IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 23, NO. 12, JUNE 15, 2011
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Adiabatic Mode Conversion in Multimode Waveguides Using Chirped Computer-Generated Planar Holograms Ming-Chan Wu, Fu-Chen Hsiao, and Shuo-Yen Tseng
Abstract—We discuss the design of a mode converter based on optical analogy of adiabatic passage via a level crossing in multimode waveguides. Computer-generated planar holograms are used to implement the chirped gratings that mimic the laser excitation used in the transfer between quantum states of atoms and molecules. The mode coupling properties are analyzed using the coupled mode theory and shown to resemble the level crossing process. Key features of the devices are illustrated with theoretical calculations and numerical examples. The proposed mode converter shows a bandwidth in excess of 300 nm in the optical communications band. Index Terms—Coupled mode analysis, gratings, multimode waveguide.
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I. INTRODUCTION
OR THE development of modern optical communications networks, several devices utilizing mode conversion in waveguides have been proposed and demonstrated [1]–[3]. These devices typically use tilted Bragg gratings (TBGs) to produce the mode conversion, and the spectral response of TBGs [4] makes them suitable for use in wave division multiplexing (WDM) systems for long-haul communications. On the other hand, the cost of WDM systems may be prohibitive for their applications in short distance communications and optical interconnects. An attractive alternative to WDM is the mode division multiplexing (MDM) [5], in which each mode represents an independent data channel. Several basic building blocks for MDM, such as the mode add/drop multiplexer [6] and the mode generator [7], have been proposed. A common feature of these devices is that adiabatic design schemes were used. Adiabatic devices are expected to be robust against process variations and broadband [8], [9], making them cheaper to produce and more compatible with the low-cost components typically used for short distance communications. Recently, we proposed adiabatic mode conversion and mode splitting devices based on the quantum-optical analogy of stimulated Raman adiabatic passage (STIRAP) [10] in engineered
Manuscript received February 18, 2011; revised March 17, 2011; accepted March 26, 2011. Date of publication April 05, 2011; date of current version May 25, 2011. M.-C. Wu and F.-C. Hsiao are with the Department of Electro-Optical Engineering, National Cheng Kung University, Tainan 701, Taiwan. S.-Y. Tseng is with the Department of Electro-Optical Engineering, National Cheng Kung University, Tainan 701, Taiwan, and also with the Advanced Optoelectronic Technology Center, National Cheng Kung University, Tainan 701, Taiwan (e-mail:
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2011.2138127
multimode waveguides [11], [12]. These devices are robust against process variations in the same way that the STIRAP process is robust against laser pulse parameter variations. In the STIRAP conversion process, two modes are coupled by one or more intermediate modes using multiple delayed laser pulses. To mimic this process in multimode waveguides, two or more gratings are needed to generate the required coupling conditions [11], [12], and this increases the design and fabrication complexity of these devices. Moreover, these devices only support unidirectional conversion, for example, from mode A to mode B, but not from mode B to mode A. On the other hand, the idea of light power transfer in coupled waveguides via level crossing and its analogy with the Landau Zener problem of quantum physics has recently been proposed and demonstrated [13]. Implementation of such a scheme using chirped gratings for 3-dB mode conversion has also been demonstrated [14]. Such devices support bidirectional conversion and require only a single chirped grating. In this work, we present the analysis of adiabatic mode conversion in multimode waveguides by chirped computer-generated planar holograms (CGPHs) [15]. Mode converter is designed and simulated using optical analogy of adiabatic passage via a level crossing. The performance of these devices is numerically investigated. II. THEORETICAL ANALYSIS We assume a multimode waveguide supporting distinct spatial modes with complex mode amplitude and propagation constant corresponding to the th mode. Consider two spatial modes and coupled by a CGPH with coupling coefficient (real) and period , the evolution equation for mode amplitudes and obeys the following coupled mode equations [16]
(1) (2) where . Details of the calculation of in [17]. By introducing new variables and requiring simplified coupled equations in matrix form as
1041-1135/$26.00 © 2011 IEEE
can be found , , we obtain
(3)
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where defining
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 23, NO. 12, JUNE 15, 2011
. Transferring (3) to a rotating frame by , we obtain (4)
where with . Replacing the spatial variation with the temporal variation , (4) is equivalent to the time-dependent Schrödinger equation describing the interaction dynamics of a two-state system driven by a coherent laser excitation, and is the Hamiltonian. The off-diagonal term is the Rabi frequency characterizing the interaction strength, and in our case, indicates the coupling strength. is the frequency detuning between the laser carrier frequency and the Bohr transition frequency , and indicates the grating period detuning in our case. After the transformation, the propagation constants corresponding to the diabatic modes and are now 0 and , respectively. Solving for the eigenvectors of , we find two adiabatic modes, which are -dependent superpositions of and
Fig. 1. Calculated chirped CGPH pattern to implement adiabatic passage via a level crossing. Dashed lines indicate the waveguide core.
(5) (6) where . The propagation constants of the adiabatic modes are the two eigenvalues of , (7) When the evolution is adiabatic, the adiabatic propagation constants have an avoided crossing; there is no transition between the adiabatic modes, and their populations are conserved. It is apparent from (5) and (6) that if only one adiabatic mode is excited initially, the populations in the two diabatic modes and can be adjusted through by changing the CGPH parameters and . Assuming the detuning sweeps slowly from some large negative value at the input to some large positive value at the output with a constant , the propagation constants of the two diabatic modes cross when the detuning is zero, and the adiabatic propagation constants approach the diabatic ones 0 and at the input/output. In other words, because and , approaches , and approaches at the input. At the output, approaches , and approaches . Consequently, would evolve adiabatically to using , and would evolve adiabatically to using . Bidirectional mode conversion can thus be accomplished. III. DEVICE DESIGN AND SIMULATION In this section, we design a chirped CGPH on a five-moded waveguide to implement adiabatic mode conversion between modes 1 and 2 via a level crossing. A polymer ridge waveguide structure is chosen with the following design parameters: 3 m thick SiO on a Si wafer for the bottom cladding layer, 2.4 m layer of Cyclotene (BCB) for the core, the width of the waveguide is 3 m, and the length of the waveguide is 25 mm. The refractive indexes are assumed to be 1.536 for the BCB and 1.46 for the SiO . The device is designed for an input wavelength of 1.55 m and TE polarization. Subsequent analysis is performed
Fig. 2. WA-BPM simulation of mode conversion from mode 1 to mode 2. Light intensity is shown. Dashed lines indicate the waveguide core.
on the two-dimensional (2-D) structure obtained using the effective index method. The wide-angle beam propagation method (WA-BPM) [18] is used for device simulations with discretizam and m. tion steps We design the chirped CGPH with a maximum refractive to implement the index modulation of sweeping detuning as described in the previous section. The deis swept from at the input to at tuning calculated for modes 1 and 2 the output, with at 1.55 m and the TE polarization in the unperturbed waveguide. Fig. 1(a) shows the calculated chirped CGPH pattern . The chirped with the index modulation normalized to CGPH pattern is used as an effective index perturbation to the multimode waveguide. Details of the CGPH calculation can be found in [19]. The WA-BPM simulation results using mode 1 and mode 2 as the input are shown in Fig. 2 and Fig. 3, respectively. As explained in Section II, modes 1 and 2 are converted adiabatically to modes 2 and 1. This device indeed supports bidirectional conversion. From the analysis presented above, the light transfer characteristics between the guided modes in multimode waveguides with properly designed chirped CGPHs indeed resembles adiabatic light passage via a level crossing. As a result, we expect the devices to be robust against fabrication imperfections since the coupling coefficient is equivalent to the Rabi frequency, which is proven to be robust against pulse variations [13]. Here, we present simulation results on the spectral characteristics of the mode converter. In Fig. 4, using mode 2 as the input to the mode
WU et al.: ADIABATIC MODE CONVERSION IN MULTIMODE WAVEGUIDES USING CHIRPED CGPHs
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mode equations (CMEs) agree quite well with the results from the WA-BPM simulation. IV. CONCLUSION In conclusion, an optical mode converter using chirped CGPH is analyzed. Compared with previously proposed devices using optical analogy of STIRAP, this device requires only a single grating and supports bidirectional mode conversion. This device has a bandwidth in excess of 300 nm due to the adiabatic nature of the conversion process. These features make the proposed device well-suited for applications in short distance optical communications using MDM. Fig. 3. WA-BPM simulation of mode conversion from mode 2 to mode 1. Light intensity is shown. Dashed lines indicate the waveguide core.
Fig. 4. WA-BPM simulation results of modal power evolution in modes 1 and 2 using mode 2 as the input to the mode converter for different wavelengths.
Fig. 5. Coupled mode equations (CMEs) (3) solution of modal power evolution in modes 1 and 2 using mode 2 as the input at 1.55 m. WA-BPM results are also shown for comparison.
converter, we simulate the power evolution in modes 1 and 2 for different wavelengths spanning from 1.30 m to 1.65 m using the WA-BPM. From the -band to the -band, the device retains its mode conversion property very well. This robustness is expected due to the adiabatic nature of the conversion process. The level crossing occurs at different positions along the propagation direction due to different zero detuning points for different wavelengths. For a step index multimode increases linearly with the wavelength [19]. As waveguide, a result, when the input wavelength is increased, the crossing is more negmoves towards the input where the detuning ative. We also numerically solve the coupled mode equations in (4) using the values obtained from our device parameters at 1.55 m as shown in Fig. 5. The solutions from the coupled
REFERENCES [1] J. M. Castro, D. F. Geraghtya, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express, vol. 13, pp. 4180–4184, 2005. [2] J. B. Khurgin, M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Add-drop filters based on mode-conversion cavities,” Opt. Lett., vol. 32, pp. 1253–1255, 2007. [3] D. Runde, S. Brunken, S. Breuer, and D. Kip, “Integrated-optical add/ drop multiplexer for DWDM in lithium niobate,” Appl. Phys. B, vol. 88, pp. 83–88, 2007. [4] T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Amer. A, vol. 13, no. 2, pp. 296–313, 1996. [5] S. Berdagué and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt., vol. 21, pp. 1950–1955, 1982. [6] M. Greenberg and M. Orenstein, “Multimode add-drop multiplexing by adiabatic linearly tapered coupling,” Opt. Express, vol. 13, pp. 9381–9387, 2005. [7] J. B. Park, D.-M. Yeo, and S.-Y. Shin, “Variable optical mode generator in a multimode waveguide,” IEEE Photon. Technol. Lett., vol. 18, no. 20, pp. 2084–2086, Oct. 15, 2006. [8] A. A. Rangelov, U. Gaubat, and N. V. Vitanov, “Broadband adiabatic conversion of light polarization,” Opt. Commun., vol. 283, pp. 3891–3894, 2010. [9] S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev., vol. 3, pp. 243–261, 2009. [10] K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys., vol. 70, pp. 1003–1025, 1998. [11] S.-Y. Tseng and M.-C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett., vol. 22, no. 16, pp. 1211–1213, Aug. 15, 2010. [12] S.-Y. Tseng and M.-C. Wu, “Mode conversion/splitting by optical analogy of multistate stimulated Raman adiabatic passage in multimode waveguides,” J. Lightw. Technol., vol. 28, no. 24, pp. 3529–3534, Dec. 15, 2010. [13] F. Dreisow, A. Szameit, M. Heinrich, S. Nolte, A. Tünnermann, M. Ornigotti, and S. Longhi, “Direct observation of Landau-Zener tunneling in a curved optical waveguide coupler,” Phys. Rev. A, vol. 79, p. 055802, 2009. [14] P. S. Chung and M. G. F. Wilson, “Optical mode conversion using chirped gratings,” Electron. Lett., vol. 17, pp. 14–15, 1981. [15] S.-Y. Tseng, S. Choi, and B. Kippelen, “Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,” Opt. Lett., vol. 34, pp. 512–514, 2009. [16] H.-C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express, vol. 17, pp. 11710–11718, 2009. [17] S.-Y. Tseng, C. Fuentes-Hernandez, D. Owens, and B. Kippelen, “Variable splitting ratio 2 2 MMI couplers using multimode waveguide holograms,” Opt. Express, vol. 15, pp. 9015–9021, 2007. [18] G. R. Hadley, “Wide-angle beam propagation using Padé approximation operators,” Opt. Lett., vol. 17, pp. 1426–1428, 1992. [19] S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, “Implementation of discrete unitary transformations by multimode waveguide holograms,” Appl. Opt., vol. 45, pp. 4864–4872, 2006.
