A compact and high quality terahertz photonic crystal switch is proposed in silicon. The switch operates based on the self-imaging principle in multi-mode photonic crystal waveguide of triangular lattice. The finite-difference time-domain method and the plane wave expansion method are used to verify and analyze the characteristics of the proposed switch. Numerical simulation results agree well with the theoretical expectation. This kind of device is potentially important for terahertz application and might be a new breakthrough to design other kinds of photonic crystal switches.
©2006 Optical Society of America
Terahertz (THz) radiation, which bridges the gap between optical region and microwave, offers significant scientific and technological potential in many fields. It has attracted much attention in last decade. With the realization of THz generator and detector [1,2], THz technology develops rapidly and now involves from semiconductors, medical images, military detection, to high speed communication, chemical and biological sensing [3–6]. However, if we still utilize existing waveguide technology to the THz radiation wave, the size of the device would be extremely huge. Owing to silicon (Si) is transparent to waves below 10 THz , an integrated and compact Si-based platform for THz signal propagation, guidance, manipulation and readout would be preferable.
Fortunately, photonic crystal (PC) opens a new opportunity for this critical issue because of its capability to control the electromagnetic wave in a small region of photonic band gap (PBG). Therefore, many researchers seek their new ways in the THz regime from photonic crystal technology [8–12]. Recently, well-established fabrication technologies and advanced numerical methods have enabled the sophisticated engineering of PC dispersion characteristics and PC devices, like power splitter and all-optical switch [13–16]. However, to gain robust performance of devices, most works are assumed to operate under a single mode condition. In fact, multi-mode interference (MMI) devices are important components for photonic integrated circuits due to their simple structure, low loss and large optical bandwidth [17,18]. The operation of the MMI devices is based on the self-imaging principle, a property of multi-mode waveguide, that an input field profile is reproduced at regular intervals along the propagation direction. Kim et al. reported that the self-imaging principle is available in the multi-mode photonic crystal waveguide (PCW) of square lattice . But they do not tell us whether it still holds true in a triangular lattice multi-mode PCW.
In this paper, we show that, in the triangular lattice PCW structure, the self-imaging principle is still valid. Numerical computation with the finite-difference time-domain (FDTD) method and the plane wave expansion (PWE) method  are used to simulate and analyze such phenomenon in the multi-mode PCW. Simulation results agree well with the theoretical expectation. As an application, a THz PC switch is proposed using two-dimensional (2D) Si PC of triangular lattice based on the self-imaging and wave interference principles.
2. Guided mode property
Figure 1 shows the structure of the Si-based THz PC switch with a MMI region. The structure is a high-resistivity Si slab with air holes forming a triangular lattice. The thickness of the Si slab is 525 μm. The high-resistivity Si is transparent at THz frequencies with a refractive index of 3.42 . In Fig. 1, the PC lattice constant is a = 65.7 μm while the air hole radius is r = 21.0 μm. In this configuration, the ratio of the air hole radius to the lattice constant is r/a = 0.32. The structure consists of two types of PCWs, i.e. PCW1 and PCW2, created by removing one row and four rows of the air holes along the Γ-K direction of the crystal, respectively. PCW1 functions as input waveguides (W1, W2) and output waveguides (W3, W4), while PCW2 serves as the MMI region. To calculate the dispersion properties of the structure, an effective refractive index of 2.8 is obtained according to the method reported in reference . Based on the band gap diagram analysis of the THz PC structure by using the 2D PWE method, there is no band gap for TM mode. For TE mode, a band gap is found between normalized frequencies of 0.26(a/λ) and 0.34(a/λ) as shown in Fig. 2. The corresponding wavelength ranges from 193.2 μm (1.553 THz) to 252.7 μm (1.187 THz). It should be noticed that all the THz waves discussed in this paper are TE mode.
Before launching an input THz field into the multi-mode PCW2, the properties of guided modes in the PCWs are calculated, which strongly depend on the number of modes, propagation constants, and field patterns. Figure 3 shows the calculated dispersion curves of guided modes in the input/output waveguides (PCW1) and the MMI region (PCW2), respectively. The computational super-cells for the PCW1 and the PCW2 are on the right side of Figs. 3(a) and (b), respectively. Further investigation shows that the propagation constant increases when the air holes become smaller.
Figure 3(a) indicates that the PCW1 supports one guided mode when the frequency is located between 0.305(a/λ) and the top of the band gap, while Fig. 3(b) indicates that the PCW2 supports two guided modes from 0.311(a/λ) to 0.323(a/λ). In order to have a strong confinement of light in our structure, we choose an operating frequency of 0.317(a/λ), corresponding to a wavelength of 207.2 μm (1.448 THz). Table 1 lists the mode properties of the PCW2 at a specified frequency.
3. Self-imaging phenomenon in the PCW2
According to the self-imaging principle in conventional multi-mode waveguides , the input single mode field will mirror itself at the length of L = p(3Lπ), where p (= 0, 1, 2, …) denotes the periodic nature of imaging along the multi-mode waveguide, Lπ is defined as the beat length of the two lowest-order modes and is expressed as:
It should be emphasized that for the square lattice PCW and the conventional waveguide [20, 23], the boundaries of the multi-mode area are perpendicular to the propagating direction. But in our case, there is a 60° acute angle between the boundary and the propagating direction. In other word, the PCW2 is an inclined MMI region and its boundary lies at an angle of 60° to the propagating direction. Therefore, we define L tr as the length of the PCW2 as shown in Fig. 4. For L tr = L = 22.5a, a length of L z = 26a in z direction can be easily derived from plane geometry.
Another point we concerned is the phase relationship between the input and the output PCWs. By transferring the configuration of Fig. 1 directly to the FDTD computational domain, the phase shift of the THz wave can be obtained after propagating one repeat distance L tr = 22.5a (L z = 26a). We assume that the original phases of the THz waves from the W1 and W2 are φ 1-in = φ 2-in = 0, the phases in the W3 and W4, induced by the THz wave coupled from the W1, are obtained to be φ 13 = π/2 and φ 14 = π. Similarly, the phases in the W3 and W4, induced by the THz wave coupled from the W2, are obtained to be φ 23 = π and φ 24 = π/2.
4. Switch design
In this section, a THz PC switch in the 2D Si is proposed based on the self-imaging principle. Since the THz wave is an electromagnetic wave, interference phenomenon of waves is still valid in the THz regime. Therefore, when the two THz waves with same polarization (TE mode) and amplitude (A 0) interfere, the total wave amplitude (A) can be expressed as:
where Δφ is the phase difference. From Eq. (2) we can see that A = 0 when Δφ = mπ (m = 1, 3, 5 …) and A = 2A 0 when Δφ = nπ (n = 0, 2, 4 …).
Figure 5 shows the schematic diagram of the proposed THz PC switch based on the structure of Fig. 1. The length of the MMI region is 26a and the shape of the MMI region is trapezium. This is to ensure a symmetric field distribution when the THz wave is coupled into any one or two of the W1 and W2. In our simulation, we suppose that all the incident waves (signal wave and control wave) are with same wavelength (0317 a/λ) and polarization (TE mode). As an example, we regard the THz waves coupled into the W1 and W2 as the signal and control waves, respectively. In order to realize the switching function by wave interference, a certain phase difference Δφ in must be introduced between the signal wave and the control wave. If we set Δφ in appropriately to satisfy Δφ = mπ (m = 1, 3, 5 …) or Δφ = nπ (n = 0, 2, 4 …), the switch can shift the input signal coupled into the W1 to the output waveguide W3 (BAR state) or W4 (CROSS state). The desired Δφ in is listed in Table 2. In Table 2, A 3 and A 4 are the wave amplitudes in the W3 and W4, respectively.
5. Simulation results and discussion
We use the 2D FDTD method to prove and evaluate the properties of the THz PC switch. Since a finite structure is considered, the whole computational domain is surrounded by perfectly matched layers to absorb the outgoing waves. The Gaussian modulated pulse is launched at the input waveguides with a half width of 0.5a.
First, we transfer the configuration of Fig. 4 to the FDTD computational domain for numerical calculation. When the Gaussian modulated pulse with the frequency of 0317(a/λ) is launched into the W1, the distribution of time-averaged Poynting vector can be obtained after sufficient time steps and a single image is reproduced at the length of 26a in the propagation direction, which validates our assumption. Second, we transfer the THz PC switch structure of Fig. 5 to the FDTD computational domain for property evaluation. Figure 6 shows the steady-state magnetic field distributions of the switch. We can see that once the control wave is introduced, the field distribution in the MMI region is changed and hence the output state of the device can be controlled. When the phase difference between the signal wave and the control wave is π/2, the signal wave will be exported to the W3 (BAR state) as shown in Fig. 6(a). If the phase difference is changed to 3π/2, the signal wave will be outputted to the W4 (CROSS state) as shown in Fig. 6(b).
To evaluate the characteristics of the switch, normalized output powers in the W3 and W4 are calculated and listed in Table 3. P 3-B and P 4-B indicate the normalized output powers in the W3 and W4 when the switch is at the BAR state, respectively while P 3-C and P 4-C are the normalized output powers in the W3 and W4 corresponding to CROSS state, respectively. Theoretical extinction ratio for the BAR state, defined as 10log(P 3-B/P 3-C), is 16.5 dB and for the CROSS state, defined as 10log(P 4-C/P 4-B), is 14.7 dB. The crosstalk at the BAR state, defined as 10log(P 4-B/P 3-B), and at the CROSS state, defined as 10log(P 3-C/P 4-C), are -14.6 dB and -16.6 dB, respectively.
The applicability of the self-imaging principle in the multi-mode PCW consists of triangular lattice has been validated. Numerical simulation results agree well with the theoretical expectation. As an example of application, a THz PC switch based on the self-imaging principle is proposed and characterized. Theoretical calculation shows that the transmission is up to 95.2% with an average crosstalk of -15.6 dB. This kind of device is potentially important for THz application and might be a new way to design other kinds of PC switches.
This work was supported by the National Natural Science Foundation of China (Nos. 90401008, 60577001), the Research Fund for the Doctoral Program of Higher Education (No. 20040558009), and the Program for New Century Excellent Talents in University (No. NECT-04-0796).
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