can be reduced by matching the fiber lengths more precisely. The measured standard phase deviation of the proposed system is similar to that of the homogeny system [14]. The homogeny system needs three lasers and its stability against ambient temperature fluctuation is reported to be poor. The measured standard phase deviation of the proposed system is slight worse than that of the balanced self-heterodyne system [13]. The balanced self-heterodyne system requires two photoconductive antenna (i.e., photomixers) to generate identical phase noise for noise cancellation. Both of the homogeny and balanced self-heterodyne systems suffer from a reduced THz signal power.
In the conventional self-heterodyne systems, imperfect overall delay balance control increases the phase noise in the LIA output signal originating from the phase noise in the laser sources [9]. To investigate the effect of imperfect overall delay balance control in the proposed self-heterodyne system, the phase response of the LIA output signal is measured with an intentional imperfect overall delay balance control. As depicted in Fig. 6(a), the length of the optical path (l5) is changed from 2 m up to 4 m, while the lengths of the other three optical paths (l2, l4, l7) are fixed at 2 m. At the same time, the generated THz-wave frequency is increased from 0.3 THz to 1.6 THz. Figures 6(b)-6(d) show the standard deviation of the measured phase response of the LIA output signal for l5 of 2 m, 3 m, and 4 m, respectively. Measurement results show that the standard deviation of the measured phase response of the LIA output remains almost identical even though the difference of the optical path length is increased. The spikes of the standard deviation observed at the frequencies of 551, 746, 985, 1092, 1156, 1201, 1223, 1407, and 1597 GHz are attributed to the reduced signal level due to absorption of the THz signal by water vapor [15].
Fig. 6 (a) Simplified diagram of the proposed system to investigate the effect of imperfect overall delay balance control. Measured phase standard deviations of the system with a different length of the optical path (l5) as a function of the generated THz wave frequency. (b) l5 = 2 m, (c) l5 = 3 m, (d) l5 = 4 m. l4 is fixed at 2m. The time constant of the LIA is 1 ms.
The proposed system has robust phase stability against the ambient temperature variations. To confirm the stability of the system, the phase response of the LIA output signal of the proposed system is measured indoors for 10 hours. For this experiment, the wavelengths of the lasers are set to 1545 nm and 1547.39 nm, which generate a terahertz signal with a frequency of approximately 300 GHz. Considering Eqs. (9) and (18), the amount of the phase drift is proportional to ΔT and ωTHz and thus increases as ωTHz is increased. However, the coefficient of the phase drift depends on the lengths of various optical paths involved. The phase drift shown in Fig. 7 is attributed to the variation of the ambient temperature in the laboratory. While the experimental setup is open to air without any strict temperature control, the phase variation is as small as approximately 10 degrees. This value is similar to that of the conventional self-heterodyne system using short fiber components [16] (approximately 15 degrees) under strict temperature control and is much smaller than that (approximately 50 degrees) measured at room temperature without strict temperature control [11]. The phase variation of the proposed system is also comparable to that of the conventional self-heterodyne system, where optical paths are implemented using a photonic integrated circuit with the critical optical path lengths as small as 2 mm [16].
Finally, the proposed system is used to measure the thickness of photocopy paper using the phase response of the system. Since the photocopy paper is a weak absorption material, measurement of the thickness of the photocopy paper from the amplitude response of the self-heterodyne THz systems is fairly difficult [8]. The wavelengths of the lasers are set to 1545 nm and 1549.79 nm, which generate a terahertz signal with a frequency of approximately 600 GHz. The photocopy paper is placed between the O/E converter and the photomixer. The phase response of the proposed system measured for different numbers of papers is shown in Fig. 8(a). As shown in Fig. 8(a) and (b), the SNR of the proposed system degrades with increasing number of papers. From the linear fitting shown in Fig. 8(b), the phase delay per a piece of paper (Δθ) is calculated to be 33.93 degrees, which can be used to estimate the thickness of the photocopy paper.
Fig. 8 Photocopy paper thickness measurement results. (a) Phase response in time domain. (b) Phase response as a function of the number of photocopy paper. The time constant of the LIA is 1ms.
The thickness of the paper can be expressed as
where Δθ and k0 are the phase delay per a piece of paper and the wave number of the THz wave in air, respectively, and n is the refractive index of the paper [8]. The refractive index of the paper is approximately 1.44 [15]. Consequently, the estimated thickness of the paper is 107.1 um, which is close to the reported value [15].5. Conclusion
A self-heterodyne system with a phase noise compensation technique to reduce the effect of the phase noise induced by the laser sources without overall delay balance control is proposed and experimentally demonstrated. The proposed system shows reduced phase noise even without any overall delay balance control. The proposed system also shows reduced phase drift against the ambient temperature variations. On the basis of the low phase noise characteristic and the robustness of the phase response against the ambient temperature variations, the proposed self-heterodyne system with a phase noise compensation technique is attractive for practical THz measurement applications requiring high phase sensitivity.
Acknowledgments
This work was supported in part by grants from the Bio-imaging Research Center program at GIST, the Brain Research Program (NRF-2014M3C7A1046050) through the NRF of South Korea, and a project titled ‘Development of Ocean Acoustic Echo Sounders and Hydro-Physical Properties Monitoring Systems’ funded by the MOF, South Korea.
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