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OPTICS LETTERS / Vol. 38, No. 21 / November 1, 2013
All-optical tunable filters based on optomechanical effects in two-dimensional photonic crystal cavities Yaoxian Zheng,1,2 Quanqiang Yu,1,2 Keyu Tao,1,2,3 and Zhengbiao Ouyang1,2,4 1
College of Electronic Science and Technology, Shenzhen University, Shenzhen 518060, China 2 THz Technical Research Center of Shenzhen University, Shenzhen 518060, China 3 4
e-mail:
[email protected] e-mail:
[email protected] Received August 2, 2013; revised September 19, 2013; accepted September 19, 2013; posted September 19, 2013 (Doc. ID 194970); published October 23, 2013 All-optical tunable filters are basic elements for various micro-optical circuits. Obtaining all-optical tunability remains a challenge for micro-optical circuits. Optical forces with significant effects in nanophotonic systems provide new ways for wavelength tuning. In this Letter, the optomechanical effects in two-dimensional photonic crystal cavities are investigated. Simulations based on the finite element method demonstrate that forces arise in single and coupled cavities with movable rods inside. The optical force controls the positions of the movable rods and, thus, the resonance wavelength of the cavity, based on which tunable filter is designed. The operating wavelength of the cavity or the filter for the signal can be tuned by a control light with a different frequency. The results have potential applications for various integrated circuits. © 2013 Optical Society of America OCIS codes: (130.7408) Wavelength filtering devices; (230.5298) Photonic crystals; (130.5296) Photonic crystal waveguides. http://dx.doi.org/10.1364/OL.38.004362
All-optical tunable filters are basic elements for various micro-optical circuits, e.g., microtunable lasers, microoptical signal detectors, and micro-optical signal analyzers. Conventionally, there are many ways to obtain tunability, e.g., applying the grating diffraction effect, thermoelectric effect, nonlinear effect, electro-optical effect, or magneto-optical effect. However, it remains a challenge for all-optic tuning in micro-optical circuits. Fortunately, optical forces, which have been proposed to construct novel optomechanical systems, e.g., optical wavelength converters and efficient optical-to-mechanical energy converters, can provide efficient methods for alloptical frequency tuning in micro-optical circuits [1–6]. Photonic crystal cavities can exhibit an extremely high quality factor with a very small cavity volume, which attracts considerable interest for micro-optical circuits [7,8]. For rods inside two-dimensional (2D) photonic crystal cavities, they can be moved by optical forces that are greatly enhanced in the cavities, thus efficiently changing the resonance properties of the cavities. Optical forces are more significant in photonic crystal cavities than in other kinds of resonant cavities because of their high quality factors. Therefore, using optomechanical effects to control light by light is far more efficient than traditional methods, such as nonlinear optical effects, for nanophotonic components [9,10]. In this Letter, we investigate the properties of optical forces inside 2D photonic crystal cavities and propose a novel all-optical tunable filter based on optomechanical effects. Only TE polarized electromagnetic waves of the system have been considered in this Letter. We first consider a 2D photonic crystal cavity with a movable rod inside, as shown in Fig. 1(a). The dielectric constants of the background material and the green rods are 1 and 11.56, respectively. The lattice constants of the photonic crystals and the radius of the green rods are a 1 and 0.2 μm, respectively. The dielectric constant of the movable rod inside the single cavity, shown in Fig. 1(a) in red (center), is 6.6 with its radius of 0.1 μm. Resonant 0146-9592/13/214362-04$15.00/0
frequencies have been calculated, and only one TE mode is found inside the cavity at the range of the TE band gap frequencies (87–126 THz). When the red rod is in the center of the cavity, the resonant frequency of the cavity is 106.0 THz. When the position of the red rod varies along the x axis [Fig. 1(a)], the resonant frequency changes, as shown in Fig. 1(c). The optical force can be calculated based on the energy conservation law. The eigenmode frequency of the resonant system changes with the position of the red rod, leading to a change in the total energy of the system. The energy variation can be viewed as the result of the work done by the photon force, thus the mechanical force on the rod can be written as [11–14] F −
∂U 1 ∂ω − U; ∂l ω ∂l
(1)
Fig. 1. (a) Two-dimensional photonic crystal with a movable rod in red inside the point defect cavity. The red rod can move along the x axis. (b) Ez (z-component of the electric field) of the cavity eigenmode in the x–y plane. (c) Eigenfrequency and quality factor as functions of the red rod’s position. Parameter l is the displacement of the red rod to the center of the cavity. © 2013 Optical Society of America
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where U is the stored electromagnetic field energy inside the cavity, and l is the distance between the red rod and the center of the cavity. The sign of parameter l indicates the direction: positive means the red rod moves to the right and negative to the left. Neglecting the material loss of waves, and taking only the radiation loss of the cavity, we know that the optical power loss of the system (P loss ) is equal to the incident optical power (P) when the system is in an equilibrium state. So, the quality factor (Q) of the cavity can be written as Qω·
U U ω· : P loss P
(2)
From Eqs. (1) and (2), we can calculate the power-toforce-conversion coefficient (PFCC), γ, as γ
F Q ∂ω − 2 : P ω ∂l
(3)
The PFCC, which can also be regarded as the optical force on the x axis movable rod per unit input power, can be calculated from Eq. (3) as shown in Fig. 2. From Fig. 2 we can see that the PFCC has the largest value of 19.4 nN∕mW when the red rod is near the edge of the cavity. Moreover, the quality factors remain high when the position of the red rod changes [Fig. 1(c)]. Optomechanical potentials can be calculated based on Z (4) E p Fdl: The results obtained show that the red rod tends to stay at the center of the cavity (l 0 μm), consistent with the principle of energy being the lowest in a mechanical system. Referring to the definition of the PFCC given by Eq. (3), from Fig. 2 we can see that the optical force is so large that it can be practically measured. This is due to the resonance enhancement of the high quality factor cavity. This system can be used for force measurement by applying mechanical force against the optical force on the rod [15]. To further investigate the optical forces, two coupled resonators are considered, as shown in Fig. 3(a). Such a structure can be used to modify and enhance the
Fig. 2. Relation between the PFCC (γ) and the position of the movable rod in a single point defect cavity.
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interaction between light and matter in cavities by manipulating their mutual coupling coefficient [16]. In Fig. 3(a), there are two point defects (red rods near center) inside the two cavities. Suppose the red rods are movable along the x axis. With the same parameters as that for Fig. 1(a), both z-odd symmetry (E z is antisymmetric) mode and z-even symmetry (E z is symmetric) mode are generated inside the coupled cavities. Consider that the two movable rods move along the x axis at the same time toward opposite directions. Note that the optical forces acting on the two movable rods are mirror symmetric and the lack of symmetry of the cavity structure will decouple the cavity modes. The eigenmode frequencies in different positions of the movable rods are shown in Fig. 3(c). Figure 4 shows the PFCC in the twocavity system. In view of the definition of the PFCC, given by Eq. (3), from Fig. 4 we can see that the optical force changes with parameter l. The parameter l is the distance between the left-side movable rod and the center of the left-side cavity. For the right-side movable rod, its distance to the center of the right-side cavity is −l. A practical tunable filter is proposed and demonstrated in Fig. 5. In the system, the eigenfrequencies are different from the results obtained above, but there can still exist two modes (even and odd modes) in it. The system is designed so that the even and odd modes can be separated. In this Letter, we suppose that the signal and control are even and odd modes, respectively (it does not matter if the signal and control light are swapped to be odd and even modes, respectively). They first transmit and interact with each other. Afterward, they are separated. The even and odd modes transmit, respectively, to the bottom and right ports because the waves from the two cavities exhibit constructive (destructive) interference for odd modes in the right (bottom) port, due to differences in phase change in the two waveguides for the two modes, while they exhibit destructive (constructive) inference for odd modes in the right (bottom) port, which can be seen from Figs. 6(a) and 6(b), eliminating the cross talk between the signal and the control.
Fig. 3. (a) Two cavities with movable rods (in red) inside form a coupled system. The red rod can move along the x axis at the same time toward the opposite direction. (b) Ez (z component of the electric field) of the antisymmetric and symmetric eigenmodes in the x–y plane that can be generated in the system. (c) Eigenfrequency and quality factor versus the position of the red rods. Parameter l is the displacement of the right-side movable rod to the center of the right-side cavity.
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Fig. 4. Influence of the position of the movable rods on the PFCC (γ) in the system with two coupled point defect cavities.
Here, we only consider the case that l is between 0.3 μm and 0.4 μm. Both modes will generate optical forces pointing to the center of the two cavities. The whole system would work like this: without the control light, the signal-produced force and external mechanical forces or the elastic restoring force are in equilibrium, and the red rods stay at the position of l 0.4 μm. With the control light, there is an extra force. It moves the red rods to the position of l 0.3 μm, thus tuning the resonance frequency of the cavities. As long as the control light exists in the resonant system, the red rods will maintain the position of l 0.3 μm. Moreover, when there is no longer a control light, the red rods move back to their initial position (l 0.4 μm). From Fig. 5(b) we can see that the cavity resonant frequency is tuned from 110.40 to 108.95 THz, with the rods moved from l 0.4 μm to 0.3 μm. Figures 6(a) and 6(b)
Fig. 5. (a) Schematic of the tunable filter. The control light is applied to change the positions of the movable rods. (b) The resonant frequency is tuned from 110.40 to 108.95 THz when the rods move from l 0.4 to 0.3 μm.
show that the signal (control) goes to its own output port when there is only the signal (control). Figures 6(a) and 6(c) show that transmission channel of the signal is blocked by the control light. Figures 6(a) and 6(d) show, further, that the transmission frequency of the signal is tuned by the control light. Moreover, Figs. 6(c) and 6(d) demonstrate that the transmission for the signal is blocked in one frequency but opened in another frequency for the same frequency of the control light. We note that the transmission of the signal is high with or without the control, but that of the control is changed. However, it is still above 60% when both the signal and control exist. The rest of the control light is reflected back to the input side. Since the resonance frequency for the control is changing, the wavelength of the control light should be varied for high efficiency controlling and tuning. A practical way is to use control light with broadband, multifrequency, or tunable frequency, so that a wider tuning range of the filter can be achieved. Moreover, the linewidth of the signal transmission spectrum, i.e., that of the filter, can be further reduced by increasing the quality factor of the resonant cavities. Next, we consider the fabrication details of the structure. When applied in three dimensional systems, we can use a photonic crystal slab with a woodpile photonic crystal base and a woodpile photonic crystal cover with two periods in thickness, which corresponds to eight layers of logs; enough for the required confinement of waves in the third dimension. The length of the movable rods should be less than the height of the slab, to prevent the rods from contact with the base and the cover, so that the movement of the rods will not be hampered by the base or cover. For mechanical balance an elastic system needs to be used to provide external mechanical force. A practical solution is to use foamed elastic materials,
Fig. 6. (a) Field distribution E z when there is only the signal with the frequency of 110.40 THz for the red rods in the initial position (l 0.4 μm), showing large output signal power. (b) Field distribution E z in the x–y plane when there is only the control light (odd mode, 103.62 THz) in the x–y plane. (c) Field distribution Ez for both the signal (110.40 THz) and the control (103.62 THz); the red rods have been moved by the control light to l 0.3 μm, in which the cavities are tuned by the control light, showing small output signal power, i.e., the signal channel is blocked. (d) Field distribution E z for both the signal (108.95 THz) and control (103.62 THz) when the controlproduced force is acting on the red rods and it maintains the position of l 0.3 μm.
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such as ROHACELL foam [17,18] with a dielectric constant of about 1, to fill around the red rods. The foam can be paved in a long smooth guiding groove whose axis is along the direction of the motion of the red rods, so that just long thin strips of foam are necessary and a small optical force can produce a large displacement to the red rods. Similar to a spring instrument, the whole foam strip is free except that one of its ends is fixed with the base of the system and another end binds with the red rods. As the magnitude of the optical force produced by the control light is in direct proportion to its power, it gives an adjustable force to overcome the elastic restoring force from the foamed material for moving the red rods. As an example to see the scale of the control power in the system, we consider the simple structure in Fig. 3(a) and take the foamed material in the groove as ROHACELL WF 71, whose Young’s modulus is 114.5 Mpa [18]. We consider a strip of 0.4 μm × 0.3 μm × 6.2 μm made by this kind of foam with its left end fixed at the left-hand-side of the system and its right end bound with the left red rod. Further, there is another strip of the same foam material with the same size, but with its left end being bound with the right red rod and its right end being fixed at the righthand side of the system. The elastic constant of the strips can be found to be 0.509 N∕m. So, a force of 50.9 nN is necessary for the movable rod to move 0.1 μm. Then, from Fig. 4 we can obtain γ −23.68 nN∕mW for the displacement of 0.1 μm. Here, the position of the left red rod is supposed to move from 0.4 to 0.3 μm. The corresponding input control power can be found from Eq. (3) to be P F∕γ 2.15 mW for the coupled cavity system shown in Fig. 3(a). Such a level of optical input power is practical for applications. The movable cylindrical dielectric rods can also be replaced by rods with a different shape, such as a cube or a sphere, if they work better. Furthermore, a coupler is to be used for coupling the signal and control light into the input port. In conclusion, we have demonstrated an all-optical tunable filter, using coupled photonic crystal cavities, with movable rods inserted in the cavities. By analyzing the eigenfrequencies of the cavities, the optical forces acting on the movable rods and filtering properties of the system were calculated. Our theoretical model of this system shows the possibility of tuning the resonant frequency of the cavities using control lights with a different
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frequencies. Also, the tuning of filter frequency is efficient and wide. We are thankful for the support from the National Natural Science Foundation of China (NSFC) (Grant Nos. 61275043, 61171006, 61107049, and 60877034), the Guangdong Province Natural Science Foundation (NSF) (Key project, Grant No. 8251806001000004) and the Shenzhen Science Bureau (Grant No. 200805, CXB201105050064A). References 1. P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007). 2. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, Nature 456, 480 (2008). 3. M. Li, W. H. P. Pernice, and H. X. Tang, Nat. Photonics 3, 464 (2009). 4. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009). 5. D. Van Thourhout and J. Roels, Nat. Photonics 4, 211 (2010). 6. P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, Nat. Commun. 3, 846 (2012). 7. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, Nature 425, 944 (2003). 8. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005). 9. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, Phys. Rev. Lett. 97, 023903 (2006). 10. H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, Phys. Rev. B 78, 165129 (2008). 11. M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, Opt. Express 13, 8286 (2005). 12. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, Opt. Lett. 30, 3042 (2005). 13. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, Nat. Photonics 1, 416 (2007). 14. W. H. P. Pernice, M. Li, and H. X. Tang, Opt. Express 17, 1806 (2009). 15. A. Mizrahi and L. Schachter, Opt. Lett. 32, 692 (2007). 16. M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998). 17. D. Eaves, Handbook of Polymer Foams (Smithers Rapra, 2004). 18. J. Wang, H. Wang, X. Chen, and Y. Yu, J. Mater. Sci. 45, 2688 (2010).