Compact beam splitters in coupled waveguides using shortcuts to adiabaticity

Transcription

1 Journal of Optics PAPER Compact beam splitters in coupled waveguides using shortcuts to adiabaticity To cite this article: Xi Chen et al 08 J. Opt View the article online for updates and enhancements. This content was downloaded from IP address on 5/03/08 at 0:50

2 Journal of Optics J. Opt. 0 (08) (9pp) Compact beam splitters in coupled waveguides using shortcuts to adiabaticity Xi Chen,3, Rui-Dan Wen, Jie-Long Shi and Shuo-Yen Tseng,3 Department of Physics, Shanghai University, Shanghai, People s Republic of China Department of Photonics, National Cheng Kung University, Tainan 70, Taiwan xchen@shu.edu.cn and tsengsy@mail.ncku.edu.tw Received 8 September 07, revised 4 January 08 Accepted for publication 6 February 08 Published 4 March 08 Abstract There are various works on adiabatic (three) waveguide coupler devices but most are focused on the quantum optical analogies and the physics itself. We successfully apply shortcuts to adiabaticity techniques to the coupled waveguide system with a suitable length for integrated optics devices. Especially, the counter-diabatic driving protocol followed by unitary transformation overcomes the previously unrealistic implemention, and is used to design feasible and robust and 3 beam splitters for symmetric and asymmetric three waveguide couplers. Numerical simulations with the beam propagation method demonstrate that these shortcut designs for beam splitters are shorter than the adiabatic ones, and also have a better tolerance than parallel waveguides resonant beam splitters with respect to spacing errors and wavelength variation. Keywords: shortcuts to adiabaticity, coupled waveguide, multiple beam splitter (Some figures may appear in colour only in the online journal). Introduction With the development of integrated optics [ 6], directional couplers, switches, and beam splitters in various optical waveguide structures play significant roles in optical circuits and communications. Such devices also provide a platform for optical simulation of NOON states in quantum optics [7, 8]. Among them, the design of beam splitters, which divide an input signal into several outputs with a specific ratio, is quite demanding and usually achieved with Y-branch junctions and directional couplers. Two major schemes, i.e. resonant coupling and adiabatic coupling are normally exploited in the designs. The resonant coupling with parallel waveguides is quite simple, but the output power changes periodically with propagation distance, and the efficiency is sensitive to the fluctuation of wavelength and refractive index. To remedy it, adiabatic coupling in two-by-two couplers [] and three waveguide couplers [3 5] is proposed to achieve a stable output power with wavelength insensitivity and fabrication tolerance. However, the adiabatic devices require a long length to satisfy the adiabatic criteria [6], which might be 3 Authors to whom any correspondence should be addressed. problematic, especially when minimizing integrated optics devices. The analogy between quantum mechanics and wave optics provides an elegant tool to design optical waveguide arrays and even photonic lattices, as shown in a review by S Longhi [9]. Various adiabatic (three) waveguide coupler devices [0 0], including high-tolerance directional couplers and broadband beam splitters, are proposed based on the optical analogy of a stimulated Raman adiabatic passage (STIRAP) or coherent tunnelling adiabatic passage (CTAP) in quantum optics; see the current review [, ]. Recently, shortcuts to adiabaticity (STA) have been proposed to speed up adiabatic state evolution in quantum systems [3]. Similar to superadiabatic evolution in quantum physics [4], the STA schemes achieve the reproduction of the same final populations, or the same final state, as the adiabatic process but in a shorter time. Two main shortcut protocols that have received a lot of attentions are the inverse engineering by Lewis- Riesenfeld invariants [5, 6] and counter-diabatic driving (or quantum transitionless driving) [7 33]. Both STA techniques have been fully exploited for light manipulation in optical two and three waveguide structures with applications /8/ $ IOP Publishing Ltd Printed in the UK

3 Figure. Schematic diagram for the asymmetric (a), symmetric (b) triple wells, and the analogous asymmetric (c), symmetric (d) optical waveguides with designed coupling J and mismatch of propagation constants Δ and U, corresponding to the waveguide spacing D and width W j ( j L, M, R). in mode transformers [34 36], directional couplers [38 44], and others [45, 46]. However, the counter-diabatic driving requires an additional coupling [3], which is generally difficult or even impossible for practical implementations [47]. Particularly, the existence of non-adjacent or imaginary coupling coefficients makes the scheme unrealistic in three waveguide beam splitters. Currently, we first have managed to reduce the quantum three-level systems to effective two-level problems at large intermediate-level detuning, or on resonance, apply counter-diabatic driving along with the unitary transformations and eventually modify the pump and Stokes pulses for achieving fast and high-fidelity population transfer [48, 49]. The implementation of counterdiabatic driving but without additional coupling could provide new applications in optical three-waveguide couplers and splitters. In this paper, we apply the technique of STA to design compact three waveguide beam splitters, instead of two waveguide beam splitters [38 40]. What we consider here are the coupled asymmetric and symmetric three waveguides, with the corresponding coupled mode equations satisfying SU() and U3S3 Lie algebra. We first calculate directly the counter-diabatic terms to achieve the shortcut. Instead of resorting to the non-adjacent coupling or imaginary coupling coefficient, a proper unitary transformations [47] that modifies the geometrical parameters of the waveguides allows us to get the desired results. Numerical simulations by the beam propagation method (BPM) demonstrate that these shortcut beam splitters are more compact than those designed by (fractional) STIRAP and its variants, and are more robust against spacing error and wavelength variation as compared to the conventional parallel waveguides resonant beam splitters.. Model and hamiltonian Our model consists of three coupled waveguides, as shown in figure, where the refractive index and geometry of the waveguides are allowed to vary along the propagation direction z. Under the scalar and paraxial approximation and the assumption of weak coupling, the variations in the guided-mode amplitudes of individual waveguides, Y [ A,A,A 3] T, with propagation distance is described by the coupled-mode equations as follows, i dy d z H 0 () z Y, where the Hamiltonian is (by setting º ) U +D J H0 () z J 0 J. () 0 J U -D Here, the matrix element J (real) denotes the coupling coefficients, the diagonal elements U and Δ describe the mismatch of propagation constants for the fundamental modes of the individual waveguides. In the simplest resonant case with Δ U 0, we have three parallel identical (resonant) waveguides coupler and the guided-mode amplitudes of each waveguide can thus be analytically calculated as Y( z) [-isin( Jz), cos( Jz), -i sin( Jz) ] T with the initial condition (0,,0) T. So we can achieve beam splitters at the length of L p ( J) andalso 3 one at the length of

4 L arcsin( 3 ) ( J). However, such a resonant scheme is usually sensitive to the parameter fluctuations, e.g. spacing error or input wavelength variation, as compared to the adiabatic passage [50], which is also true in coupled two waveguides [38]. Alternatively, we can apply the fractional STIRAP for beam splitting in optical waveguides with controllable coupling J satisfying the adiabatic criteria [4], even in the more general case, D ¹ 0 or U ¹ 0.Theadia- batic schemes are applicable to achieve robust beam splitters, but require relatively long device lengths. So, in order to combine the advantages of resonant and adiabatic techniques, we apply STA to design short and robust and 3 beam splitters using the common asymmetric ( D ¹ 0 and U 0) and symmetric (Δ 0 and U ¹ 0) coupled optical waveguides, as shown in figure. Specifically, we utilize the counter-diabatic driving and unitary transformationtodesignfeasiblecouplingcoefficients and mismatch. This is enabled by the fact that the asymmetry and symmetry of such structures respectively satisfy SU() and U3S3 Lie algebra, thus allowing us to apply the schemes in [5]. 3. Counter-diabatic driving and unitary transformation In this section, we shall begin with an introduction to the concept of counter-diabatic driving [8, 9], or quantum transitionless algorithm [30, 3]. This approach suggests that a supplemental counter-diabatic interaction Hamiltonian H cd can be added to cancel the diabatic transition of the reference Hamiltonian H 0. That is, the exact state evolution of the supplemented Hamiltonian H 0 + H cd would now coincide with the adiabatic states (eigenstates) of the reference Hamiltonian H 0 in arbitrarily short times, implying the acceleration of the slow adiabatic state transfer. Mathematically, the supplemented Hamiltonian H(z) in this problem of coupled waveguides can be constructed as H(z) H 0 (z) + H cd (z) with Hcd i å n zfn( z) ñáfn( z), where fn () z ñ (n,, 3) are the instantaneous eigenstates of H Asymmetric three waveguides For the asymmetric three waveguides (U 0), the instantaneous eigenstates of Hamiltonian () are given by ( cosj) f,3 ( z) ñ sinj, ( ) ( cosj) -sin j f ( z ) ñ cosj, 3 ( ) sin j with the eigenvalues E,3 E0 º D + J and E 0. The counter-diabatic term is obtained as H cd () z 0 -ij ij 0 -ij, ( 4) 0 ij 0 where j arctan( J D). The dot denotes the derivative with respect to z. Obviously, the implementation of such an interaction is problematic, since the coupling coefficients become imaginary. To overcome the difficulty, we apply the unitary transformation, U () z e -ial 3 (Λ 3 represents the matrix element of Lie algebra SU(), see appendix), so that the Hamiltonian, H () z U zh() zu() z -iu () zu() z, becomes H D-a ei a ( J -ij / ) e-i a( J + ij 0 ei a / ) ( J - ij / ). ( 5) 0 e-i a ( J + ij / ) -D + a In order to make the off-diagonal terms ( J cos a + j sin a ) ij ( sin a - j cos a ) real, one has to impose the condition, a() z arctan( j E0 sin j). Substituting it into equation (5), wefinally write down the following Hamiltonian: D J H J 0 J, ( 6) 0 J -D where J j + E sin 0 j, ( 7) sin j E0 3 cos jsin j E0j E0 cot jj j" D [ + + ( - ) ]. j + E0 sin j ( 8) Because we apply the unitary transformation to cancel the unfeasible imaginary coupling coefficients, the dynamics are now governed by the Hamiltonian H ()and z is described by Y () z U () z Y() z. In order to achieve the exact target state beginning with the common initial state, i.e. H () 0 H (), 0 H ( L) H ( L), the initial and final boundary conditions, U () 0 UL ( ) 0, should be guaranteed. 3.. Symmetric three waveguides Similarly, for symmetric three waveguides (Δ 0), the instantaneous eigenstates are also calculated as cosj f,3 ñ cosj, ( 9) cosj fñ 0, 0 ( ) - with the eigenvalues E,3 E0( cos j ) and E E0 cos j, where j arctan( J U) and E 0 U + 8J. The counter-diabatic term in this case takes the 3

5 Figure. Cross-sectional schematic of the polymer channel waveguide structure considered for device design. form H cd 0 -ij ij 0 -ij. ( ) 0 ij 0 Again, the unitary transformation, U () z e -iag 4 (G 4 is defined in appendix), with a() z arctan( j E0 sin j) is applied to obtain the following Hamiltonian: U J H J 0 J, ( ) 0 J U with U J j + E0 sin j, ( 3) sin j[ E0 3 cos jsin j + E 0j E0 cot + ( jj - ) j" ]. j + E0 sin j ( 4) With these supplemented and unitarily transformed Hamiltonians (6) and (), the system evolution will now follow U fñ exactly, where fñ could be any of the instantaneous eigenstates of the reference Hamiltonian in (), (3), and (9). It is clear that the evolution of these states are governed by a single parameter j, so we are now ready to choose the right ansatz of j to realize short beam splitters by determining the coupling coefficients J and mismatch of propagation constants Ũ or D through (7), (8), (3), and (4). 4. Beam splitters: implementation and simulation In this section, we turn to illustrate the design of short beam splitters in a realistic optical waveguide platform using the shortcuts mentioned above, and perform simulation of the beam splitters using a beam propagation method (BPM) [5]. The BPM code used in the simulations solves the scalar and paraxial wave equation using the finite difference scheme with the transparent boundary condition and is well-suited for the simulation of weakly guiding waveguide structure considered in this work. The cross-section of the polymer channel waveguide structure considered here is shown in figure,where 3 μm thicksio (n.46) on a Si (n 3.48) wafer is used for the bottom cladding layer, the core consists of a.4 μm layerof BCB (n.53), and the upper cladding is epoxy (n.50). This weakly-guided coupled waveguide structure can be welldescribed by the scalar and paraxial wave equation, and the evolution of the guided modes in the coupler can be accurately described by equation (). A input wavelength of λ.55 μm and the TE polarization is considered, and the device length is chosen to be L 000 μm. In the simulation, the default widths for the left, middle, and right waveguides are chosen to be W R W M W L μm. The relation between the mismatch D (or Ũ) and width difference δw W L,R W M can be approximated by a linear relation [], and the relation between the coupling coefficient J and waveguide separation D in a symmetric coupled waveguide is well fitted by the exponential relation J J0exp[ -g ( D - D0)] [5]. Wealsoassumethat the exponential relation can be used to obtain an estimation of the coupling coefficient in the asymmetric coupled waveguide. The waveguide spacing D and width difference dw are then adjusted along the propagation direction to satisfy the designed functions of J, D or Ũ. 4.. beam splitter First of all, we design the beam splitter using asymmetric three waveguides. We assume that the light transfer begins with the middle waveguide, Ψ (0) (0,, 0) T, and ends up in the left and right waveguides with equal power, Y ( L) (-, 0, ) T, along the instantaneous eigenstates f() z ñ in equation (3). So at the edges we have to impose the following boundary conditions: j( 0) 0, j( L) p, j ( 0) j ( L) 0, ( 5) to make the initial and final states coincide with the instantaneous eigenstates of Hamiltonian, and also to guarantee that the counter-diabatic terms vanish at z 0 and z L, i.e., H cd (0) H cd (L) 0. More boundary conditions j " () 0 j " ( L) 0, are required to satisfy U () 0 UL ( ) 0. Additionally, we also impose j ( L) 0 to avoid the divergence of the function D at z L. Now we can interpolate the function j(z) by choosing the 6 polynomial ansatz, j () z j ajzj å 0, where a j is solved analytically by using the boundary conditions above. Regarding the function E 0 (z), we simply choose the polynomial ansatz, E z j ajzj 0() å 0,toobtainE0() z z L - z L3, by setting the boundary conditions E0( 0) E0( L) 0, E0( L ) ¹ 0. ( 6) 4 Noting that the boundary condition at z L/ is used, simply to make E 0 (z) nonzero and nonnegative. With the help of functions j(z) and E 0 (z), equations (7) and (8) give the coupling coefficient J and mismatch of the propagation constant D designed by counter-diabatic driving shown in figure 3(a). Basedonthe linear relation mentioned above, the waveguide widths, Wj ( j L, M, R), are further shown in figure 4(a), where waveguide widths W M μm (solid black line), 4

6 Figure 3. Function of designed coupling coefficient J and mismatch of propagating constants D or Ũ in asymmetric (a), (c) and symmetric (b), (d) triple-well waveguide couplers for beam splitter (a), (b) and 3 beam splitter (c), (d). The device lengths are L 000 μm. Figure 4. Waveguide parameters for and 3 beam splitters, where (a), (c) waveguide widths W j ( j L, M, R) and (b), (d) waveguide separation D. Parameters: WM mm (solid black line), WLR, WM + dw, depending on D and Ũ in the asymmetric (a), (b) and symmetric (c), (d) triple-well waveguide couplers. Dashed blue and dotted red lines represent the cases of and 3 beam splitters, respectively. 5

7 Figure 5. Beam propagation simulations of (a), (b) and 3 polymer beam splitters in the asymmetric (a), (c) and symmetric (b), (d) triple-well waveguide couplers, where white lines indicate the waveguide cores. The device cross-section is shown in figure. WLR, WM dw and δw is proportional to D due to the asymmetric structure. Figure 4(b) also depicts the waveguide separation D (dashed blue line), whered > D μm with J mm and g.409 mm -. In the symmetric three waveguides, we choose the protocol for beam splitter from Y () 0 ( 0,, 0) T to Y ( L) (, 0, ) T along the instantaneous eigenstate f3() z ñ in equation (9). In this case, the boundary condition (5) becomes j( 0 ) p, j( L) 0, ( 7) and the other previously imposed boundary conditions in the case of the asymmetric waveguides are also applied. The functions j(z) and E 0 (z) are again interpolated by polynomial ansatz, and the designed J and Ũ are shown in figure 3(b). As a consequence, the geometric parameters, waveguide widths Wj ( j L, M, R) and waveguide separation D for this symmetric waveguides are shown by dashed lines in figures 4(c) and (d), where WLR, WM + dw are undistinguishable and δw is proportional to Ũ due to the symmetric structure. In figures 5(a) and (b), BPM simulation demonstrates beam splitters using asymmetric/symmetric three waveguides with the shortcut design. The fundamental mode of the middle waveguide is excited at z 0, and the power is gradually divided into the left and right waveguides with the same amplitudes and finally ends up with the /:/ ratio at z L. Clearly, beam splitters can be achieved with short length in both asymmetric and symmetric configurations. However, when the length becomes shorter, the dramatic change of waveguide width difference δw (corresponding to mismatch of the propagating constant D or Ũ) makes such a device lossy due to excitation of the leaky modes. In this sense, the symmetric beam splitter might be more efficient than that of the asymmetric one, because the geometry of the symmetric beam splitter is in general smoother, as shown in figures 5(a) and (b) beam splitter Next, we shall design the 3 beam splitter using asymmetric three waveguides. We start from the initial state, Y () 0 ( 0,, 0), T to the final state Y ( L) (- 3, 3, 3) T, along the instantaneous eigenstate f() z ñ in equation (3). To do this, we set the boundary condition, j( 0) 0, j( L) arctan, ( 8) and the other boundary conditions on the first, second and third derivatives used previously are again imposed here. Similarly, we can also design the symmetric 3 beam splitter, starting from the initial state, Y () 0 ( 0,, 0),to T the final state Y ( L) ( 3, 3, 3) T, along f3() z ñ in equation (9). In this case, the boundary condition should be 6

8 modified as j( 0 ) p, j( L) arccos( 3 ), ( 9) and the others are again imposed. With the boundary conditions, the function j(z) can be interpolated by simple polynomial ansatz. In addition, the function E 0 (z) is also constructed as done previously for the beam splitters. Figures 3(c) and (d) illustrate the designed J, D, and Ũ using asymmetric and symmetric three waveguides, respectively. Again, the geometric parameters, waveguide widths W j ( j L, M, R) and waveguide separation D, for both asymmetric and symmetric cases are also displayed by dotted red lines in figure 4. BPM simulation results are finally illustrated in figures 5(c) and (d) showing complete 3 beam splitting using the shortcut design. Similarly, when the length becomes shorter, the dramatic change of waveguide width difference d W (corresponding to mismatch of the propagating constant D or Ũ) for both the symmetric and asymmetric beam splitters, as shown in figures 5(c) and (d), will make such a device lossy due to excitation of the leaky modes Stability Finally, we shall shed light on the efficiency of the designed and 3 beam splitters. Errors in the waveguide geometry translate to errors in the coupling coefficient J and mismatch Δ (or U), which in turn change the splitting ratios. In practical applications, the coupling coefficient J is proportional to the waveguide width through a quadratic relation [5], and is related to the waveguide spacing through the exponential relation described earlier. Also, we model the error in the mismatch by changing the input wavelength because it is the major contributor in Δ (or U) variations as compared to geometrical effects [, 3]. To quantify the robustness of such devices against the fluctuation of input wavelength and spacing error, we define d S- + S + S3 -, ( 0 ) for beam splitter, and d 3 S- + S - + S , ( ) for 3 beam splitter, where S j ( j,, 3) are the normalized power in each waveguides. Ideally, the power splitting ratio at the final length z L should be /:/ for and /3:/3:/3 for 3 beam splitters, respectively. So the quantities δ and δ 3 denote the deviation from the ideal target value. Figure 6 shows that the beam splitters designed here are insensitive to the wavelength fluctuation and spacing error. In general, the designed beam splitters with short length are more robust as compared with the resonant parallel waveguides couplers. (see also [50], in which the robustness of counter-diabatic driving and resonant scheme is discussed). However, the performance of beam splitters is in general better than that of 3 beam splitters, since the values of geometric parameters for beam splitters are larger, see figure 4, and the relative errors in wavelength input and spacing separation are smaller. In addition, the exponential relation between the coupling coefficient J and waveguide separation D is more valid in a symmetric coupled waveguide. Therefore, the symmetric structure works better in both and 3 beam splitters than the asymmetric one. As a matter of fact, there is, in practice, a physical constraint on shortcut design in optical waveguide couplers. When the device length becomes shorter, dramatic changes in waveguide spacing and width will be required, which leads to the excitation of leaky modes and energy loss during propagation. While the lengths of the demonstrated adiabatic (three) waveguide coupler devices are on the order of 0 to 0 mm [0 8], it is clear that our proposed devices at L 000 μm indeed provide shortcuts to adiabatic light transfer in weaklyguided coupled waveguides. Although the adiabatic devices are more robust than the proposed designs, the current designs are an order of magnitude shorter than the adiabatic ones and show better tolerance than the parallel waveguides resonant coupling designs. We also stress that the adiabatic schemes rely on the evolution of the dark mode consisting of the linear superposition of [, 0, 0] T and [ 0, 0, ] T [3], as a result, the middle waveguide is not populated during the evolution, thus limiting their application to the design of 3 beam splitters. On the other hand, the STA based design can achieve an arbitrary ratio (three) waveguide beam splitter by simply modifying the boundary conditions of the instantaneous eigenstates. The design procedures described in this work can be applied in general to the design of beam splitters in high index-contrast platforms, such as silicon-on-insultaor (SOI), as long as the coupled-mode formalism is valid [36, 37], leading to more densely packed and compact structures. 5. Conclusion In summary, we have proposed short and robust and 3 beam splitters using shortcuts to adiabaticity. In detail, the geometry of asymmetric and symmetric three waveguides, including the waveguide spacing and widths, are designed in terms of the counter-diabatic driving and unitary transformation to realize the shortcut designs for beam splitters. These results have been verified by BPM simulations. More interestingly, the compact beam splitters obtained here is shorter than the conventional adiabatic ones, but have higher stability, with respect to spacing error and input wavelength variation, than the parallel resonant waveguides. Here, we concentrate on the counter-diabatic driving and its applications in the symmetric and asymmetric three coupled waveguides. However, one can hybridize the inverse engineering and optimal control theory to design the optimal beam splitting with minimizing the influence of spacing error and wavelength variation [39, 44]. Of course, compact multiple beam splitting can be also designed in N coupled waveguides by using STA and Lie transforms [5]. In a word, all of these results are relevant to the non-adiabatic control of light propagation in optical devices, especially based on quantumoptical analogy in optical waveguides, which might have 7

9 Figure 6. (a,b) δ as a function of input wavelength (a) and spacing error (b) for beam splitters; (c), (d) δ 3 as a function of input wavelength (c) and spacing error (d) for 3 beam splitters. potential applications in integrated optics, optical communications, and integrated quantum computing [53]. Acknowledgments This work was partially supported by the NSFC (47493), Shuguang program (4SG35), the Program of Shanghai Municipal Science and Technology Commission, the Program for Professor of Special Appointment (Eastern Scholar), and MOST of Taiwan (05--E MY3, M-0-00). Appendix: Lie algebra The Hamiltonian () for asymmetric triple-well waveguide satisfies SU() Lie algebra in terms of the following matrix: L L 0 0 0, i i 0 -i, 0 i 0 0 L ( A) with the commutation relations, [ L, L ] il 3, [ L, L 3] il, [ L3, L ] il. On the other hand, the Hamiltonian () for the symmetric triple-well waveguide satisfies U3S3 Lie algebra in terms of the following matrix: 0 0 G 0, i 0 G i 0 i, 0 -i 0 0 G3 0-0, G ( A) 0 0 with the commutation relations, [ G, G] ig3, [ G, G ] ig, [ G, G ] ig, [ G, G ] ig, [ G, G ] ig. 3 ORCID ids Xi Chen https: /orcid.org/ Shuo-Yen Tseng https: /orcid.org/

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