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We discuss, by a finite-difference time-domain method, a photonic crystal structure with two adjacent parallel line-defect waveguides to provoke highly-efficient directional emission of terahertz waves. The output field is analogous to a two-point-source interference pattern. Numerical simulations show that the maximum directional emission efficiency is up to 75%. Moreover, the efficiency is insensitive to transverse alignment error and angular alignment error. This structure is potentially important for high-efficiency coupling between freely propagated terahertz waves and Si waveguide systems.
©2007 Optical Society of America
1. Introduction
Recently, directional emission from photonic crystals (PCs) has attracted a lot of attention due to the discovery of enhanced transmission and directional emission from PC waveguides (PCWs), which was predicted theoretically [1] and demonstrated independently in experiment [2]. However, the reported directional emissions are inefficient because of reflection and impedance mismatch at the PCW exit. And the directional emissions in previous works also depend strongly on the PC structure termination. To address these drawbacks, many different solutions have been proposed, such as decreasing the input wavelength, using a positive surface corrugation, increasing the refractive index of surface layer, optimizing the modulated surface structure, or adding a surface corrugation with a grating-like layer to input surface of the PCW [3-5]. These directional emissions are achieved by exciting surface modes, which couple to the continuum of radiative modes via a surface corrugation. Alternatively, Chen et al. reported that directional emission can be obtained by adding two point defects near the PCW exit (producing a kind of triple-point-source) [6]. Very recently, theoretical investigations were made on the highly-efficient directional emission behaviors via self-collimation effect [7,8]. But a problem still remaining in the reported works is that the coupling efficiency is highly sensitive to the misalignment between the input light beams and the narrow PCWs [1-7]. Fortunately, self-collimation can address this issue because it does not require a physical boundary for lights confinement and enable high-efficiency coupling and routing lights without defects [9-13]. So a strict alignment for coupling lights into a narrow PCW can be avoided [11]. Considering that terahertz (THz) radiation bridges a gap between far-infrared (photonics) and millimeter wave (electronics), it offers significant scientific and technological potentials in communicating, medical imaging, chemical testing and biological sensing [14-16]. For future application, a compact THz platform will be favorable. One issue that needs to be solved is efficient coupling between freely propagated THz waves and waveguide-based devices in Si. Current methods typically rely on thick substrate lenses and/or bulky free-space optics such as off-axis paraboloidal mirrors and the like [17,18]. So a new method for high-efficiency THz waves coupling is important.
In this work, to combine the advantages of directional emission and self-collimation, we present a model of directional emission for coupling at THz frequency regime provoked by a PC structure, which works under a self-collimation condition. Highly-efficient directional emission can be obtained with a large alignment tolerance regardless of the detail of the PC structure termination.
2. Model for directional emission
As shown in Fig. 1, we consider a finite PC structure with a width of 11a in y-direction and a length of 20a in x-direction. The finite PC structure consists of a two-dimensional square lattice of air holes in a high-resistivity (larger than 104 Ω•kcm) Si slab with an effective refractive index of 2.94. The hole radius r=0.31a=15.5 µm, where a=50 µm is the lattice constant. Two PCWs are introduced in the structure by removing two rows of air holes along y-direction. Figure 2 shows the equifrequency contours (EFCs) of the second band of the PC for the TM-polarization modes (electric field parallel to the hole axis). From Fig. 2, it can be seen that the EFCs can be approximated by squares for frequencies 0.28≤a/λ≤0.34 (corresponding to wavelength 150 µm≤λ≤180 µm), which means that the self-collimation phenomenon occurs when THz waves in this region propagate along the y-direction, where λ is the wavelength in free space.
Fig. 1. Schematic diagram of the proposed structure. RQ is a measurement line of directional emission efficiency.
3. Simulations and analyses of directional emission
To characterize the transmission properties, efficiency of the directional emission is quantified by calculating the power transmission coefficients (normalized by the input power) within -1.31a≤x≤1.31a at the position of y=26a, which forms an angle of ∠RPQ=10° and is in the far-field region of the PCW termination, as shown in Fig. 1. Figure 3 shows the spectrum of transmission versus wavelength when the normal incidence THz TM-polarization Gaussian wave, placed at x=0, is launched into the PC structure. The width of the Gaussian wave is 150 µm. From Fig. 3, it can be seen that a maximum efficiency of up to 75% can be achieved when the wavelength is 166 µm (1.81 THz). We define an acceptable bandwidth as within that the efficiency of the directional emission is larger than 80% of the maximum efficiency. With this definition, from Fig. 3, we can obtain that the proposed structure is applicable in the acceptable bandwidth 156 µm≤λ≤180 µm. To simplify the discussion, in the following sections, we restrict our attention to the wavelength of 166 µm. It is worth noting that the transmission properties are similar to all wavelengths in the acceptable bandwidth. To calculate the propagation of the THz wave in the PC structure, a finite-difference time-domain (FDTD) method is used. Figure 4 shows the FDTD simulated field distribution when the normal incidence THz wave, placed at x=0, is 1.81 THz. The incident THz wave is efficiently coupled into the PC structure and guided with almost no diffraction in the perfectly periodic PC structure via self-collimation. It is confined in the two PCWs due to index contrast (the average index in the defect region is higher than that of the surrounding region). At the exit surface, the output fields interfere constructively, giving rise to a directional emission.
Fig. 3. Transmission versus wavelength, showing that the maximum efficiency is achieved when the wavelength is 166 µm.
Fig. 4. Steady-state field distribution in the PC structure. The dashed rectangular region is for later analysis.
To analyze the directional emission in more detail, power density obtained as time average at y=11.31a (PC structure interface) is calculated, as shown in Fig. 5. From Fig. 5, it can be seen that there are two main peaks (at x=±0.8a) with a distance of 1.6a. Compared with the main peaks, the side lobes can be neglected. So the output field in Fig. 4 can be regarded as a two-point-source interference pattern. The simulated two-point-source interference field distribution is shown in Fig. 6(a). The distance between the two point sources is 1.6a and the frequency of the THz wave is 1.81 THz. The dashed rectangular region in Fig. 4 is shown in Fig. 6(b) for comparison. From Fig. 6, we can find that the steady-state field distributions surrounded by the dashed lines are similar to each other. This confirms our supposition that the output field in Fig. 4 is analogous to the two-point-source interference pattern. The two point sources, which evoke a directional emission regardless of the detail of the PC structure termination, can be regarded as the output fields emitted by the two PCWs. A similar directional emission is reported in Ref. 6, but it is a consequence of triple-point-source interference.
Fig. 5. Transverse power density obtained as time average at y=11.31a. Two main peaks appear at x=±0.8a.
Fig. 6. (a). Steady-state field distribution interfered by two point sources with 1.6a distance. (b) The dashed rectangular region in Fig. 4.
4. Evaluations of the alignment errors
Directional emission is considered as a candidate for enhanced coupling and connecting [1, 2, 6, 7]. From application point of view, the efficiency of coupling and connecting should be alignment-insensitive. Generally, the effect of longitudinal alignment error on the efficiency can be neglected, thus we only study the effect of transverse alignment error and angular alignment error. Since the symmetry of the PC structure, we just discuss the normal incidence Gaussian beam position varied positively for the transverse alignment error and the incident Gaussian beam angle θ (deviation of the incident THz wave from the normal of interface) varied clockwise for the angular alignment error. Figure 7 shows the transverse power densities, for these two kinds of alignment errors, obtained as time average at y=26a. It can be found that the alignment errors affect the efficiency of the directional emission, but not as severely as one might expect. With an increase of the alignment error, the output power density at the center peak diminishes slowly, while the power density of the side lobes increases gradually. In addition, there is nearly no fluctuation in the position of the maximum power density in the central peaks. As proposed before, we define an acceptable alignment error as within that the efficiency of the directional emission is larger than 80% of the maximum efficiency. With this definition, considering the symmetrical characteristic of the PC structure, we conclude that the acceptable alignment errors are -1.0a≤x≤1.0a and -5°≤θ ≤5° for transverse alignment error and angular alignment error, respectively. It is believed that the alignment-insensitive characteristic is due to the self-collimation, which collimates light propagation insensitive to the divergence of the incident beam [9] and enables high-efficiency coupling and routing light without physical boundary (line defect) [11]. So if the bulk Si after y=26a is truncated and replaced by a THz slab waveguide, the structure will be a strong candidate for efficient, broadband coupling between guided and freely propagated THz waves.
Fig. 7. Transverse power densities obtained as time average at y=26a. (a) for transverse alignment error and (b) for angular alignment error.
5. Conclusion
A directional emission from two adjacent parallel line-defect PCWs is investigated. The output field is analogous to the two-point-source interference pattern. The calculated maximum efficiency of the directional emission is 75% at 1.81 THz and the efficiency is insensitive to the transverse alignment error and angular alignment error. The acceptable bandwidth is from 156 to 180 µm, while the acceptable alignment errors are -1.0a≤x≤1.0a and -5° ≤ θ ≤ 5° for transverse alignment error and angular alignment error, respectively. These results suggest that it would be a promising device for high-efficiency coupling between freely propagated THz waves and dielectric waveguide systems with large alignment tolerance.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 60625404, 60577001, 90401008) and the Research Fund for the Doctoral Program of Higher Education (Grant No. 20040558009).
References and links
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