1. INTRODUCTION

Applications of terahertz (THz) waves have been rapidly expanding in industrial and biomedical fields using newly developed THz sources and detectors [1–3]. In medical applications, development of flexible waveguides for delivery of THz waves is expected in endoscopic applications such as detection of early cancers using a THz spectroscopy technique [4]. Various types of waveguides, including dielectric waveguides [5], photonic crystal waveguides [6–9], and metal wire waveguides [10–12], have been proposed for THz-wave delivery. A hollow-core optical fiber is another option and is especially suitable for endoscopic applications due to its flexibility and relatively low transmission loss. In hollow optical fibers, a THz wave is strongly confined within the hollow core, and this will be a large advantage, especially in medical endoscopic applications.

Several types of hollow-core fibers have been developed: metal hollow optical fibers [13,14], dielectric tube waveguides [15,16], and metal hollow-core fibers with an inner dielectric layer [17–20]. The transmission properties of these THz hollow-core waveguides have been mainly evaluated using a Fourier-transform infrared spectrometer with a far-infrared light source and a multilayered THz beamsplitter to extend the spectral range down to the THz region. Wavelength-tunable THz sources based on parametric oscillation have also been used to obtain the transmission-loss spectra of hollow-core THz waveguides. Propagation mode properties of hollow optical fibers have been also discussed based on the loss spectra and output beam profiles measured with a THz laser [21]. However, quantitative analysis has been difficult because information about the phase of the transmitting THz wave is lost in these measurements.

More recently, with the advent of relatively low-cost femtosecond light sources, a variety of THz time-domain spectroscopy (THz-TDS) systems have been commercialized and are becoming popular [22]. Some groups have evaluated hollow THz waveguides with THz-TDS systems and showed transmission mode properties and spectral losses [23,24]. Bao et al. showed the loss spectra of dielectric tube waveguides measured by their THz-TDS. However, the frequency range was lower than 1 THz because of the limited signal-to-noise ratio (SNR) and difficulty in coupling THz wave into the waveguides [25]. Due to the high coupling loss between the THz-emitting antenna and hollow-core fibers, which is caused by the large divergence angles of the emitted THz beam and extremely small numerical aperture of hollow optical fibers, a clear loss spectrum of hollow-core waveguides longer than roughly 15 cm has not been reported in the THz region with frequencies higher than 1 THz. In this study, we optimized our THz-TDS system for evaluation of hollow-core waveguides and succeeded in obtaining clear transmission spectra of a metal hollow-core waveguide with a length of 42 cm. From the measured results on transmission loss spectra and mode properties, we show that the combination of THz-TDS systems and hollow-core fibers is useful for THz remote spectroscopy and experimentally demonstrate the feasibility of the combined system.

2. EXPERIMENTAL SETUP

Figure 1 shows a schematic of the THz-TDS system for evaluation of hollow-core optical fibers. Femtosecond pulses with a width of 50 fs and a central frequency of 800 nm were split into two beam paths: pumping light with an average power of around 120 mW for THz generation and probing light of 10 mW for THz wave detection. A delay was given to the probing light to obtain the temporal waveform of the detected THz pulses. We used two types of optical delay lines as shown in Fig. 1 to make the system capable of measurement of hollow optical fibers that were long enough for practical uses. One of the delay lines was a movable mirror mounted on a motorized linear translation stage that gave the maximum scanning range of around 1300 ps. This time delay corresponds to an optical path of 39 cm, and therefore, with only this delay line, the length of fibers that could be measured was limited to shorter than 15 cm considering additional optical paths for launching and detecting THz waves for the fiber as shown in Fig. 1. Then we added an optical delay line providing additional delay of 1300 ps. By inserting this delay, we obtained the temporal waveforms from the fiber with length up to 45 cm. We used a THz detector composed of a ZnTe electro-optic crystal. The THz wave intensities were detected as a current corresponding to the birefringence of the probe light generated by the Pockels effect induced from the detected THz waves. This balanced detection by using an electro-optic detector gives higher dynamic range, broader sensitivity in the THz region, and less sensitivity to misalignment, and these characteristics are important for measuring a hollow optical fiber with some transmission losses.

 

Fig. 1. Schematic of measurement setup.

Download Full Size | PDF

We used a hollow optical fiber made of flexible polycarbonate tubing with a thin silver layer on the inside in our experiment. In our previous experiment, we found that the thickness of the silver layer was around 200 nm and confirmed that it was thicker than skin depth of silver, which is about 65 nm at 1.5 THz [17]. The fiber with an inner diameter of 3 mm and a length of 42 cm was kept straight in this experiment. The bore of the fiber was purged with nitrogen gas, and the humidity in the entire optical path was kept to less than 3.0% to suppress the effect of water vapor absorption. Emitted THz pulses were focused into the hollow optical fibers by using a polymethylpentene polymer lens with a focal length of 10 cm.

3. ANALYSIS OF PROPAGATION MODES IN HOLLOW OPTICAL FIBERS

A. Propagation Loss

Before measurement of fiber transmission, a reference pulse form was obtained with the fiber removed from the optical axis, as shown in the inset of Fig. 1. The overall light path length was the same as that used for measurement of optical fibers. Figure 2 shows a temporal waveform of a reference pulse in which the hitting time of the THz pulse was set to zero. Figure 3 shows a frequency spectrum of the reference pulse shown in Fig. 2 calculated using Fourier transform. In the calculation, a Hamming window was applied to a time region from $-10$ to 60 ps, and the noise level was set to 0 dB. The sharp dips on the spectrum come from the absorption of water vapor in air, and it was found that the frequency region that can be measured with this system was around 0.1 to 3 THz.

 

Fig. 2. Temporal waveforms of THz waves of reference.

Download Full Size | PDF

 

Fig. 3. Frequency spectrum of THz waves of reference.

Download Full Size | PDF

Figure 4 shows a temporal waveform of the THz pulse after being transmitted in the fiber with a 3-mm diameter and 42-cm length. Compared to the waveform shown in Fig. 2, it was found that the amplitude of the THz pulse was attenuated by transmission in the fiber and that dispersion of the fiber broadened the pulse. Figure 5 shows a frequency spectrum of the pulses transmitted in the fiber. The dynamic range of the measurement system is higher than 60 dB at 1 THz, and fiber transmits a wide THz region from around 0.2-3 THz.

 

Fig. 4. Temporal waveforms of THz waves after fiber propagation.

Download Full Size | PDF

 

Fig. 5. Frequency spectrum of THz waves after fiber propagation.

Download Full Size | PDF

Figure 6 shows a measured loss spectrum that is the difference between the two transmitted power spectra shown in Figs. 3 and 5. A clear loss spectrum with a reasonably high SNR was obtained between 0.2 and 2.5 THz due to the THz-TDS system being optimized for evaluation of hollow-core fibers. The loss increased at frequencies lower than 0.5 THz because the size of the focused THz spot increased at longer wavelengths, which caused larger coupling losses. The loss peaks at around 1.2, 1.4, and 1.7 THz were thought to be due to interferences between modes propagated in the fiber, and the lowest loss of 4.0 dB was obtained at 1.3 THz. These interference fringes of hollow optical fibers were, as far as we know, clearly observed for the first time due to this THz-TDS system and the high coherency of THz waves in the system.

 

Fig. 6. Measured loss spectrum of hollow optical fibers.

Download Full Size | PDF

B. Analysis of Propagation Modes

When the Gaussian beam of the THz wave of the linear polarization is incident into a circular hollow waveguide made of metal, it is well known that the ${\mathrm{TE}}_{11}$ mode with a normalized lateral phase constant of $u=1.84$ is excited as the fundamental mode. The ${\mathrm{TM}}_{11}$ mode ($u=3.83$) and ${\mathrm{TE}}_{12}$ mode ($u=5.33$) are the second and third modes, respectively. Therefore, we calculated a theoretical transmission loss taking the interference between the ${\mathrm{TE}}_{11}$ and ${\mathrm{TM}}_{11}$ modes into account.

Coupling efficiencies $\eta$ between an incident Gaussian beam with a radius $w_0$ and each transmission mode are calculated using the following equation, where $2T$ is the diameter of the fiber, $k_0$ is the wavenumber in vacuum, and $J_0$ and $J_1$ are the zero and first-order Bessel functions, respectively:

Attenuation constants $\alpha$ and phase constant $\beta$ of the modes in the metal hollow-core waveguide with complex refractive index $n–jk$ are derived as

We define the amplitude $A$ of each mode after transmitting along a fiber with a length $z$ as

The averaged power $P$ of the two modes after transmitting in the fiber is calculated using the following equation, taking the mode interference into account:

Theoretical transmission losses of the mixed ${\mathrm{TE}}_{11}$ and ${\mathrm{TM}}_{11}$ mode calculated from Eq. (6) are shown in Fig. 7. The peaks in the theoretical spectrum coincide well with those in the measured spectrum. From these results, we confirmed propagation of the ${\mathrm{TE}}_{11}$ and ${\mathrm{TM}}_{11}$ modes in the hollow optical fibers.

 

Fig. 7. Measured and theoretical loss spectra of hollow optical fibers.

Download Full Size | PDF

C. Short-Time Fourier Transform

Dominant propagation modes could be identified through propagation mode analysis based on the loss spectra, as shown previously. In the next step, we conducted further analysis of the propagation modes using a short-time Fourier transform to identify other higher-order modes that have little effect on the loss properties. In the calculation of short-time Fourier transform, a Hamming window with a width of 5 ps was moved in steps of 0.25 ps on the temporal waveforms. As a result, a map of frequency spectra after receiving THz pulses was obtained, as shown in Fig. 8.

 

Fig. 8. Frequency spectra map of transmitted THz pulse.

Download Full Size | PDF

The cutoff frequency ${\omega }_c$ of the mode having a mode constant $u$ in the hollow optical fiber is expressed as

For the fiber diameter $2T$ of 3 mm, the cutoff frequency of the ${\mathrm{TE}}_{11}$ mode is derived as around 0.06 THz. Then the group velocity $v_g$ of the pulse in the fiber is

Theoretical group delays of the ${\mathrm{TE}}_{11}$ and ${\mathrm{TM}}_{11}$ modes calculated using Eq. (9) are also shown in Fig. 8. Comparing the measured values with the theoretical ones, one can see a clear trajectory of the ${\mathrm{TE}}_{11}$ mode in the measured one, and another trajectory coinciding with the ${\mathrm{TM}}_{11}$ mode appears at small delay times. From these results, we confirmed that the other modes were not propagated in the fiber.

4. GROUP VELOCITY

One of the advantages of TDS is that one can obtain information on phases of THz waves in addition to that on amplitudes. We derived the $v_g$ of THz-wave pulses propagated in the fiber from phases obtained from the Fourier transform of the temporal waveform of transmitted pulses.

The group velocity $v_g$ is derived by Eq. (10) from propagation constant $\beta$ and frequency $\omega$,

For the fiber with a length $L$, the phase difference of the THz pulse before and after propagating in the fiber is expressed as $\beta L$, and that of the reference pulse transmitted in the free space is $k_{0}L$. Therefore, propagation constant $\beta$ is derived from the phase of the reference measurement ${\phi }_{\mathrm{ref}}$ and that of the fiber measurement ${\phi }_{\mathrm{fiber}}$ as

We calculated the $v_g$ of the THz wave transmitted in the fiber as a function of frequency by using Eq. (10) with phase information obtained from the temporal waveforms shown in Figs. 2 and 4 with Eq. (11). For comparison, theoretical curves of the $v_g$ values of the ${\mathrm{TE}}_{11}$ and ${\mathrm{TM}}_{11}$ modes calculated from Eq. (8) are also shown in Fig. 9. The $v_g$ values were normalized by the speed of light. The measured $v_g$ dropped in the low frequency region due to the mode cutoff, and the measured curve coincided well with the theoretical $v_g$ of the ${\mathrm{TE}}_{11}$ mode at the low frequencies. This phenomenon was well explained by the result shown in Fig. 8, where the fiber supported only the ${\mathrm{TE}}_{11}$ mode in the frequency range lower than 1 THz. In contrast, some fluctuations and spike noises that were due to mode transitions and mode hopping were observed in the frequency region higher than 1 THz.

 

Fig. 9. Normalized group velocity.

Download Full Size | PDF

5. DEMONSTRATION OF REMOTE THZ-TDS WITH HOLLOW OPTICAL FIBER

Finally, by using the THz-TDS system for evaluation of hollow optical fibers, we remotely measured an absorption spectrum of a pill containing theophylline, known to show an absorption peak in the THz region. In the experiment, a pill with a thickness of 1 mm and diameter of 8.1 mm containing 24 mg of theophylline was used as a sample. The sample also contained lactose as a diluent base, cellulose as a binder, and other materials that did not affect the absorption peak of theophylline in the THz region. Figure 10 shows a schematic of the measurement setup used for remote spectroscopy. A hollow optical fiber with an inner diameter of 3 mm and length of 35 cm was used, and the sample was put at the distal end of the fiber. A gold-coated mirror was set behind the sample to reflect the transmitted THz wave; therefore, the THz pulses passed through the sample twice, and they were directed to the detector using a beamsplitter. The sample thickness of 1 mm was chosen to obtain reasonably high SNR in the measurement of absorption spectra. We also measured the sample’s absorption without the fiber for comparison. In this measurement, the sample was set at the focal point of the lens of the setup.

 

Fig. 10. Schematic of measurement setup used for remote spectroscopy.

Download Full Size | PDF

Figure 11 shows a measured absorption spectrum of the sample measured by using a hollow optical fiber. A clear absorption peak at 0.98 THz was observed, and the peak frequency coincided with the data in a previous study [26]. We confirmed that the intensity of the peak was the same as that measured without the fiber. The small peaks in the frequency range of 1.2–1.5 THz seemed to be absorption peaks of components other than theophylline because they also appeared in the spectra measured without the fiber. The SNRs defined by the peak intensity at 0.98 THz and the noise level at around 1.2 THz were 7.4 for the system with the fiber and 32 for the one without the fiber. From this experiment, we confirmed the feasibility of a remote THz-TDS system using hollow optical fibers.

 

Fig. 11. Absorption spectrum of theophyllyne.

Download Full Size | PDF

6. CONCLUSION

By using a THz-TDS system specially designed for evaluation of hollow optical fibers, the propagation properties of THz pulses in hollow optical fibers were investigated. A loss spectrum of a 42-cm-long hollow silver optical fiber was measured in the frequency range from 0.2 to 2.5 THz. On the loss spectrum, we found some loss peaks that were due to interference between propagation modes. These interference peaks were, as far as we know, clearly observed for the first time by carefully optimizing the system for measurement of the fiber having enough length to exhibit interference of propagated modes. Those modes were identified as the fundamental ${\mathrm{TE}}_{11}$ and second-order ${\mathrm{TM}}_{11}$ modes by theoretical evaluation. The existence of these two modes was also supported by the maps of frequency spectra calculated from the short-time Fourier transform. From information on the phases of THz waves transmitting in a hollow optical fiber, we also derived the group velocity dispersion of the fiber and found that the group velocity is determined mainly by the fundamental ${\mathrm{TE}}_{11}$ mode.

We showed that, from these experimental and theoretical results, a hollow optical fiber is appropriate as a delivery medium of THz waves used in time-domain spectroscopy. To show its feasibility, we conducted remote spectroscopy of a theophylline pill and successfully obtained the absorption spectrum having absorption peaks around 1 THz. In this investigation, straight metal hollow-core fibers were used for simplicity. We are currently working on evaluation of the mode properties in bent waveguides. We have also been developing metal hollow optical fibers with an inner dielectric coating that show much lower losses than metal-only hollow fibers. The test results using these fibers will be reported elsewhere.