Transmission and dispersion relation of THz waves in two-dimensional photonic crystal (PC) composed of metal rods are studied by using finite-difference time-domain simulation and THz time-domain spectroscopy measurement. The PC is embedded in a parallel metal plate waveguide with an air gap between the PC and one of the plates. The photonic-band-gap well-defined at small air gap narrows systematically with opening the air gap and disappears when the air gap is 2 ∼ 3 times the rod height, where the two-dimensional nature of PC is destroyed. The mechanical tunability of photonic band structure would be useful in functional THz device.
© 2012 OSA
Artificial periodic structure known as photonic crystal (PC) allows us to design high performance or functional devices such as lasers, sensors and optical switches [1–3]. It also provides us a superior platform for the fundamental physics of light-matter interaction . On the other hand, special attention has been recently paid to the application of THz waves in biological and pharmaceutical sciences, and security , to which the use of PCs are expected to improve the performance of the devices. Although THz PCs have been already studied, aiming at a low loss waveguide [6–8] and a high-quality resonator [9–11], highly functional PCs like a mechanical tunable filter  or an optical modulator  are still under development.
If one focuses on one-dimensional (1D) or two-dimensional (2D) PCs due to their easy fabrication, electromagnetic wave confinement in another dimension is necessary. In THz frequency range, metal has rather low absorption and is attractive as wave confinement material in addition to PC material itself. The confinement is achieved by a waveguide structure like parallel metal plate waveguide (PPWG) . Bingham et al.  have studied a 2D PC embedded in PPWG, confining THz wave with polarization parallel to the PC rod-axis (transverse magnetic (TM) mode). Recently, metal grooves equivalent to 1D PC inside a PPWG has been investigated by Lee et al . They reported the THz transmission properties with control of air gap between the metal plates and the tunability of the stop bands. The results indicate that a metal PC with PPWG is a possible highly functional THz device, and stimulated us to explore the tunability of the band structure in 2D metal PC in PPWG.
It is known that 2D PCs embedded in PPWGs with no air-gap show a cut-off phenomenon . The existence of an air gap allows a wave propagation and photonic band formation in the frequency region below the cut-off. Then the important issue in the 2D PC is how the air gap affects the band structure. In the limit of small air gap, the system may be viewed as a 2D PC. However opening air gap introduces the non-uniformity along the rod-axis direction and destroys the 2D nature. So the different band structure may appear at wide air gap. The other THz property to be investigated is the dispersion relation, which has not been reported but provides an important information on the wave propagation such as the group velocity. The tunability of group velocity is attractive to control the THz wave propagation on demand.
In this paper, we report the THz propagation properties in the 2D metallic PC with the air gap in detail. The mechanical tunabilities of the band structure and the dispersion relation of photonic band are confirmed by both simulations and THz time-domain spectroscopy (TDS) measurements.
The simulated PC structure is illustrated in Fig. 1. A 2D square lattice of Au rods with 35 μm height is embedded in a PPWG of Au to confine THz wave along the rod axis. The device has a controllable air gap between the top surfaces of the rods and the upper metal plate.
The commercial FDTD software (Mizuho Information & Research Institute Inc., EMERGE) was used to simulate the THz wave propagation. A 200 fs-width Gaussian-pulse or a continuous wave (CW) was used as the excitation source. The polarization of THz electric field was set to be parallel to the rod axis, namely the TM polarization. The electrical conductivity of Au was assumed to be 4.5×107 Ω−1m−1. THz electric field spectrum, EPC, was obtained by the Fourier transformation of temporal wave form after 9 mm propagation from the source. The power transmission, T, is defined as follows:
The Fourier transformation also provides the phase, ϕPC, of the THz wave propagating in the PC, which was used for constructing the band diagram. If the reference phase of Eref associated with the fundamental TEM mode of PPWG is known, the wave vector, kPC, is approximately given by the expression:Eq. (2), L, ϕPPWG, c and ω mean the PC-length, reference phase, speed of light and angular frequency, respectively. The third term in right-hand side of Eq. (2) is necessary to compensate the subtraction of ϕPPWG assuming TEM mode which is close to free space propagation. The straightforward application of Eq. (2) is possible for the ω-kPC relation possessing the minimum frequency at the Brillouin zone edge (Γ, X or M). For the second photonic band (PB2), we added π phase shift in standing waves between the first photonic band (PB1) and PB2 .
Figure 2 shows an example of T for an air gap of 30 μm. The lattice constant, a, and the rod radius, R, are 120 μm and 30 μm, respectively. The air gap leads to the disappearance of the cut-off (fc ∼ 2.5 THz) observed without the air gap (not shown) and a band structure emerges for f < fc. In the particular case shown here, the first band gap appears in the range 1∼1.4 THz in the Γ-X direction and 1.2∼1.4 THz in the Γ-M direction. We note here that fc is higher than c/2a of the fundamental transverse-electric-field mode in a rectangular waveguide with the width of a. In the case of PC, the effective a taking into account R is shorter than a and leads to higher fc. By considering R = 30 μm, the minimum waveguide width is 60 μm, which corresponds to the fc of 2.5 THz.
The two lowest pass bands are assigned as shown in Fig. 2. For the Γ-M direction, PB2 would be a mode uncoupled to the excitation wave, following the band diagram of square dielectric-rod PC . Based on the reported band diagram , the plausible bands between 1.5 and 2.2 THz are the third (PB3) and the fourth (PB4) bands, both of which are the modes coupled to the excitation. Because PB3 and PB4 are not well separated, we describe them as PB3+PB4. Rather low T of PB3+PB4 below 2 THz is also reported in a square dielectric-rod PC , of which the physical origin is unclear.
Electric field component, Ez, of the THz wave propagating along the Γ-X direction in the PC with the 30 μm air gap are computed with the CW excitation at selected frequencies. The Ez patterns in xy plane are shown in Figs. 3(a) and 3(b) for PB1 at 0.3 THz and in Figs. 3(c) and 3(d) for PB2 at 1.5 THz (the circles are the Au rods). We present the Ez pattern at the center of the air gap (z =50 μm) in Figs. 3(a) and 3(c), and that at the center of the rod region (z =17.5 μm) in Figs. 3(b) and 3(d). In PB1, strong Ez confinement in the air gap region is found from Figs. 3(a) and 3(b), whereas Figs. 3(c) and 3(d) show that the field distributes also in the rod region at the PB2 frequency. In each band, the electric field distribution along z direction is homogeneous in both air gap and rod regions (not shown). This means that a resonance condition at the band edge does not depend on a structure parameter along z direction and the device can be regarded as a 2D-PC made of dielectric rods. Therefore the stop band frequency in a permittivity contrast given by an air gap distance would be determined by a and R.
The region between the high frequency edge of PB1 along the Γ-M direction and the low frequency edge of PB2 along the Γ-X direction gives the full photonic band gap (PBG) (see Fig. 2). Figure 4 shows the PBG map, at several air gap widths, made by scanning R with a fixed a of 120 μm. The band edge is defined as the frequency where T is equal to −10 dB. The PBG is represented by the range between the same symbols for each R/a and the air gap. Obvious PBG appears for the air gap smaller than 50 μm. Narrowing the air gap gradually widens the PBG, which originates from an increased effective permittivity contrast made by the metal rods and air. The widest PBG is obtained at R/a = 0.25 (R =30 μm), which we focus on hereafter.
The air gap dependences of T along Γ-X and Γ-M directions were calculated and summarized in Figs. 5(a) and 5(b) as contour maps. The air gap was varied by a step of 10 μm and the frequency resolution is 8 GHz. We assign the band as described in the figures. The controllability of the band structure by the air gap is clearly demonstrated. With opening the air gap, PB2 (PB3+PB4) exhibits a suppression (discontinuity) around 80 μm (70 μm) air gap in Γ-X (Γ-M) direction.
To understand the nature of the band structure, Ts for narrow (10 μm) and wide (150 μm) air gap cases are presented in Figs. 6(a) to 6(d). In the limit of small air gap, the device can be viewed as 2D PC (see the schematic drawing in Fig. 6(a)) showing remarkable stop bands around 1.25 THz. The PC with 150 μm air gap exhibits complex band structure with several narrow stop bands. The band structure resembles that of 1D metal grooves in PPWG . So the rod array in the y direction was replaced by the metal rectangular parallelepiped of the same height, and Ts of such 1D corrugated PPWGs were simulated as shown by the broken lines in Figs. 6(a) to 6(d). For the case of 150 μm air-gap, the corrugated PPWGs well reproduce the Ts of PC for both directions. In contrast, for the 10 μm air gap, the corrugated PPWGs cannot explain Ts of the band structure particularly in the higher bands. For the air gap larger than 150 μm, we checked that T is well reproduced by the 1D corrugated PPWG. It can be thus suggested that the device characteristic changes from 2D PC to 1D corrugated PPWG with opening air gap. The difficulty of adiabatic interpolation between 1D and 2D PCs would induce the anomaly in the band structure with air gap control around 80∼110 μm (70 μm) for Γ-X (Γ-M) direction, where the uniformity (and therefore the 2D nature) of the system along the rod-axis direction is highly broken.The non-uniformity along the rod-axis direction can permit a band-edge resonance condition depending on the rod height and/or the air gap distance. According to Lee et al. , several stop-band-positions of corrugated PPWG are determined by the structure parameters of groove height and air gap distance. As evidenced by Figs. 6(c) and 6(d), the same resonance conditions are shared between the corrugated PPWG and the PC. For example, in Fig. 6(c), the stop band around 1 THz is caused by a cancel of THz field between the straight propagated wave in air gap and the wave detouring around one groove (see the band gap A in ref. ). The Bragg reflection with the period of lattice constant leads to the stop band around 1.25 THz. The resonance by a standing wave with the wavelength corresponding to the distance between metal plates is responsible for the stop band around 1.6 THz.
The fabrication process of the PCs is as follows. A dry etching of Si substrate with 250 μm thickness was performed by an inductively-coupled-plasma to form the PC pattern (see Fig. 7(a)). The etching mask was an Al film patterned by conventional photolithography method. After the removal of the Al mask by wet etching, 1.2-μm-thick Au film was deposited on the surface of Si with a rotating angled holder to ensure the deposition on the side wall of the rods. Figure 7(b) is the photograph of the fabricated PC (side view) with a and R being 120 μm and 30 μm, respectively. The rod height is 40 μm which is slightly deeper than that in the FDTD simulation. The area of the rod array is 9 mm × 15 mm. The bottom plane of the rod array acts as one of the metal plates of PPWG. The other plate of PPWG was a Au-coated Si substrate prepared separately as shown in Fig. 7(c).
THz-TDS schematic set up is shown in Fig. 7(c). THz pulses were generated by a low-temperature (LT) grown GaAs interdigitated photoconductive antenna (PCA) optically excited by a mode-locked Ti:sapphire laser. This laser produces ultrafast pulses with a central wavelength of 800 nm and a duration of 100 fs at a repetition rate of 80 MHz. The generated THz pulses were focused to the minimum beam waist of 2 mm at 1 THz by a pair of off-axis parabolic mirrors. The PC was attached by a tapered input coupler  made of Al for the efficient coupling between free-space THz wave and the PC (see the expanded view of the PC in Fig. 7(c)). The other LT-GaAs PCA with a Si lens triggered by time-delayed pulses of the laser was placed just behind the output facet of the PC. The polarization of the THz wave was parallel to the PC-rod axis. The Γ-X and Γ-M propagations were separately measured by using the devices as shown in Fig. 7(c). The air gap was mechanically controlled with an accuracy of about ±10 μm.
Experimental contour maps of T is shown in Figs. 8(a) and 8(b). Experimental values of T were calculated following Eq. (1) with Eref separately measured by using PPWG without PC. The photonic bands marked by the black circles are the artifact caused by the insufficient separation between the signals and the noise floor. Since it was difficult to achieve complete zero gap in the experiments, the contour map was made for the air gap wider than 10 μm. The measured air gap dependences of the transmittance along two directions are qualitatively in good agreement with the FDTD results (Figs. 5(a) and 5(b)), demonstrating the mechanical tunability of the photonic band structure.
The ω-kPC curves for the experiment and the FDTD simulation calculated by Eq. (2) are shown in Figs. 9(a) to 9(d). The curves of higher order bands are given only for the simulation on the Γ-X direction with rather small air gap. For the other cases, it is difficult to apply straightforwardly Eq. (2), or the spectra are not enough wide and/or strong to calculate the dispersion. Both in the simulation and the experiments, all the curves of PB1 apparently deviate from the light line with k approaching X or M. The frequency at X or M in each curve coincides with the band-edge indicated by transmission spectrum (see Figs. 5 and 8) except for the case of 110 μm air gap in the Γ-M direction, where the separation between PB1 and higher band is unclear. The narrowing of the air gap systematically decreases the slope of the dispersion curve due to the increased occupancy of the metal, indicating that the control of the dispersion relation by the air gap is realized.
We can confirm a qualitative consistency between the experimental results and the simulation, particularly for the wide air gap. A very good agreement is obtained in the Γ-X direction for the air gap wider than 70 μm. The deviation between the experiment and the FDTD simulation for the narrow air gap would be due to the non-uniformity of the air gap over the whole sample area.
Although it is difficult to construct the experimental band diagram for the higher order photonic bands, since the observed bandwidth is rather narrower than expected, informative results on the dispersion can be extracted by looking at the group velocity as shown in Figs. 10(a) to 10(d). Figures 10(a) and 10(c) represent the experimental time domain traces at the 20 μm air gap along the Γ-X and Γ-M directions, respectively. We call THz pulse arriving before 25 ps fast component and that after 25 ps slow component. The Fourier spectra of these components are shown in Figs. 10(b) and 10(d). It is clearly seen that the fast component reproduces the main part of PB1 while the slow component forms PB2 (or PB3+PB4) and the edge of PB1, indicating that the group velocity in PB2 (or PB3+PB4) and near the edge of PB1 is slower than that in the region far from the edge in PB1, consistently with the band diagrams in Figs. 9(c) and 9(d).
THz transmission and dispersion relation of photonic band of the 2D metal PC embedded in PPWG have been investigated in detail by FDTD simulations and THz-TDS measurements. PBG systematically widens with narrowing air gap below 50 μm. In addition to the mechanical tunability of PBG, we have found the discontinuous change of device characteristic from 2D PC into 1D corrugated PPWG with opening air gap. The experimental results have confirmed our findings in FDTD simulation.
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