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Abstract
A terahertz continuous wave system is demonstrated for thickness measurement using Gouy phase shift interferometry without frequency sweep. One arm of the interferometer utilizes a collimated wave as a reference, and the other arm applies a focused beam for sample investigation. When the optical path difference (OPD) of the arms is zero, a destructive interference pattern is produced. Interference signal intensity changes induced by the OPD changes can be easily predicted by calculations. By minimizing the difference between the measured and the calculated signal against the OPD, the thicknesses of sub-100-μm-thick samples are determined at 625 GHz.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz) waves, defined as electromagnetic waves in the range of 0.1 THz to 10 THz, have intrigued much attention due to their potential usage in imaging, spectroscopy, and communications technologies [1,2]. THz waves are transparent in a wide variety of non-conducting materials, such as clothing, paper, and plastic while being strongly attenuated in metals and liquid water. Also, the energy of THz waves is not sufficient to ionize DNA or other biological molecules. These properties of the THz waves have advantages for applications in medicine, pharmacy, security and other industrial fields [3–7]. One of the promising industrial applications of THz technology is non-destructive inline thickness measurement of paint films on automobiles, ships, airplanes and other vehicles [8]. Unlike the conventional methods that use eddy currents and the ultrasound, THz technology could provide non-contact and remote measurements [9].
Signal sources for THz technology can be classified into THz pulsed waves and THz continuous waves (CWs). THz systems based on pulsed waves typically utilize time-of-flight technique [10,11] for thickness measurement and tomography. In this technique, reflected THz pulses from an object at different axial positions are temporally separated in the detected THz time-domain signals when the temporal separation is larger than the pulse width in the time-domain. When the pulses are overlapped, several numerical methods have been applied to determine the thicknesses of sub-100-μm-thick samples [12–14]. Even though THz systems using pulsed waves been extensively studied for thickness measurement and tomography, its industrial application is still limited due to the system size, cost, and complexity mainly due to bulky and costly laser systems.
To overcome the issues, THz CW systems can serve as alternative systems. We have developed dual mode lasers (DMLs), including two distributed feedback laser diodes for compact and cost-effective THz CW sources [15]. Using this source, the thicknesses of a commercial PDP-glass and Teflon disks were determined in transmission configurations at the mm scale [16,17]. However, reflection configurations are suitable for in-line measurements. So far, there have been efforts for thickness measurement and tomography using THz CW in reflection configurations. For example, THz tomography using frequency-modulated CW (FMCW) method is attractive due to the fast data acquisition time, compactness of the system, and a wide A-scan range. However, the technique has a low axial resolution (typically order of 1-10 mm) due to a limited frequency modulation range [18]. For a better axial resolution for THz CW tomography and thickness measurement, THz optical coherence tomography (OCT) can be a considerable solution [19,20]. The minimum resolvable depth using this technique is reported to 0.61 mm in the air using frequency sweep from 400 to 800 GHz [20].
As the axial resolution of the aforementioned THz CW techniques is improved by the increased spectral bandwidth, the time required for the frequency sweep is a significant issue to obtain a higher axial resolution of the system. So far, efforts have been made to reduce the time required for frequency sweeping for a high-speed THz tomography [21,22]. Yee et al. showed that the axial resolution of 0.510 mm is obtained with a high-speed broadband frequency sweep ranging from 0.5 to 1.5 THz [21].
To summarize, there is a trade-off between the high axial resolution which requires broadband measurements and the fast data acquisition time in the present THz CW techniques. We tackle this issue by using phase shift interferometry. The interferometer has one arm with a focused THz wave and the other arm with a collimated beam as typically used in the THz OCT system. When the optical path lengths of the two arms are matched, a destructive interference is produced by the additional phase shift approximately π for the focused beam. The shift is called Gouy phase shift that is universally observed in a focused beam [23–25]. The destructive interference is disrupted by small change in the optical path difference (OPD) of the two arms. The measured interference signal change can be compared to the calculated signal using a theoretical model presented in this paper. By minimizing the difference between the measured and the calculated signals against the OPD, the sample thickness is determined without frequency sweep. In this paper, we demonstrated a THz CW technique using phase shift interferometry for measuring sub-100-μm-thick samples without frequency sweeping.
2. Experiment set-up
The experiment set-up is illustrated in Fig. 1. It consists of a tunable laser source, THz emitter, THz detector and a Michelson interferometer. A tunable laser diode (8164A Lightwave measurement system, Agilent technologies Inc.) and a laser diode driver (Modular controller 8000, Newport Inc.) are used for generating two laser beams with different wavelengths. The beams are injected into a home-made transmitter (Tx) module via an optical fiber for the THz wave generation [26]. The Tx module includes a bow-tie antenna integrated uni-traveling carrier photodiode (UTC-PD) photomixer and a hemispherical silicon (Si) lens. The frequency of the generated THz wave is determined by the frequency difference of the two laser beams. THz waves emitted from the UTC-PD are collimated with a beam expander. The collimated THz waves are divided into two collimated waves by a beam splitter. We employ 100-μm-thick, high resistivity Si wafer as a beam splitter to minimize the Fabry-Perot interference effect [27]. One part of the wave goes to the reference mirror (the reference arm) and the other part is focused onto a sample mirror (the sample arm) using an off-axis parabolic mirror. The reference mirror is placed on a motorized stage (M-ILS100PP, Newport Inc.) driven by a controller (ESP300, Newport Inc.). THz waves reflected from the mirrors are focused to a receiver (Rx) with an off-axis parabolic mirror. For the Rx, a Schottky barrier diode (SBD) (WR1.5ZBD-EXT, Virginia Diodes Inc.) is used. A lock-in amplifier is used to amplify the detected signals. For the lock-in detection, the bias voltage to the Tx is modulated with a 40 kHz square wave.
Fig. 1 Schematic of the experiment set-up. A beating signal generated from a tunable laser system is injected into the transmitter (Tx) for THz generation. The THz waves are divided by a beam splitter. The beams reflected from the two mirrors are focused on the receiver (Rx).
3. Results and discussion
Before we discuss our method, let us briefly consider the characteristics of the conventional sweep-source (SS) OCT technique in our experimental condition as a reference.
First, interference patterns are measured at different positions of the reference mirror. The measured THz frequency is tuned from 550 to 750 GHz for all measurements. The black line in Fig. 2(a) shows the frequency spectrum when the optical path lengths of the THz waves from two arms are different more than 1 cm. Oscillations induced by the path length difference of the arms are clearly observed. By Fourier-transforming the signal shown in Fig. 2(a), the OPD of the two arms is obtained as shown in Fig. 2(b). When the reference mirror position is changed to set the OPD to about 3 mm, the number of oscillations in the frequency spectrum is reduced as shown by the blue line in Fig. 2(a). The blue line in Fig. 2(b) shows the corresponding Fourier-transformed signal. As clearly visualized in Fig. 2(b), reflected THz waves at different axial positions can be separated in Fourier-transformed signal of the spectrum and this can be utilized for thickness measurement and tomography. The axial resolution of the SS-OCT is theoretically described as the following formula [28]
where, λ0 is the center wavelength, Δλ is the full width at half-maximum (FWHM) of the swept- source signal, and n is the refractive index of the object. The inset of Fig. 3. depicts the calculated spectral bandwidth dependent axial resolution of the experimental system. For example, when the center frequency is 600 GHz and FWHM is 100 GHz, the corresponding theoretical depth resolution is 1.31 mm. As the spectral bandwidth increases, the axial resolution exponentially improves. To obtain axial resolution of 100 μm, a frequency sweep range greater than 700 GHz is required.Fig. 2 (a) Interference patterns measured at different locations of the reference mirror. The black line shows the pattern when the OPD is more than 1 cm. The blue line presents the spectrum when the OPD is changed to about 3 mm. (b) OPD of the two arms obtained from FFT of the interference patterns. The color code is the same as that used in Fig. 2(a).
Fig. 3 The FWHM of the pulse-like waveform obtained from FFT of the frequency spectrum as a function of frequency sweep range. Inset shows the calculated axial resolution of the system against spectral bandwidth.
In practice, the axial resolution can be measured by the FWHM of the pulse-like waveshape as shown in Fig. 2(b). The measured values of FWHM of the pulse-like waveforms such as in Fig. 2(b) for various frequency sweep ranges are shown in Fig. 3. The graph clearly shows that the depth resolution is improved when the frequency sweep range is increased. However, by increasing the sweep range to achieve an enhanced depth resolution, the measurement time is excessively increased to produce a 3D image or to measure the sample thickness in a large area.
We propose a new approach to enhance the depth resolution in the conventional THz OCT system without frequency sweep. The technique utilizes the sensitivity of the interference signal change to the OPD change for the determination of the thicknesses of sub-100-μm-thick samples. When an electromagnetic beam passes through a focus, the beam experiences phase shift given as [23–25]
where is the characteristic frequency above which the phase shift reaches π, is the focal length of a focusing optical element and is the beam radius. In our measured conditions, the characteristic frequency is 48.7 GHz. Therefore, nearly π phase difference is acquired between the THz beams from the reference arm and the sample arm in all measured frequencies. The electric fields at each arm are expressed by where is electric field of the reference (sample) arm, is the wavenumber, is the angular frequency of the wave, is the optical path of the sample arm and is the OPD of the two arms. The calculated interference signal power is given asThe main point of our technique is to generate destructive interference signal as a reference. For the destructive interference generation, π phase difference between the two arms is required. One can get the phase difference using Gouy phase shift of the focused beam as described before. In this case, OPD should be zero. As an alternative, OPD can be tuned to induce the π phase difference between the two arms. To compare the sensitivity of the aforementioned systems, the spectrum change as a function of OPD change is calculated and shown in Fig. 4 in two different conditions using Eqs. (3)-(5). The typical system for OCT using a focused beam for the sample arm and a collimated beam for the reference arm is shown in Fig. 4(a). As a comparison, we also investigate an interferometer using collimated beams for both arms as illustrated in Fig. 4(b).
Fig. 4 Schematic of (a) an interferometer with a focused beam and a collimated beam. (b) An interferometer with collimated beams for both arms. (c, d) Signal change as a function of OPD in the experimental conditions of (a) and (b), respectively.
Detected signal powers only from the reference mirror are shown in Figs. 4(c) and 4(d) (black lines). The signal drop at approximately 560 GHz is due to the THz wave absorption by water vapor. With the measured spectrum, the intensity change against OPD change is estimated. First, the measured signal is smoothed with a FFT filter using OriginPro 2015 to remove the undesirable spikes in the spectrum, which mainly originated from multiple reflections within the Si lens included in the Tx module. Then take the square root of the measured signal and the value is used for the E1 and E2 in Eqs. (3) and (4), respectively. Then, the interference pattern can be calculated as a function of OPD using Eqs. (3)−(5). When the interferometer has a focused beam and a collimated beam, signals form the two arms will destructively interfere when OPD is zero as shown in the red line of Fig. 4(c). However, any changes in either the phase or the amplitude of the THz beams will lead to the disruption in the destructive interference and a large signal would be measured. The signal is increased four orders of magnitude when the OPD is changed from 10 to 100 μm at 625 GHz. All the spectra are well distinguished with implies that sub-100-μm axial resolution can be obtained by measuring the change in the power of the interference pattern. As previously described, in the conventional THz OCT method, to obtain a sub-100-μm axial resolution, frequency sweep more than 700 GHz is required. In our technique, the frequency sweep range is 250 GHz and is irrelevant to the axial resolution, as discussed in the following section. In case of the interferometer with collimated beams for both arms, the signal is 2.5 times decreased when the OPD is changed from 10 to 100 μm at 625 GHz. The interferometer using Gouy phase shift gives us orders of magnitude higher sensitivity.
We now present a process to determine the sample thickness using Gouy phase shift interferometry. In Fig. 5, the measured spectra at different reference mirror positions are given and are compared with the simulated spectra. The simulation is performed in a similar way with the previous section except that, in this case, the signal from the reference arm is used for E1 and signal from the sample arm is used for E2 to reflect experimental conditions.
Fig. 5 Interference patterns as a function of the reference mirror location change. (a) The measured frequency spectra and (b) the simulated spectra.
When the reference mirror position is changed up to approximately 110 μm, the signal power increases at all measured frequencies. In Fig. 5(a), when the reference mirror position is further changed, signal power at high frequencies decreased. The main features of the measured spectra are reproduced in the simulated results as shown in Fig. 5(b). The discrepancy between measured and simulated spectra is attributed to several factors such as the accuracy of the motorized linear stage, signal power fluctuations, imperfections of the alignments of the optical elements. More specifically, the guaranteed accuracy of the motorized linear stage is ± 2.0 μm, the signal power fluctuates in time within ± 2%, and the signals reflected from the reference mirror at different positions are not identical. All the factors contribute to the inaccuracy of the sample thickness determination.
Figure 6(a) shows the signal difference between the measured signal Imeasured and the calculated signal Icalculated. A spectrum is acquired when the stage is changed by 90 μm from the reference position. In the calculation, OPD is changed from 80 to 100 μm with a 1 μm step. Signal difference is linearly changed with OPD changes at all measured frequencies. This implies that frequency sweeping is not required to achieve an enhanced axial resolution in this technique. Then we define an error as
Here, is fixed to 625 GHz and is the error calculation range. The error as a function of OPD changes, in case of the error calculation range is zero, is shown in Fig. 6(b). We determine the calculated OPD as 93 μm by searching the minimum point of this curve. The process is expressed asErrors for different error calculation ranges are also calculated. All the resulting curve shows the similar behavior as Fig. 6(b), and the calculated OPD is not changed depending on the error calculation range as shown in the inset of the Fig. 6(b). This clearly shows that our technique provides sub-wavelength resolution without frequency sweeping.Fig. 6 (a) Difference between the measured spectrum when the stage is change by 90 μm from the reference position and the calculated spectra with different OPDs. (b) OPD dependent error. Inset shows the obtained OPD is insensitive to the measured spectral bandwidth.
The same processes are performed for signals obtained at other linear stage positions. The calculated OPDs as a function of the actual OPDs are depicted with blue circles in Fig. 7.
Fig. 7 Calculated OPD change by changing the linear stage position (blue circles) or by attaching adhesive tapes on the sample mirror (black squares) against actual OPD change.
Finally, we measure the thicknesses of adhesive tape layers attached to the sample mirror. When the sample is attached in front of the sample mirror, the OPD induced by the sample is defined as
where nsam is the refractive index of the sample, nair is the refractive index of air, and lsam is the sample thickness. The sample thickness is measured with an electronic digital micrometer with ± 2 μm accuracy and the refractive index of the sample is obtained with a typical THz time domain spectroscopy measurement in a transmission configuration. The detailed process of obtaining the refractive index of the thin samples is described elsewhere [29]. The actual OPDs for the measured adhesive tapes are 69, 103, and 138 μm, respectively. The calculated OPDs are 70, 98, 139 μm, respectively. Note that this enhanced axial resolution of the experiment configuration is not related to the frequency sweep range, and single frequency measurement is sufficient for determining the thickness of sub-100-μm-thick samples. In addition, the technique does not suffer from 2π phase ambiguity present in the conventional CW system which must be solved for measuring various thickness ranges [15–17]. As we compared our method to THz OCT for the resolution enhancement, a comparison table of the two methods are given in Table 1. Note that the technique is applied to measure the thicknesses of single-layered materials. Applying a variable focal length lens in the sample arm and measuring the interference signals at different foci of the lens inside the samples might enable to determine the thicknesses of multi-layered samples.
Table 1. Comparison of THz OCT and Gouy phase shift interferometry
This point detection technique can be extended to a compact raster scanning technique for practical applications for thickness measurements and tomography by adapting our home-made DML [15] as a THz source and a THz beam scanner consist of a two-dimensional Galvano mirror and a telecentric f-theta lens [26] in the sample arm.
4. Conclusion
A terahertz CW system for thickness measurement using phase shift interferometry is proposed and demonstrated. When the optical path lengths of the two arms in the interferometer are matched, a destructive interference pattern is produced as the focused beam experience Gouy phase shift. A Slight change in amplitude and/or phase of the reflected THz waves in one arm leads to a disruption of the destructive interference and a large signal change is induced. By minimizing the difference between the measured and the calculated signal against OPD, thicknesses of sub-100-μm-thick samples are determined at 625 GHz. Note that no frequency sweep is required in our technique while the conventional SS-OCT technique requires frequency sweep more than 700 GHz to obtain an axial resolution of 100 μm. By adapting this technique, the data acquisition time can be substantially reduced while obtaining sub-wavelength axial resolution which is critical for practical applications such as in-line thickness measurements and THz tomography.
Funding
Ministry of Trade, Industry, and Energy (MOTIE) and Ministry of SMEs and Startups (MSS) of the Korean Government (S2524372); Electronics and Telecommunications Research Institute (ETRI) (19ZR1200, 19ZH1700).
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