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In this work, we present the design of an integrated photonic-crystal polarization beam splitter (PC-PBS) and a low-loss photonic-crystal 60° waveguide bend. Firstly, the modal properties of the PC-PBS and the mechanism of the low-loss waveguide bend are investigated by the two-dimensional finite-difference time-domain (FDTD) method, and then the integration of the two devices is studied. It shows that, although the individual devices perform well separately, the performance of the integrated circuit is poor due to the multi-mode property of the PC-PBS. By introducing deformed airhole structures, a single-mode PC-PBS is proposed, which significantly enhance the performance of the circuit with the extinction ratios remaining above 20dB for both transverse-electric (TE) and transverse-magnetic (TM) polarizations. Both the specific result and the general idea of integration design are promising in the photonic crystal integrated circuits in the future.
©2009 Optical Society of America
1. Introduction
A photonic crystal (PC) is an optical material in which the refractive index is periodically modulated [1-2]. Two-dimensional PC slab structures have recently gained a lot of consideration as they allow for the realization of various active and passive devices [3-7], which have potential use in photonic integrated circuits (PIC). Among them, the polarization beam splitter (PBS) is one of the key functional components in PIC where polarization needs to be controlled. The PBS is usually realized by using a Mach-Zehnder interferometer [8], an asymmetric Y-branch [9], or a multimode interference device [10]. The device length of these PBSs is usually hundreds of microns, limiting the enhancement of integration density. Recently, PBSs based on PCs have exhibited ultra-compact device lengths [11-13], a great advantage in the use of PC optical circuits.
Another important device for PC integrated circuits is the waveguide bend [14-15]. In order to connect different optical units effectively, it is crucial that the waveguide bend can guide light through sharp corners with low loss. It is frequently discussed that the transmission is mostly determined by the bend structure itself, and various low-loss sharp bend structures with different guiding mechanisms have been proposed [14-19]. However, in optical circuits design, the interference between devices should also be considered, which makes the bend reflection and single-modedness especially crucial. In the rest part of this paper, we illustrate this point by studying the integration of a PC-PBS and a low-loss 60° waveguide bend, both based on a two-dimensional photonic crystal. In Section 2 and Section 3, we introduce the design and performance of the PC PBS and a low-loss waveguide bend separately. In Section 4, we investigate the integration of these two devices and analyze the mechanism of high transmission efficiency. We explain that the transmission will be determined not only by the bend structure, but also by the mode characteristics of the PC-PBS. Finally, we propose an optimized PC-PBS structure, which effectively improves the transmission of the integrated unit.
2. PC PBS based on directional coupler
Among all kinds of PC-PBSs, the one based on the PC directional coupler combines a compact device length with flexible integration with other PC devices [11], so we choose it as part of our integrated circuit. The schematic view of the PC-PBS is shown in Fig. 1. Our design is based on a two-dimensional photonic crystal consisting of a triangular lattice of air holes in dielectric, which illustrates the fundamental principles of such devices in a simple context. The design could potentially be realized directly in three-dimensions using a low-index contrast waveguide like a Ga(Al)As heterostructure slab, whose weak vertical confinement allows it to reproduce results from two-dimensional structures [16, 20]. One could also use three-dimensional PC structures that emulate two-dimensional PCs [2, 25]. More generally, the same underlying principles should be applicable to many other structures, such as high-contrast photonic-crystal slabs [2]. The refractive index of our dielectric material was chosen to be n=3.32, corresponding to the “effective” index of the Ga(Al)As heterostructure. The coupler consists of two W1 waveguides (formed by a missing row of holes), both along Γ-K direction. The period of the triangular lattice is a=450 nm, and the radius of the air holes is r=0.36a. The air holes separating the two waveguides have smaller radii r1=0.28a. It has been proved that introducing smaller airholes in the middle of the coupler can enhance the coupling strength and thus decrease the device length [11]: the smaller holes decrease the strength of the photonic bandgap between the waveguides, and hence increase the coupling. The mechanism of splitting TE and TM polarized light can be simply explained based on coupled-mode theory [21]. In the coupled waveguides, the possible modes are approximately “supermodes”: even (same phase) and odd (180° out of phase) combinations of the guided modes of the two isolated waveguides. (This is an approximate description of the true propagating modes of the parallel waveguides, which is accurate as long as the field overlap of the individual waveguide modes is small.) The even and odd supermodes couple during propagation, shifting the power of incident light between the two waveguides after a beat length given by:
Fig. 1. Schematic view of the PC-PBS. The length of the coupler is 61a. The period of the triangular lattice is a=450 nm, and the radius of the air holes is r=0.36a. The refractive index is 3.32. The air holes separating the two waveguides have smaller radii of r1=0.28a.
where keven and kodd are the propagation constants of the even supermode and odd supermode at the operating frequency, respectively. The transverse-electric (TE, electric field in the plane) light is confined in the waveguide by the photonic band gap (PBG) [2], while the transverse-magnetic (TM, magnetic field in the plane) light is confined in the dielectric-cored waveguide by an index-guiding effect [2, 22-23]. The different guiding mechanisms lead to different propagation properties. Figure 2(a) and Fig. 2(b) show the photonic band structures of the direction coupler for TE and TM light, respectively. Note that we only show the frequency range within the photonic band gap (the PBG range is 0.228<a/λ<0.347). For TE light, the even and odd supermodes cross at a/λ=0.3, where keven = kodd. So the beat length at that frequency is infinity, and the TE light is approximately decoupled between the two waveguides. (There is still some small coupling due to the fact that the supermodes are not exactly sums of the isolated waveguide modes, but this is a higher-order effect in the overlap between the waveguide modes.) Meanwhile, the even and odd supermodes of TM light do not cross at the same frequency, so there is a finite beat length for TM. The length of the coupler is 67a, and it is designed to be the half beat length of the TM light. Thus, the TE light at decoupled frequency will always propagate along the TE branch with very small coupling into the neighbor waveguide, while the TM light will beat one time into the TM branch. The transmission spectrum of TE and TM light calculated by two-dimensional finite-difference time-domain (2D-FDTD) method is shown in Fig. 3(a) and Fig. 3(b), respectively.
Fig. 2. The photonic band structures of the PC-PBS for (a) TE light and (b) TM light. In Fig. 2(a), the decoupling frequency is indicated by the horizontal dotted line. There is a high order odd supermode at the decoupling frequency. The Magnetic field distributions of the two modes are shown in the inset and indicated by arrows.
At the decoupling wavelength λ=1.50 um, The TE and TM light are split effectively with extinction ratio both above 20dB.
3. The low-loss 60° waveguide bend
Although plenty of low-loss waveguide bend structures have been proposed [14-19], it is not appropriate to use those bend structures directly in our case, because we treat the period and filling factor of the PC PBS as already fixed. In order to avoid the loss introduced by the junction between the border of the PBS and bend, the period and radius of the photonic crystal of the waveguide bend should be the same as that of the PC PBS. Under this limitation, we design a new low-loss waveguide bend structure for our circuit by a relatively effective method [24]. The proposed low-loss 60° waveguide bend structure is shown in Fig. 4. The refractive index, the lattice period and the air hole radius are the same as the PC PBS. We consider the process of propagation through the bend as following: first, light propagates along the straight waveguide in the Γ-K direction, then it couples with and turns into the mode in the Γ-M direction, and finally couples back to the mode in the Γ-K direction again. Because the photonic bandgap prohibits light from scattering out of the waveguide, only reflection need be considered, and it has been shown that the bend can be accurately described by a one-dimensional (1d) model determined only by the propagation constants (k) of the modes propagating in each direction [14]. In general, for a single-mode mirror-symmetric bend, one can achieve high transmission over broad frequency ranges by a resonant process analogous to 1d electron resonant scattering [2, 14]. Furthermore, the 1d model predicts the broadest bandwidth of high transmission when the propagation constant in the Γ-M direction is matched with that in the Γ-K direction. Intuitively, the transmission can be very high because the lightwave experiences the least difference between propagating along the straight waveguide and through the bend. We perform the design by following steps. First, we should choose a mode in the waveguide bend, whose mode pattern is similar with the guided mode in the straight waveguide. Then, we alter the band structure of the waveguide bend by adjusting the size and position of the air holes near the bend corner in order to have a wider frequency region in which the wave number of the mode in the bend matches with that of the straight waveguide. Here, we make the air hole at the inner corner smaller, and add one airhole at the outer corner of the bend. The two air holes have the radii of r′ = 0.75r, and they are moved 0.3a oppositely along the symmetric axis of the bend (also the Γ-K direction). The photonic band structures of the straight waveguide and the diagonal waveguide in the bend are shown in Fig. 5 (a) and Fig. 5(b), respectively. Note the guided modes in the frequency region a/λ = 0.278~0.3 indicated by red lines. They are both even modes polarized along the propagation direction. The magnetic field distributions of the two modes are shown in the insets. Then we calculate the wave numbers of these two modes in this frequency region. For the straight waveguide, the band is folded one time, so the wave number is k ΓK≈3.71π ~ 4.40π μm-1; for the waveguide bend, the band is folded three times, and the wave number is k ΓM ≈3.86π ~ 4.11π μm-1. The mode patterns and wave numbers of these two modes match well.
Fig. 5. The photonic band structures of (a) the straight waveguide and (b) the waveguide bend. The red lines indicate the high-transmission frequency region. The supercell and the mode pattern are also shown as the inset.
For the waveguide bend, there is another mode (shown as dotted line) in this frequency region. The magnetic field distribution of this mode is also indicated in the inset of Fig. 5(b). It is a high-order odd mode polarized in both Γ-K and Γ-M directions, and its mode pattern is very different from the mode in the straight waveguide. The wave number does not match that of the mode in the straight waveguide, either. So the light from the straight waveguide will mostly couple with the mode in the bend with matched mode pattern and wave number, and the mode with different mode pattern, parity and wave number is excited only weakly. The transmission spectrum of the optimized bend waveguide calculated by 2D-FDTD method is shown in Fig. 6. The high transmission region ranges from 1.496μm ~ 1.612μm, which corresponds to the expected frequency region in the band structure where the propagation constants are matched. The bandwidth of over 90% transmission is 116nm, which is a significant improvement over the primitive waveguide bend without tuning. (The primitive waveguide bend has a bandwidth only about 20nm.) This bend structure with high transmission, wide bandwidth and the same PC parameters with the PC-PBS is a good candidate in our integrated circuits use.
4. Integration of the PC PBS and low-loss bend
The schematic view of the integration of the PC PBS and the low-loss bend is shown in Fig. 7.
Fig. 7. Schematic view of the integration of the multi-mode PC-PBS and the waveguide bend. The length of the PC-PBS is 61a.
We directly add our low-loss bend to the TE branch. The TM branch remains straight, because there is no band gap for TM light in our PC structure, and the TM light will suffer great loss when encountering the sharp bend. In order to evaluate the performance of the integrated devices, we calculate the transmission spectrum of the circuit by 2D-FDTD method. A pulse centered at λ=1.55 μm with Gaussian spatial distribution is launched at port A as the incident light source for both TE and TM light. A time monitor, shown as the short line in Fig. 7, is located at each port to detect the input and output power. The output power is divided by the input power to calculate the transmission efficiency. The monitor at port A only measures the power coupled into the coupler, and the reflecting power is excluded from the calculation. The transmission spectrum of the integrated circuit is shown in Fig. 8. There are many peaks in the spectrum, with an average efficiency of less than 60%. At the decoupling wavelength λ=1.50μm, the transmission efficiency is only 53%. The simulation result shows that the performance of the integrated circuit is degraded by simply putting the two devices together.
To explain this, we have to start from the mode property of the PC PBS. In Fig. 2(a), The decoupled frequency of TE light is a/λ=0.3, as indicated by the horizontal dotted line. The decoupled supermodes (left inset) have the similar mode pattern, parity and wave number with the guided mode in the bend, but there is another high order supermode at the decoupled frequency (right inset), whose mode property is intrinsically different from the transient guided mode of the bend. As a result, a large amount of the lightwave is reflected by the bend rather than propagating through it. Moreover, the group velocity of this high-order mode is much slower than the other guided modes according to the dispersion curve. Because slow light reflects much more easily at junctions (such as the bend or the ends of the directional coupler), the reflected light will resonate in the PBS like in a Fabry-Perot cavity, leading to the oscillations in the transmission spectrum of Fig. 8.
According to the above analysis, the multi-mode property of the PC-PBS causes the low transmission of the circuit. To solve the problem, we propose a novel single-mode PC-PBS structure, as shown in Fig. 9. The round airholes between the two waveguides are deformed to an elliptical shape. The semi major axis and semi minor axis of the elliptic air holes are 0.39a and 0.21a, respectively. This modification also enhances the coupling strength of the waveguides, and decreases the length of the PBS to 56a. For TE light, it affects the photonic band structure by pulling the decoupled point (a/λ=0.29) down to the single mode region so that it is isolated from the high order supermode, as shown in Fig. 10. The transmission spectrum is calculated with the same 2D FDTD method, and shown in Fig. 11. The resonance peaks disappear, and the average efficiency rises to above 90%. At the decoupling wavelength λ=1.55μm, the transmission efficiency is as high as 97%.
Fig. 9. Schematic view of the integration of the single-mode PC-PBS and the waveguide bend. The length of the PC-PBS is 56a.
The Poynting vector distributions of TE light at decoupled wavelength of multi-mode PBS and single-mode PBS are shown in Fig. 12(a) and Fig. 12(b), respectively. The mode mismatching and interference of the multimode PBS cause the strong reflection in the neighboring waveguide and low transmission through the bend. As for the single-mode PBS, most power propagates through the bend. We also calculate the Poynting vector distribution of TM light at the same wavelength, as shown in Fig. 12(c). The figure shows that although TM light has a greater loss due to the lack of PBG effect (the TM light can radiate at places where periodicity is broken, such as the end of the PBS), the bend does not affect the guiding of TM light (since the TM field amplitude at the bend is almost zero). The extinction ratio for both polarizations is better than 20 dB.
Fig. 12. The poynting vector distributions of the integrated circuit at decoupled wavelength for (a) TE light with multi-mode PC-PBS (b) TE light with single-mode PC-PBS and (c) TM light with single-mode PC-PBS.
5. Conclusion
In this paper, we study the integration of a photonic crystal polarization beam splitter (PC-PBS) and a low-loss 60° waveguide bend. It shows that multimodedness in PC-PBS will influence the transmission efficiency when it is integrated with a low-loss waveguide bend. More generally, the importance of the single-mode condition has been emphasized in a number of other phenomena involving resonant transmission processes [2]. We propose a deformed single-mode PC-PBS structure, which significantly improve the transmission efficiency of the circuit by eliminating coupling to higher-order modes. Such enhancement of transmission efficiency has potential applications for practical use, and the design method should be applicable to photonic-crystal integrated-circuit design in a variety of geometries not limited to the specific structure considered here.
Acknowledgments
Project supported by the National Nature Science Foundation of China (Grand No.10634080, No.60838003 and No.60677046), the Chinese National Key Basic Research Special Fund (2006CB921705), and the National High Technology Research and Development Program of China (Grant No. 2006AA03Z403).
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