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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 24, DECEMBER 15, 2010 3529 Mode Conversion/Splitting by Optical Analogy of Multistate Stimulated Raman Adiabatic Passage in Multimode Waveguides Shuo-Yen Tseng and Ming-Chan Wu Abstract—We propose and describe mode converter/splitter based on optical analogy of multistate stimulated Raman adiabatic passage in multimode waveguides. Computer-generated planar holograms are used to implement the coupling coefficients that mimic the optical pulses used in the transfer among quantum states of atoms and molecules. The mode coupling properties in multimode waveguides are analyzed using the coupled-mode theory and shown to resemble the multistate stimulated Raman adiabatic passage process. Key features of multistate systems are illustrated with theoretical calculations and numerical examples. Mode converter and splitter are designed based on the theoretical analysis and verified using beam propagation simulations. Index Terms—Coupled mode analysis, gratings, multimode waveguide, optical beam splitting, optical planar waveguides. I. INTRODUCTION I NTEGRATED optical devices are important elements for the development of all-optical networks. One important class of devices is the mode conversion and mode splitting devices. They can be used for mode-division multiplexing in optical interconnects and short-distance communications [1]. For long-distance communications, they can be used to perform key functions such as filtering, power splittering, and multiplexing [2]–[4]. For most of these devices, mode coupling is mediated by system perturbations resulting from periodic index variations, and their characteristics are highly sensitive to the device parameters. More tolerant coupling schemes would be desirable in terms of processing cost reduction. Recently, similarities between quantum mechanics and wave optics have been exploited in the studies of a rich variety of integrated optical devices [5]. Of particular interest is the optical analogy of the stimulated Raman adiabatic passage (STIRAP) phenomena [6]. STIRAP refers to the adiabatic transfer of population between two energy levels in a three-level system via Manuscript received August 10, 2010; revised September 15, 2010; accepted October 21, 2010. Date of publication October 28, 2010; date of current version December 03, 2010. This work was supported in part by the National Science Council of Taiwan under Contract NSC98-2221-E-006-016. S.-Y. Tseng is with the Department of Electro-Optical Engineering, National Cheng Kung University, Tainan 701, Taiwan, and the Advanced Optoelectronics Technology Center, National Cheng Kung University, Tainan 701, Taiwan (e-mail: [email protected]). M.-C. Wu is with the Department of Electro-Optical Engineering, National Cheng Kung University, Tainan 701, Taiwan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2010.2089785 two delayed optical pulses, and it is a robust process insensitive to the precise pulse shapes. In integrated optics, such systems can be mimicked by a coupled three-waveguides structure with spatially varying coupling coefficients for adiabatic power exchange between two waveguides [7]–[9]. Similar schemes have been considered on three-moded waveguide systems with two spatial gratings for slow light propagation [10], [11]. A three-moded adiabatic mode converter based on optical analogy of STIRAP has also been proposed recently [12]. The remarkable feature of such systems is their high tolerance to system parameter variations, because the process is based on adiabatic evolution of a system eigenmode. This feature greatly relaxes the demands on geometrical control as required in conventional devices. Due to the complexity of optical networks, devices involving guided modes need to be considered. As shown multiple in the literature [13], [14], the extension of STIRAP to multistate systems is nontrivial. An important feature of the three-moded STIRAP is that the intermediate mode is not populated during the conversion, owing to the existence of the dark mode which does not involve the intermediate mode [12]. In a multimode system, however, such a dark mode does not exist [13], and some intermediate modes may acquire significant populations during the conversion. The coupling dynamics are more complex and provides richer physics for us to explore. In this paper, we study optical mode conversion/splitting based on the multistate STIRAP process. We consider a modes coupled multimode waveguide structure with by multiplexed long-period gratings with variable coupling strengths. Mode coupling in multimode waveguides can be achieved with tilted gratings [4]. In this work, the gratings are calculated as computer-generated planar holograms (CGPHs) using our previously developed algorithm [15] to create optical analogies of the multistate STIRAP. Coupled-mode theory is used to analyze the mode coupling properties of these devices and to show their resemblance to the multistate STIRAP process. The key difference between three-moded STIRAP and multistate STIRAP is discussed and illustrated using beam propagation simulations. Mode converters/splitters are designed using optical analogies of the multistate STIRAP. II. THEORETICAL ANALYSIS We consider a waveguide supporting distinct forward-propagating modes and use the coupled-mode theory [16] to analyze the effect of CGPHs on the mode evolution in the waveguide. The CGPHs are multiplexed long-period 0733-8724/$26.00 © 2010 IEEE 3530 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 24, DECEMBER 15, 2010 Fig. 1. Schematic of the chainwise-connected system providing the quantum is the analogy of light transfer in a multimode waveguide with CGPH. Rabi frequency of the pulse coupling states jmi and jm i. +1 gratings that couple the propagating modes depending on the grating shapes and periodicity [15]. We also make the usual assumptions that the waveguide is single-moded in the transverse direction, and the effective index method can be used to separate the transverse and lateral components of the electric field. In a -moded multimode waveguide with of mode CGPH, the evolution equation of mode amplitude can be derived as (1) is the spatially varying coupling coefficient bewhere tween modes and . Details of the derivation of (1) and the calculation of coupling coefficients can be found in [17]. Assuming the CGPH only couples the nearest-neighboring modes, a matrix form can be used for (1) (2) is a tri-diagonal matrix where .. . .. . .. . .. . .. . .. . .. . (3) , and denotes the transpose. The matrix is Hermitian [15]. Replacing the spatial variation with the temporal variation , (2)–(3) are used to describe the probability amplitudes of the -state multilevel STIRAP system using the Schrödinger equation under the rotating-wave approximation [13], in which represents the probability amplitudes of the states being populated, and is the Rabi frequency of the pulse coupling states and . Throughout the paper, is assumed to be real and positive without loss of generality. The light transfer among the modes is equivalent to the chainwise-connected system shown schematically in Fig. 1. As a result, the unique features of STIRAP can be observed in a multimode waveguide with CGPH. We can solve for the eigenmodes of , which, under the adiabatic condition (i.e., varies slowly with ), evolve adiabatically without energy exchange among them. For the purto , an adiabatic pose of adiabatic mode conversion from mode , which is equal to at the input and mode at the output, needs to exist. For the purpose of adiabatic mode splitting from to , and , an adiabatic mode , which is equal to at the input and modes , and at the output, needs to exist. With the existence of these modes, the coupling coefficients can be engineered to implement the mode conversion/splitting. Intuitively, for a transition from to via the intermediate states, the pulse sequence (coupling coefficients) coupling the states needs to be applied in a manner such that the earlier the transition, the earlier the pulse arrives. In the STIRAP process, the order of the pulse sequence is reversed: the later the transition, the earlier the pulse arrives [6]. As will be shown in the discussions below, this counterintuitive coupling scheme is required for the STIRAP process. Based on literatures on multilevel systems [13], [18], we first briefly review three-moded systems and then extend the analysis to -moded systems. A. Three-Moded Waveguides When , the equation reduces to the one that describes the three-moded adiabatic mode converter with modes , and . We briefly restate the main results from [12]. An adiabatic mode of with a zero eigenvalue can be found as (4) When the eigenmode is excited initially, the system will remain in it if the coupling coefficients are varied slowly. From (4), if the two variable coupling coefficients and are applied in a counterintuitive scheme, where precedes , then approaches at the input. If is incident in the waveguide, then the eigenmode will be excited, which corresponds to of the unperturbed waveguide. At the output , the eigenmode has adiabatically evolved to , corresponding to of the unperturbed waveguide. As a result, will exit the waveguide. This is the dark mode that facilitates adiabatic conversion of to without populating . We note that in a waveguide with number of modes , an optical analogy of STIRAP involving only three modes , and can be implemented by two gratings coupling modes and for the conversion from to without populating . In the general case presented below, all modes are involved, which represents an optical analogy of the multistate STIRAP process. B. N-Moded Waveguides We only consider the case when is odd because an adiabatic mode does not exist for with an even [13]. A zero eigenvalue exists for with the corresponding adiabatic mode [13], [18] (5) (6) (7) TSENG AND WU: MODE CONVERSION/SPLITTING BY OPTICAL ANALOGY 3531 Fig. 2. Cross-sectional schematic of the ridge waveguide structure used for the device design and simulation. where is a normalization factor. It can be shown that if the and are arranged counterincoupling coefficients tuitively ( precedes ), is equal to at the input at the output [13]. Hence, adiabatic mode conversion and to can be achieved. All even modes are not popufrom lated in the conversion as shown in (7). However, the intermediate odd modes are populated during the conversion. This is STIRAP where the infundamentally different from the termediate state is not involved in the dark mode shown in (4). For example, the adiabatic mode for a five-moded waveguide is Fig. 3. Counterintuitive coupling scheme for mode conversion using optical analogy of the three-moded STIRAP. (8) It is obvious that the even modes and are not involved in the conversion. When the coupling coefficients and are arranged counterintuitively, is equal to at the input and at the output. Comparing (8) with (4), we observe that the intermediate mode is populated, and its population is directly related to the ratios of and to and . This offers us freedom in designing devices with mode splitting functionalities as will be shown in the next section. Fig. 4. (a) Calculated CGPH pattern to implement the counterintuitive coupling coefficients in Fig. 3. (b) WA-BPM simulation of the mode conversion from mode 5 to mode 1 using optical analogy of the three-moded STIRAP. Dashed lines indicate the waveguide core. III. DEVICE DESIGN AND SIMULATION In this section, we design CGPHs on a five-moded waveguide to implement multistate STIRAP for mode conversion/splitting. A polymer ridge waveguide structure as shown in Fig. 2 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 50 mm. The refractive indexes of the layers are given in Fig. 2. The device is designed at a wavelength of 1.55- m input wavelength and the TE polarization. We use a finite-difference mode solver [19] to verify that this waveguide geometry indeed supports five guided modes. Subsequent analysis is performed on the two-dimensional (2-D) structure obtained using the effective index method. The wide-angle beam propagation method (WA-BPM) [20] is used for device simulations with discretization steps m and m. A. Mode Conversion With We first design a mode converter converting to using optical analogy of the three-moded STIRAP discussed as the intermediate mode, we in Section II.A. Choosing design the CGPH with a maximum refractive index modulation to implement the counterintuitive of coupling scheme between modes 1, 3 and 3, 5 as shown in and Fig. 3. The Gaussian shaped coupling coefficients are chosen to mimic the optical pulses used in the transfer among quantum states of atoms and molecules [6]. The exact shape and parameter of the coupling coefficients are not critical as long as the conditions for adiabatic evolution is satisfied [6], [13]. Fig. 4(a) shows the calculated CGPH pattern with the . The CGPH pattern index modulation normalized to is used as an effective index perturbation to the multimode waveguide. Details of the CGPH calculation can be found in [15]. as the input is The WA-BPM simulation result using shown in Fig. 4(b). Fig. 5 shows the power in each guided mode along the propagation distance calculated by mode projection is using the WA-BPM results. As explained in Section II.A, converted adiabatically to with very little excitation of the intermediate mode , and the even modes are not involved in the conversion. B. Mode Conversion With Next, we design a mode converter using optical analogy of the multistate STIRAP discussed in Section II.B and illustrate system described above. its key differences from the 3532 Fig. 5. Evolution of mode power in the waveguide obtained by mode projection using the WA-BPM results in Fig. 4(b). Fig. 6. Counterintuitive coupling scheme for mode conversion using optical analogy of the five-moded STIRAP. A CGPH with a maximum refractive index modulation of is designed to implement the coupling and as shown in Fig. 6. The coefficients coupling coefficients are arranged in a way such that when and arises, the subsystem comprising all intermediate modes is already dressed [14], [21]. Under this condition, is coupled simultaneously to the intermediate modes, which . In such a scheme, the odd intermediate then couples to modes can be suppressed by choosing large and for is reasons given below. As shown in (8), the population in and to and . When related to the ratios of , the population in is much less than and , and as a result, the population in is suppressed throughout the conversion. Large coupling coefficients would correspond to large refractive index perturbations in the CGPH design [15]. In the design shown in Fig. 6, we deliberately set and to be equal to and to show that the is involved in the multistate STIRAP. intermediate mode The calculated CGPH pattern is shown in Fig. 7(a). Again, the exact shapes of the coupling parameters are not critical for the conversion. JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 24, DECEMBER 15, 2010 Fig. 7. (a) Calculated CGPH pattern to implement the counterintuitive coupling coefficients in Fig. 6. (b) WA-BPM simulation of the mode conversion from mode 5 to mode 1 using optical analogy of the five-moded STIRAP. Dashed lines indicate the waveguide core. Fig. 8. Evolution of mode power in the waveguide obtained by mode projection using the WA-BPM results in Fig. 7(b). as the input is The WA-BPM simulation result using shown in Fig. 7(b). Fig. 8 shows the power in each guided mode along the propagation distance. As explained in Section II.B, is converted adiabatically to . However, the adiabatic mode used in the conversion involves , which is excited in the conversion process as shown clearly in the figure. The excitation of intermediate states might be detrimental for atomic and molecular systems if the intermediate states have large decay rates. In multimode waveguides, we can use it to our advantage to perform mode splitting functionalities as shown in the next subsection. C. Mode Splitting With Here, we illustrate mode splitting from to , and . A CGPH with a maximum refractive index modulation of is designed. The coupling coefficients and having the shape of the are shown in Fig. 9, with and having sum of two delayed Gaussian pulses, and the shape of a Gaussian pulse. Their exact shapes can be found in the literature [22]. Referring to (8), with such an arrangement, TSENG AND WU: MODE CONVERSION/SPLITTING BY OPTICAL ANALOGY 3533 Fig. 11. Simulated output power in modes 1 and 5 as a function of CGPH etch depth using mode 5 as the input to the mode converter. Fig. 9. Counterintuitive coupling scheme for mode splitting using optical analogy of the five-moded fractional STIRAP. Fig. 10. Evolution of mode power in the mode splitter using WA-BPM simulation. at the input, corresponding to ; at the output, , corresponding to equal distribu, and . This is an analogy of the fractional tions among STIRAP (f-STIRAP) in which coherent atomic superpositions are created [22]. The final state is determined by the vanishing tails of the coupling coefficients, and the exact shapes of the coupling coefficients are not critical. By adjusting the ratios of the coupling coefficients, the split ratios can be varied arbitrarily. Using the coupling coefficients in Fig. 9, the power in each as the guided mode using the WA-BPM simulation with is split into input is shown in Fig. 10. As expected, the input , and at the output. This design example shows that the optical analogy of f-STIRAP in multistate system can be applied to mode splitting by choosing appropriate coupling coefficients. We note that beam splitting by f-STIRAP is recently demonstrated using three coupled waveguides [23]. Our design example extends the optical analogy of f-STIRAP to multistate systems. IV. DISCUSSION The CGPH in Figs. 4(a) and 7(a) consists of multiplexed long-period gratings with etch depth variations along the propa- gation direction. The fabrication of binary CGPH as surface relief patterns with fixed etch depth has been reported elsewhere [15], [24]. To fabricate the adiabatically varying etch depth in the proposed devices, grayscale lithography, such as the one reported for the fabrication of three-dimensional adiabatic waveguide tapers [25], would be a possible choice, and it allows wafer-scale processing of the devices. From the analysis presented in the previous sections, the light transfer characteristics among the guided modes in multimode waveguides with properly designed CGPHs indeed resembles the STIRAP phenomena. As a result, we expect the devices to be robust against variations in the shapes of the designed coupling coefficients due to fabrication imperfections since the coupling coefficients are equivalent to the pulses considered in the STIRAP process, which is proven to be robust against pulse variations. Detailed analysis on the robustness of STIRAP against pulse intensity variations and delays can be found in the literature [6], [22]. Here, we present simulation results on the variations of the mode conversion characteristics due to etch depth variations of the CGPH pattern. One of the main advantages of devices based on adiabatic mode evolution, despite their relatively large sizes, is their large fabrication tolerance. Using the mode converter introduced in Section III.B, we analyze the robustness of the proposed devices against fabrication errors. From the modal power evolution plot shown in Fig. 8, it is clear that the device is robust against device length variations resulting from typical fabrications errors. , is directly related to the Errors in the CGPH etch depth, effective index modulation, which in turn affects the coupling coefficients. For the ridge waveguide structure in Fig. 2, the efand follows an almost linear fective index modulation /nm as estirelation with a slope of mated by the effective index method. In Fig. 11, using mode 5 as the input to the mode converter, we simulate the power in modes 1 and 5 at the output as a function of etch depth. For a wide variation of over 300 nanometers, the device retains its mode conversion property very well. This robustness is expected due to the adiabatic nature of the conversion process. On the other hand, mode conversion devices relying on grating-assisted mode coupling, while being more compact than the proposed device, require stringent control of the etch depth. 3534 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 24, DECEMBER 15, 2010 V. CONCLUSION In conclusion, we have proposed optical mode converter/splitter using optical analogy of multistate STIRAP. This scheme allows adiabatic mode conversion/splitting in multimode waveguides without strict restrictions on the coupling parameters. Mode conversion using multistate STIRAP would excite the intermediate modes, while the three-moded STIRAP converts the modes without the excitation of the intermediate mode. The properties of multistate STIRAP allow for the design of mode splitters using optical analogy of the fractional multistate STIRAP. The theoretical calculations are verified with beam propagation simulations. REFERENCES [1] J. B. Park, D.-M. Yeo, and S.-Y. 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[24] 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, no. 4, pp. 512–514, 2009. [25] A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. W. Prather, “Fabrication and characterization of three-dimensional silicon tapers,” Opt. Exp., vol. 11, no. 26, pp. 3555–3561, 2003. 2 Shuo-Yen Tseng received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1999 and the M.S. and Ph.D. degrees in electrical engineering from the University of Maryland, College Park, USA, in 2002 and 2006, respectively. In 2007, he joined Georgia Institute of Technology as a post doctoral fellow in Prof. Bernard Kippelen’s Research Group where he studied organic photonics. In August 2008, he joined the faculty at the National Cheng Kung University, Tainan, Taiwan, as an Assistant Professor in the Department of Electro-Optical Engineering. His research interests include optical communications, numerical simulation, nanotechnology and integrated photonics. Dr. Tseng is a member of the Optical Society of America (OSA). Ming-Chan Wu received the B.S. degree in physics from Soochow University, Taipei, Taiwan, in 2003 and the M.S. degree in optics and photonics from the National Central University, Jhongli, Taiwan, in 2007. He is currently a Research Assistant at the Department of Electro-Optical Engineering, National Cheng Kung University, Tainan, Taiwan.