Propagation of Single-Mode and Multi-Mode Terahertz Radiation Through a Parallel-Plate Waveguide

Transcription

1 Journal of the Korean Physical Society, Vol. 53, No. 4, October 2008, pp Propagation of Single-Mode and Multi-Mode Terahertz Radiation Through a Parallel-Plate Waveguide Eui Su Lee, Jin Seok Jang, Sang Hoon Kim, Young Bin Ji and Tae-In Jeon Division of Electrical and Electronics Engineering, Korea Maritime University, Busan (Received 15 July 2008) We investigated the propagation of single- and multi-mode terahertz (THz) radiation through a parallel-plate waveguide for various gap sizes of the plates and polarizations of the incoming THz eld. Single TEM (TM 0) and TE 1 modes within the 4-THz frequency range propagated through a waveguide with a plate separation of 103 m. Multi TE and TM modes were observed for a 360-m separation. High-order modes have a very high group velocity dispersion near the cuto frequencies, which causes extensive pulse reshaping and broadening, in addition to multimode interference in the time and frequency region. The majority dominant mode is a combination of the TE 1 (97.96 %) and the TM 0 (81.61 %) modes because the lowest mode has a small absorption and an incident even eld pattern. Because each mode has dierent transmission and coupling coecients, absorption and propagation constant, the theoretical calculation for the total TM or TE modes requires a summation of the components. The theoretical calculation and the measured data t well in the frequency and time domain. PACS numbers: Bs, Ja, Et, Bz Keywords: Terahertz, Waveguide, Propagation, Parallel-plate, Mode I. INTRODUCTION Recently, ecient coupling of subpicosecond THz pulses into several waveguides has been achieved by using hyperspherical and plano-cylindrical silicon lens located at the middle of freely propagating THz beams in air [1{3]. Low-loss and single-tem-mode propagation using metallic parallel-plate waveguides (PPWGs) has been demonstrated [2{4]. Air traveled THz time-domain spectroscopy (THz-TDS) has been used as a powerful tool to scientists [5, 6]. The PPWG is also very interesting research topic for THz-TDS because most of the THz energy is concentrated in the parallel-plate gap [7,8]. When the spaces between the parallel plates of a PPWG are lled with a dense dielectric medium, the low loss and non-dispersing TEM mode is used for THz waveguide spectroscopy to study biosensing [9, 10], chemical materials [11,12] and photonic waveguide [13,14]. In addition, a PPWG can be used for THz surface plasmon coupling to a long metal sheet [15]. Plate separation is an important parameter to determine the number of modes. The parallel-plate separation and the polarization of the incoming electrical eld relative to the plate's surface determine the number of modes and the type of modes. In this study, we report the characteristics of THz radiation through a PPWG lled with air for 103-m and 360-m plate separations for TE and TM modes, which depend jeon@hhu.ac.kr; Fax: whether the incoming electrical eld is perpendicular or parallel to the plate's surface. II. EXPERIMENT The experimental setup shown in Figure 1 consists of a photoconductive GaAs transmitter and silicon-onsapphire receiver chips to generate and detect THz beam, respectively. The Si lenses located at the back side of the chips and the paraboloidal reectors align the THz beams. The PPWG is located between the two reectors. The incoming THz beam is horizontally polarized and focused into the gap of the parallel plates by using a plano-cylindrical Si lens. In the PPWG, the aluminum block is 38-mm wide, 29.7-mm long and 20-mm thick. The waist of the THz beam due to the plano-cylindrical Si lens is about 120 m. When the PPWG with a 103- m air gap is set parallel to the polarization of the THz beam, the THz pulse measured in the time domain is shown in Figure 2(a). Based on the well-known wave equations, Maxwell's equations and the boundary conditions, only TE modes can exist in the PPWG. Because of the high group-velocity dispersion near the cuto frequency, the time domain pulse is stretched and expands to more than 150 psec with negative chirp. The lower frequencies travel slower than the higher frequencies. The lower inset gure shows the extension of the oscillation from 2 psec to 13 psec. Figure 2(b) shows the spectrum

2 Journal of the Korean Physical Society, Vol. 53, No. 4, October 2008 Fig. 1. Schematic diagram of the THz parallel-plate waveguide system. mode with low-loss and negligible group-velocity dispersion [2,3]. The inset of Figure 2(b) shows the TEM-mode spectrum without any cuto frequencies. The cuto frequencies of the TM and the TE modes are given by [16] f c = mc 2d ; (1) Fig. 2. (a) Measured TE 1-mode THz pulse transmitted through a parallel plate with a plate separation of 103 m when the polarization direction of the input electric eld is parallel to the plates. Upper inset: Measured TM 0-mode THz pulse when the polarization direction of the input electric eld is normal to the plates. Lower inset: Extension of the oscillation from 2 psec to 13 psec. (b) Relative amplitude spectrum of the TE 1-mode pulse. The inset shows the spectrum of the TEM (TM 0)-mode THz pulse. of the time-domain THz pulse. The low frequency is truncated until 1.46 THz, which is the rst cuto frequency to show up. The spectrum then extends up to 4 THz. When the PPWG is rotated 90 degrees and made perpendicular to the incoming THz beams polarization, the measured THz pulse is as shown in the upper inset of Figure 2(a). The THz pulse has only TEM (TM 0 ) where m is the number of high-order modes, c is the speed of light and d is the gap between the parallel plates. When d is set 103 m, the TM and the TE modes have cuto frequencies of 1.46 THz and 2.91 THz for the rstand the second-order modes, respectively. The vertical dashed lines in the spectrum domain show the cuto frequencies. If the separation between the parallel-plates gets bigger, the cuto frequency shifts to a lower frequency, as shown in Eq. (1). When the air gap is m, the third-order TE mode comes out at 4 THz, which is the frequency limitation for 4-THz bandwidth to get a single-mode THz eld. Because the receiver chip is a dipole antenna structure to detect the incoming THz eld, only the even incoming THz eld pattern can be detected by using the antenna. If the incoming THz eld pattern is odd, the integration of the THz eld coupled to the antenna is zero. Therefore, Figure 2(b) shows the rst-order TE mode (TE 1 mode), which is an even eld pattern with a 1.46-THz cuto frequency. The calculated cuto frequency and the experimental result agree well. The second-order TE mode (TE 2 mode) does not show up in the spectrum because of the odd eld pattern. Since the TEM and the TM 2 modes are even eld patterns and the TM 1 mode is an odd eld pattern, the fundamental mode can be detected as TEM and TM 2 modes. However, the TM 2 mode has a very high absorption just after the cuto frequency, which gradually decreases to high frequencies (see the inset Figure 3(b)) and it has a slow group velocity. For these reasons, the TM 2 mode component is very small and shows up far away from the TEM mode in the time domain because the spectrum shows up only on the TEM component, as shown in inset Figure 2(b).

3 Propagation of Single-Mode and Multi-Mode { Eui Su Lee et al III. MODE ANALYSIS If there is no dispersion material in the beam's path, the phase velocity V and the group velocity V g of a particular mode depend on the wavelength. The analytic expressions for the phase velocity and the group velocity in the PPWG are given by [16] V = V ; (2) [1 (= c ) 2 1=2 ] V g = V [1 (= c ) 2 ] 1=2 ; (3) where V = p 1 and c is the wavelength at the cuto p frequency. The speed of light can be expressed as V V g. At the cuto wavelength ( = c ), the phase velocity is innite and the group velocity is zero. When the wavelength decreases from the cuto wavelength, the phase velocity and the group velocity approach the speed of light in free space. Figure 3(a) shows the phase and the group velocities for a 360-m gap and the inset curve shows the group velocities for a 103-m gap. The group velocity of each mode gets faster and approaches the speed of light with increasing frequency. The frequency-dependent absorption for each TEM and TM and each TE mode are given by [16] = 1 d < for T EM; (4) = 2 d < 1 p for T M; (5) 1 (=c 2 ) = 2 d < (= c ) 2 p for T E; (6) 1 (=c 2 ) where < is the characteristic resistance and is the intrinsic impedance of the medium. The calculated absorptions for the three modes for a 360-m gap are shown in Figure 3(b). The insets are for a 103-m gap. Because of the reactive wave impedance before the cuto frequency, the amplitude for each mode should be zero in the spectrum. Since the absorptions of higher TM modes, not TEM mode, are bigger than those of TE modes, the THz amplitudes of TM modes are more attenuated than those of TE modes. The TEM mode has no cuto frequency and its absorption gradually increases with increasing frequency; also, it has the smallest absorption coecient compared with higher TM modes in all frequency ranges. IV. MULTI-MODE PROPAGATION Figure 4(a) shows the measured TE modes for a 360- m air gap. As explained for the TE mode for a 103- m air gap, the time domain pulse is stretched to more than 160 psec. Because most of the incoming THz eld Fig. 3. (a) Phase and group velocities for the rst ve modes. The inset shows the phase and the group velocities when the plate separation is 103 m. Cuto frequencies for the TM and the TE modes are shown by the dotted vertical lines. (b) Field absorption for the rst ve modes with a plate separation of 360 m. The inset shows eld absorption when the plate separation is 103 m. couples to the PPWG, the maximum amplitude of oscillations is about 160 pa, which is 6 times bigger than that of a 103-m air gap. The solid line of the inset shows the extended gure from 5 psec to 20 psec. The high-frequency oscillation components come rst and the low-frequency oscillation components come next because the group velocity of the low-frequency components is slower than that of the high-frequency components. The spectrum clearly shows a group-velocity delay, as shown in Figure 4(b). The theoretically calculated cuto frequencies for TE modes are 0.42 THz, 0.83 THz, 1.25 THz and 1.67 THz from the rst to the fourth, as shown by the vertical dash lines in the spectral domain. Due to the evanescent wave before the cuto frequency, the spectrum of each TE mode comes out after the cuto frequency. The dominant TE 1 mode starts from 0.42 THz. Most of the spectral energy belongs to the TE 1 mode. The amplitude of TE 2 mode, which is for a 0.83-THz cuto frequency, is very small because of the odd eld pattern. However, the amplitude of the TE 3 mode is relatively large compared with those of the higher-order modes because of the even eld pattern and small absorption. Also, the TE 4 mode (1.67-THz cuto frequency)

4 Journal of the Korean Physical Society, Vol. 53, No. 4, October 2008 Fig. 5. Amplitude spectrum of the measured THz pulse from 120 psec to 160 psec (inset) at a 360-m plate separation: (a) TE modes and (b) TM modes. (c) Group velocities for the rst ve TM and TE modes. The horizontal dashed lines indicate the velocities for 120 psec (0.46 c) and 160 psec (0.39 c). The widths of the dark columns indicate the resonance bandwidths of the spectra. Fig. 4. Measurements (solid lines) and theoretical predications (dots) for a parallel-plate at a 360 m- separation: (a) TE-mode THz pulse in the time domain. The inset shows an extension of the oscillation from 5 psec to 20 psec. (b) Amplitude spectrum of the TE-mode THz pulse. The envelope of the amplitude spectrum indicates the TE 1 mode. The dashed thick circles show mixed high modes without the TE 1 mode. (c) TM-mode THz pulse in the time domain. (d) Amplitude spectrum of the TM-mode THz pulse. The envelope of the amplitude spectrum indicates the TM 0 mode. The dashed thick circles show mixed high modes without the TM 0 mode. has a dierent oscillation frequency than the TE 3 mode. There are a total of 9 modes up to the 4-THz frequency range. Because of high absorption and the slow group velocities, the higher-order modes are dicult to observe in the measured spectral domain. Figure 4(c) shows the measured TM modes when the waveguide is rotated 90 degrees. The rst shown THz pulse indicates the TEM mode and the delayed oscillation pulses indicate higher TM modes because of their slow group velocities, as shown in the upper inset gure (solid line). The maximum amplitude of the oscillations is about 30 pa, which is much smaller than that of the TE mode. The solid line of the lower inset gure shows the expanded oscillation from 40 psec to 55 psec. The two dierent-amplitude THz pulses are periodically repeating, which indicates that at least two dierent TM modes with very low group velocities are propagating. Figure 4(d) shows the corresponding amplitude spectrum for the THz pulse. The spectrum clearly shows the TEM (TM 0 ) mode (envelope of the spectrum) without a cuto frequency and higher TM modes with a cuto frequency. The small TM 1 mode, which is an odd eld pattern, comes out at 0.42-THz and the large TM 2 mode, which is an even eld pattern, comes out at 0.83 THz. In order to reveal the major higher-order modes in fre-

5 Propagation of Single-Mode and Multi-Mode { Eui Su Lee et al Table 1. Percentage of the coupled power in each mode. Mode 0(TEM) TE TM quency domain, a spectrochronography method, which an ecient tool to extend the THz time domain technique [17], is used. The insets in Figure 5 show THz pulse data from 120 psec to 160 psec for the TE and the TM modes. The resonance spectra for the limited data are shown in Figures 5(a) and (b). The amplitude of each resonance indicates mode coupling to the waveguide. As explained for the 103-m gap, only even eld patterns, such as TE 1 and TE 3 modes, can be detected on the waveguide, as shown in Figure 5(a). However, there are small TE 2 and TE 4 mode components because the system is not perfectly aligned. The odd mode components are very small compared with the signal from the even eld pattern. The TM mode data have the same situation. Because the TM 2 and the TM 4 modes are even eld patterns, they can be detected. However, the TM 4 mode has a large absorption at high frequencies, so the detection is very small, as shown Figure 5(b). Because the alignment of the cylindrical lens to the parallel-plate gap and the alignment of the PPWG to the THz beam propagating through are not complete, small amplitude of TM 1, TM 3 and TM 5 modes (odd modes) are detected in the data. The amplitude of the TM 3 mode is relatively large compared with those of other odd-mode components, such as TM 1 and TM 5. The amplitude of the TM mode in Figure 5(b) is limited to time data from 120 psec to 160 psec. The TM 3 component is only 1.52 % for the total time data (see Table 1). Because the THz pulse for the TEM mode is located at 5 psec, as shown in Figure 4(c), the TEM mode does not show up. The positions of the mode resonances do not exist at the cuto frequencies, as shown in the vertical dashed lines. They shift to high frequencies, especially for the higherorder modes. The 120 psec and 160 psec correspond to 0.46 c and 0.39 c delays and the delays are indicated to the horizontal dashed lines shown in Figure 5(c). Each resonance bandwidth is determined by the cross section between the group velocity curves and the 0.46 c and the 0.39 c lines. The dark column indicates this relationship. The column width is small in the low-frequency range because the group velocity curve is much deeper at low frequencies. The total detected complex amplitude spectrum, E out (!), is obtained from the given complex reference spectrum, E ref (!), the transmission and coupling coef- cients, the absorption and the dispersion for each mode to the waveguide. The relationship to the parameters for multi-mode propagation is given by E out (!) = E ref (!) X m T m C 2 me j(g;m o)l e ml ;(7) where T m and C m are the total transmission and coupling coecients to the entrance and the exit waveguide, respectively, m is the amplitude absorption constant, g;m is the propagation constant and o is the phase constant. The mode- and the frequency-dependent absorption and dispersion have already been shown in Figure 3. Because of the multi-mode propagation to the waveguide, the parameters are mode dependent. Therefore, each mode component should be added to get the detected complex amplitude spectrum E out (!). The dashed curves in Figures 4(b) and (d) show the theoretical calculations. There are slight dierences between the calculation and the measurement because of the limitation of the measured data. The THz pulses reection takes place after 160 psec in the measurement, that is, from the end of the transmission line of the chips. Therefore, the data take until 160 psec, but the multi-mode oscillations go on after 160 psec because of the very low group velocities. However, in the tting, the periodic oscillation and the relative amplitude in the high-frequency range are well tted in the frequency domain. Using an inverse fast-fourier transformation, the calculated THz pulses are tted well to the measured THz pulses along the time domain. Because of the good t to the measurements, only expanded scales from 5 psec to 20 psec and from 40 psec to 55 psec for the TE and TM modes are shown in the insets of Figures 4(a) and (c), respectively. In this calculation, the total contributed power in each mode is indicated in Table 1. The dominant TE mode is % TE 1 mode and 1.73 % TE 3 mode. The other odd modes or higher modes have very small contributions to the total coupled power. The dominant TM mode is % TEM mode and % TM 2 mode. The majority dominant mode is the lowest order mode as TE 1 and TEM (TM 0 ) modes because of the small absorption. V. CONCLUSION In conclusion, we have investigated single-mode and multi-mode THz propagations in PPWGs with 103-m and 360-m plate separations. When the THz eld's polarization is perpendicular to the PPWG, only the TEM mode can propagate at a 103-m plate separation. The higher order modes can't be detected because the modes have large absorption and out-of-frequency bandwidth in the measured spectrum. Also, because the integration of the eld pattern of odd modes is zero, even modes can be detected, including the fundamental mode. This TEM single mode of a THz pulse is good for THz waveguide spectroscopy applications. If the waveguide is rotated 90 degrees, only the TE 1 mode can propagate through the plate separation. It can't detect any THz signal below the cuto frequency because the wave impedance only has reactive components that can be used for a high pass lter for a THz system.

6 Journal of the Korean Physical Society, Vol. 53, No. 4, October 2008 We also demonstrated multiple mode propagation with a 360-m plate separation. Using the spectrochronography method for the tail part of the measured THz data, we conrm that the coupling of even the eld pattern is dominant. The majority dominant mode is the lowest mode of TE and TM modes. Therefore, each mode has dierent parameter weight factors, such as transmission and coupling coecients, absorption and dispersion. In order to calculate the complex output spectrum, these parameters for each mode should be added. Finally, the calculated and the measured spectra for a THz eld are in good agreement in the time domain. ACKNOWLEDGMENTS This work was supported by a Korea Research Foundation Grant funded by Korean Government (MOEHRD) (KRF D00508) and the Korea Science and Engineering Foundation (KOSEF ). REFERENCES [1] R. W. McGowan, G. Gallot and D. Grischkowsky, Opt. Lett. 24, 1431 (1999). [2] R. Mendis and D. Grischkowsky, Opt. Lett. 26, 846 (2001). [3] R. Mendis and D. Grischkowsky, IEEE Microwave and Wireless Components Lett. 11, 444 (2001). [4] R. Mendis, Opt. Lett. 331, 2643 (2006). [5] Y.-S. Jin, G.-J. Kim and S.-G. Jeon, J. Korean Phys. Soc. 49, 513 (2006). [6] U. W. Kim, S. J. Oh, I. Maeng, C. Kang and J.-H Son, J. Korean Phys. Soc. 50, 789 (2007). [7] J. Zhang and D. Grischkowsky, Opt. Lett. 19, 1617 (2004). [8] J. Zhang and D. Grischkowsky, Appl. Phys. Lett. 86, (2005). [9] M. Nagel, P. H. Bolivar and H. Kurz, Semiconduct. Sci. Technol. 20, 281 (2005). [10] J. S. Melinger, N. Laman, S. S. Harsha and D. Grischkowsky, Appl. Phys. Lett. 89, (2006). [11] N. Laman, S. S. Harsha and D. Grischkowsky, Appl. Spectroscopy 62, 3 (2008). [12] J. S. Melinger, N. Laman, S. S. Harsha and D. Grischkowksy, Appl. Phys. Lett. 89, (2006). [13] Z. Jian, J. Pearce and D. M. Mittleman, Semiconduct. Sci. Technol. 20, 300 (2005). [14] A. Bingham, Y. Zhao and D. Grischkowsky, Appl. Phys. Lett. 87, (2005). [15] T.-I. Jeon and D. Grischkowsky, Appl. Phys. Lett. 88, (2006). [16] N. Marcuvitz, Waveguide Handbook (Peregrinus, London, 1986). [17] M. M. Nazarov, L. S. Mukina, A. V. Hjuvaev, D. A. Sapozhnikov, A. P. Shkurinov and V. A. Tromov, Laser Phys. Lett. 2, 471 (2005).