Enhancement of resonant tunnelling of a wide beam through vertical subwavelength slow-light photonic-crystal waveguides (SPCWs) is considered. An assistant horizontal SPCW with a thin side wall, whose guided modes have small propagation constants, is used as an input coupler for the vertical SPCW, and the two SPCWs form a compact composite structure to enhance drastically the resonant tunnelling. An incident wide beam can excite strongly the guided modes of the horizontal SPCW, and then resonantly tunnels through the vertical SPCW efficiently. To further improve the resonant tunnelling of a wide beam, a periodic array of vertical SPCWs (with a horizontal SPCW as an input coupler) is also investigated. With this periodic structure, a wide beam can be transmitted nearly completely. When a wide beam tunnels through the vertical SPCWs efficiently, the excited fields inside the SPCWs are very strong.
© 2008 Optical Society of America
Photonic crystals (PCs) have been studied widely in various areas, such as optical integration, quantum radiation, and nonlinear optics, etc . As an important application of PCs [2–13], a slow-light PC waveguide (SPCW) can be used to enhance drastically the light-matter interaction [6–9] besides as delay lines. Efficient coupling of light into a subwavelength SPCW is an important subject [11–14]. Due to large mismatch of modes, it is difficult to efficiently couple a beam, especially a wide beam, into a SPCW. When a SPCW is of finite length, large reflection at the two open ports makes the waveguide as a Fabry-Pérot (FP) cavity . At resonance, a narrow beam can tunnel through the SPCW efficiently. To some degree, this transmission enhancement due to resonance is similar to the transmission enhancement of subwavelength holes in a metal film [15,16]. However, if the beam is wide, a large part of the beam is directly reflected by the bulky PC (at the two side walls of the SPCW) and the resonant tunnelling is of low efficiency. In this paper, the resonant tunnelling of a wide beam through SPCWs will be enhanced drastically. The main idea is to put horizontally an assistant SPCW (as an input coupler) whose guided modes are of rather small propagation constants in front of a vertical resonant tunnelling SPCW to form a compact composite structure. A wide incident beam can excite the guided modes of the horizontal SPCW strongly, and then resonantly tunnels through the vertical SPCW efficiently. Furthermore, when resonant tunnelling SPCWs are periodically placed in an array with a horizontal SPCW as an input coupler in front of them, a wide beam can be transmitted nearly completely. In this paper, only two-dimensional (2D) cases are investigated and the TE polarization (the electric field is vertical to the 2D periodic lattice of a PC) is assumed.
2. Enhancing resonant tunnelling of a wide beam through one single SPCW
The SPCWs investigated in this paper are based on a square-lattice PC, which consists of dielectric cylinders with permittivity ε=12.96 and radius r=0.3a (a is the period) in air. By removing a line of cylinders along the Γ-M direction as shown in Fig. 1(a), a single-line defect waveguide is formed. Figure 1(b) gives the dispersion curve of this waveguide calculated by the classical super-cell method based on the plane-wave expansion method  (using the software RSoft-Bandsolve). The band edge ω edge=0.26277(2πc/a) (near the Γ point) means that the dispersion curve is rather flat so that the group velocity v g=∂ω/∂k is very small and the waveguide becomes a SPCW, and the propagation constants are very small.
Large reflection exists on the interfaces between the SPCW and free space. When the SPCW is of finite length, the slow guided modes are reflected strongly at its two open ports. Some slow guide modes satisfy the FP resonance condition and then resonate strongly inside the waveguide. When the waveguide is of length L=20a, three FP resonant modes exist near the band edge ω edge. In this paper, except the dispersion curve in Fig. 1(b), all the numerical results are obtained with the time-domain finite-difference method  (using the software RSoft-Fullwave) and the grid size is a/40. The resonant frequency (RF) of the FP resonant mode nearest from ω edge is ω=0.26324(2πc/a). Two slow guided modes propagating in the opposite directions can be supported inside the SPCW at ω=0.26324(2πc/a), and the corresponding propagation constant and group velocity are β=0.022(2πc/a) and v g=.043c (obtained by discretizing its definition formula utilizing the data of the dispersion curve), respectively. Due to the slow propagation of light, the FP resonant mode has a large quality factor (Q=2.74×103). Q is obtained by calculating Q=ω/Δω, where ω is the RF and Δω is the full width at half maximum of the spectral response when a pulse line source is put inside the waveguide. Although the RFs of the other two FP resonant modes are very close to that of the above FP resonant mode, the propagation constants and group velocities of the corresponding guided modes are relatively large since the dispersion curve is very flat near the band edge ω edge, and then the Q values of the FP resonant modes are small. The two FP resonant modes with small Q values are not considered in this paper. In the following, when a FP resonant mode is mentioned, it is referred to the mode nearest the band edge ω edge possessing the largest Q value. When the waveguide becomes longer, more FP resonant modes may appear. As L increases, a slow guided mode with a smaller propagation constant may satisfy the FP resonance condition, so the FP RF moves toward the band edge ω edge gradually, and the corresponding Q value becomes larger. This is verified by numerical simulation. Figure 1(c) shows the RFs and Q values of the FP resonant modes for several different values of L. In the following, the resonant tunnelling of a Gaussian beam through a single finite-length SPCW is investigated. Transmissivity t of the SPCW is defined as the ratio between the total power transmitted through the SPCW and the input power of a down-going Gaussian beam with certain waist width w incident on the top of the SPCW. The waist of the beam is located above the axis of the SPCW and a far from the top line of cylinders. The detectors are enough wide to get the total input and transmitted powers. Reflection of a beam incident on the subwavelength SPCW includes two parts, one from the reflection at the open ports of the SPCW, and the other from the direct reflection of the bulky PC (at the two side walls of the SPCW). When the SPCW is at resonance, a narrow beam can tunnel through the SPCW efficiently. According to the power spectra of the detectors (a narrow-band incident beam is used in calculation), Fig. 1(d) shows transmissivity t for the frequencies near ω edge. As shown by curve 2 in Fig. 1(d), a very narrow beam with w=2a can resonantly tunnel through the SPCW well at RF ω=0.26324(2πc/a). The pointed peak comes from the FP resonance. If without the FP resonance, the transmissivity would be very small. However, when a wide beam is incident on the waveguide, the direct reflection by the bulky PC dominates. Curve 1 in Fig. 1(d) shows transmissivity t when the waist width w of the beam is 13a. Such a beam can approximately simulate the mode of a conventional waveguide when it is scaled to resonate at 1550nm. At RF ω=0.26324(2πc/a), transmissivity t is only about 0.19.
Now another special effect of the guided modes of very small propagation constants is investigated, which will be used to efficiently compress a wide beam through a subwavelength SPCW. An asymmetric SPCW is shown in Fig. 2(a), which is formed by making one side wall of the waveguide in Fig. 1(a) thin (leaving only two lines of cylinders). The supported slow guided modes can not be confined well by the thin side wall and become leaky. Because the propagation constants of these slow guided modes are much smaller than wave number k 0 in free space, they can be excited strongly by a normally incident beam. Even a plane wave can also strongly excite them. At ω=0.26324(2πc/a), the propagation constant of the SPCW is β=0.022(2π/a), very close to zero, and Fig. 2(b) shows the distribution of the electric filed amplitude |E z| when a plane wave is normally incident. A strong electromagnetic field is excited in the air gap of the SPCW. When a Gaussian beam is used to excite a strong field inside the SPCW, it can not be too narrow, otherwise the excitation can not be very strong. This is because when a narrow beam is expanded with plane waves, many plane-wave components can not efficiently excite the supported guided modes (as the amplitudes of their transverse wave vectors along the waveguide are much larger than those of the small propagation constants of the supported guided modes). The reason why a thin side wall is needed for the SPCW to excite a strong field can be explained as follows. When the working frequency is near the band edge ω edge of the SPCW, it is in the complete gap of the bulky PC. If the side wall is too thick, the incident beam may be reflected strongly, and then one can not obtain a strong excited field inside the SPCW. On the other hand, the side wall can not be too thin. If only one line of cylinders is left, the leakage is too strong and one can not obtain a strong excited field either. The thin side wall may be optimized further to excite a stronger filed filed by adjusting other parameters, such as the cylinder radius, which is not performed here. For other PC structures, the thin side walls may be optimized in a similar way to strongly excite the guided modes with small propagation constants.
In the following, the above asymmetric SPCW is used as an input coupler to drastically enhance the transmission of a wide beam through the symmetric resonant tunnelling SPCW shown in Fig. 1(a). Two finite-length lines of dielectric cylinders are put d=a away from the top of the vertical resonant tunnelling SPCW to construct a horizontal asymmetric SPCW as shown in Fig. 3(a), and the two form a compact composite structure. The horizontal asymmetric SPCW is of finite length. As an example, we choose the length of the horizontal SPCW as 19a, and the length of the vertical SPCW as 20a. The vertical SPCW has been investigated previously. The RF of the composite structure shifts a bit away from that of the individual vertical SPCW due to the interaction of the two SPCWs, and becomes ω=0.26329(2πc/a), and the Q value is reduced to 1.90×103. At ω=0.26329(2πc/a), the supported guided modes of the horizontal SPCW can be excited strongly by a normally incident wide beam. The excited strong field is easier to get into the vertical SPCW rather than to leak through the thin side wall of the horizontal SPCW. Thus, the incident wide beam is collected by the horizontal SPCW and then resonantly tunnels through the vertical SPCW efficiently. Curve 1 in Fig. 3(b) shows transmissivity t when a wide beam with w=13a is normally incident on the top of the structure. Resonant tunnelling of the wide beam through the composite structure is drastically enhanced as compared with that of the individual symmetric SPCW shown in Fig. 1(d). The largest t is about 0.80 at RF ω=0.26329(2πc/a). Strong resonance makes the resonant peak of the transmissivity curve very sharp. Figure 3(c) shows the distribution of |E z| at RF ω=0.26329(2πc/a). The excited field inside the horizontal asymmetric SPCW by the incident beam is strong. The field inside the vertical resonant SPCW is much stronger, and the largest |E z| is about 28 times of the amplitude of the incident beam. This is because the FP resonance further enhances the field when a wide beam is efficiently compressed into the vertical SPCW. For such an enhanced resonant tunnelling, the horizontal asymmetric SPCW is not necessarily very long. Actually the tunnelling effect will be reduced a bit if the horizontal SPCW is very long. When a beam is collected into the horizontal SPCW, the excited field is strong in a long range in the air gap of the waveguide. For a longer horizontal SPCW, the field spreads more widely. When the excited strong field propagates toward the open port of the vertical SPCW, part of the field leaks away from the horizontal SPCW through the thin side wall. If the horizontal SPCW is excessively long, the leakage may reduce transmissivity t. In addition, the horizontal SPCW should not be excessively short, otherwise a large part of the beam can not impinge on the waveguide and will be directly reflected by the bulky PC. In such a case, transmissivity t can not be large, either. The above influence of the horizontal SPCW’s length on transmissivity t is verified by numerical simulation. Thus, for the current waist width w=13a, the length of the horizontal SPCW is chosen as 19a in order to obtain large transmissivity t. Note that the narrow incident beam is not favorite for the enhancement of resonant tunnelling because many plane-wave components of a narrow beam can not excite the horizontal SPCW strongly as indicated previously. This is indicated well by transmissivity curve 2 in Fig. 3(b) for a beam with w=2a.
3. Enhancing resonant tunnelling of a wide beam through a periodic array of SPCWs
In Fig. 3(c) one can see that a large part of the reflection of the incident beam by the composite structure originates from the leakage of light at the two open ports of the horizontal asymmetric SPCW. To eliminate such leakage, we make the horizontal SPCW infinitely long. To efficiently transmit the excited field inside the horizontal SPCW, a periodic array of vertical resonant tunnelling SPCWs is used, instead of a single one. The constructed periodic composite structure is shown in Fig. 4(a). Because the excited strong field can leak through the thin side wall of the horizontal SPCW, the interval between two adjacent vertical SPCWs can not be too large, otherwise such leakage will be large and consequently the reflection will be large. For such a periodic composite structure, transmissivity t is defined as the ratio between the total transmitted power and the input power of a normally incident plane wave. When the length of the vertical SPCWs is 20a, it is found that transmissivity t becomes nearly unity when the interval between two adjacent vertical SPCWs is 9a, which is investigated as an example here. The periodic composite structure can act as a resonant cavity, and the RF and Q are 0.26311(2πc/a) and 2.15×103, respectively. Figure 4(b) shows the transmissivity curve. In calculation, the periodic boundary condition is used. Due to the strong resonant tunnelling, the transmissivity curve has a sharp peak, and the largest transmissivity t is about 0.95. Figure 4(c) shows the distribution of |E z| when a down-going plane wave is normally incident on the structure at RF ω=0.26311(2πc/a). The electric field inside the air gaps of the vertical symmetric SPCWs is very strong, and the strongest |E z| is about 33 times of the amplitude of the incident plane wave. When the interval between two adjacent vertical SPCWs is reduced to 7a, resonant tunnelling transmissivity t increases to about 0.98. If the interval is reduced further, the interaction between the vertical waveguides increases gradually and influences the transmission with a contribution mechanism different from the slow-light waveguide effect (the focus of the present paper). In practice, the incident wave is usually a beam. The periodic composite structure is especially appropriate for a very wide beam. It’s enough that the periodic composite structure is of finite width and several periods wider than the beam. For comparison, when the horizontal SPCW is removed (i.e., the thin side wall of the two lines of cylinders is removed), transmissivity t for the interval equal to 9a is shown by curve 2 in Fig. 4(b). Without the horizontal SPCW, t is very small whose maximum value is only about 0.18, and the RF of the periodic structure is ω=0.26307(2πc/a).
When the guided modes of a horizontal asymmetric SPCW have quite small propagation constants, even a wide beam normally incident on the thin side wall can strongly excite them. Using this property, resonant tunnelling of a wide beam through vertical SPCWs has been enhanced drastically. The properties of large transmission, sharp transmission peaks, and strong fields inside the SPCWs may be useful for light harvesting, filters, switches, nonlinear optical, etc. For simplicity, we have considered only the case when the vertical SPCWs are similar to the assistant horizontal SPCW (as an input coupler). The present idea can be generalized to other cases with different SPCWs and/or other types of lattices.
This work is partly supported by the National Basic Research Program (No. 2004CB719801) and the National Natural Science Foundations (No. 60688401) of China, and the Swedish Research Council (VR) (No. 2006-4048). Yi Jin thanks Yanxia Cui for her helpful discussion.
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