Confining light flow in weakly coupled waveguide arrays by structuring the coupling constant: towards discrete diffractive optics

Structuring the coupling constant in coupled waveguide arrays opens up a new route towards molding and controlling the flow of light in discrete structures. We show coupled mode theory is a reliable yet very simple and practical tool to design and explore new structures of patterned coupling constant. We validate our simulation and technological choices by successful fabrication of appropriate III-V semiconductor patterned waveguide arrays. We demonstrate confinement of light in designated areas of one-dimensional semi-conductor waveguide arrays. 2009 Optical Society of America OCIS codes: (130.2790) Guided waves; (130.3120) Integrated optics devices; (230.3990) Micro-optical devices; (080.1238) Array waveguide devices; (350.3950) Microoptics References and links 1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behavior in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). 2. J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-06-1780. 3. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46 (1973). 4. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988). 5. S. Longhi, "Transmission and localization control by ac fields in tight-binding lattices with an impurity," Phys. Rev. B 73, 193305 (2006). 6. A. Fratalocchi and G.Assanto, "Universal character of the discrete nonlinear Schrödinger equation," Phys. Rev. A 76, 042108 (2007). 7. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4752 (1999). 8. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optics Bloch oscillations," Phys. Rev. Lett. 83, 4756 (1999). 9. M. J. Ablowitz and Z. H. Musslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001). 10. P. Millar, J. S. Aitchison, J. U. Kang, G. I. Stegeman, A. Villeneuve, G. T. Kennedy, and W. Sibbett, "Nonlinear waveguide arrays in AlGaAs," J. Opt. Soc. Am. B 14, 3224 (1997). 11. S. Eisenberg , Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction Management," Phys. Rev. Lett. 85, 1863 (2000). 12. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172 (1996). 13. H. S. Eisenberg, Y. Silberberg, R. Morandotti, R. Boyd and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998). 14. A. L. Jones, "Coupling of optical fibers and scattering in fibers," J. Opt. Soc. Am. 55, 261 (1965). 15. S. M. Jensen, "The nonlinear coherent coupler," IEEE J. Quantum Electron. QE-18, 1580 (1982). 16. A. B. Aceves and M. Santagiustina, "Bistable and tristable soliton switching in collinear arrays of linearly coupled waveguides," Phys. Rev. E 56, 1113 (1997). #103693 $15.00 USD Received 4 Nov 2008; revised 30 Dec 2008; accepted 4 Jan 2009; published 17 Feb 2009 (C) 2009 OSA 2 March 2009 / Vol. 17, No. 5 / OPTICS EXPRESS 3148 17. D. N. Christodoulides and E. D. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 233901 (2001). 18. W. Królikowski and Y. S. Kivshar, "Soliton-based optical switching in waveguide arrays," J. Opt. Soc. Am. B 13, 876-887 (1996). 19. W. Królikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, "Solitonlike optical switching in a circular fiber array," Opt. Lett. 19, 320-322 (1994). 20. I. Garanovich, A. Sukhorukov, and Y. Kivshar, "Soliton control in modulated optically-induced photonic lattices," Opt. Express 13, 5704-5710 (2005), http://www.opticsexpress.org /abstract.cfm?URI=oe-13-15-5704. 21. L. Hadzievski, A. Maluckov, M. Stepic, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity,” Phys. Rev. Lett. 93, 033901 (2004). 22. A. B. Aceves, C. De Angelis, S. Trillo, and S. Wabnitz, "Storage and steering of self-trapped discrete solitons in nonlinear waveguide arrays," Opt. Lett. 19, 332-334 (1994). 23. C. Schmidt-Hattenberger, U. Trutschel, and F. Lederer, "Nonlinear switching in multiple-core couplers," 16, 294-296 (1991). 24. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "All-optical switching and beam steering in tunable waveguide arrays," Appl. Phys. Lett. 86, 051112 (2005). 25. U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Nonlinearly induced escape from a defect state in waveguide arrays," Appl. Phys. Lett. 75, 1348 (1999). 26. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726 (1999). 27. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Switching of discrete optical solitons in engineered waveguide arrays," Phys. Rev. E 70, 026602 (2004). 28. O. Bang and P. D. Miller, "Exploiting discreteness for switching in waveguide arrays," Opt. Lett. 21, 1105 (1996). 29. R. R. A. Syms, "Approximate solution of eigenmode problems for layered coupled arrays," IEEE J. Quantum Electron. QE-23, 525-532 (1987). 30. A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, "Fresnel's laws in discrete optical media", New J. of Phys. 10, 103020 (2008). 31. T. Pertsch, U. Peschel, and F. Lederer, "Discrete solitons in inhomogeneous waveguide arrays," Chaos 13, 744 (2003). 32. R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J.S.Aitchison, "Interactions of discrete solitons with structural defects," Opt. Lett. 28, 834 (2003). 33. M. Matsumoto, S. Katayama, and A. Hasegawa, "Optical switching in nonlinear waveguide arrays with a longitudinally decreasing coupling coefficient," Opt. Lett. 20, 1758 (1995). 34. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, Maïte Volatier, V. Aimez, R. Arès, C. E. Rüter, and D. Kip, "Optical modes at the interface between two dissimilar discrete metamaterials," Opt. Express 15, 4663-4670 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-8-4663. 35. A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, "Discrete negative refraction in photonic crystal waveguide arrays," Opt. Lett. 31, 1343-1345 (2006). 36. D. Modotto, M. Conforti, A. Locatelli, and C. De Angelis, "Imaging properties of multimode photonic crystal waveguides and waveguide arrays," J. Lightwave Technol. 25, 402-409 (2007). 37. R. Muschall, C. Schmidt-Hattenberger, and F. Lederer, "Spatially solitary waves in arrays of nonlinear waveguides," Opt. Lett. 19, 323-325 (1994). 38. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 11721189 (1996). 39. I. Makasyuk, Z. Chen, and J. Yang, "Band-Gap Guidance in Optically Induced Photonic Lattices with a Negative Defect," Phys. Rev. Lett. 96, 223903 (2006). 40. H. Zhuo, X. Fu, Y. Hu, and S. Wen, "Compensation of the influence of loss for a spatial soliton in a dissipative modulated Bessel optical lattice," J. Opt. Soc. Am. B 24, 2208-2212 (2007). 41. R. Burioni, D. Cassi, P. Sodano, A. Trombettoni, and A. Vezzani, "Propagation of discrete solitons in inhomogeneous networks," Chaos 15, 043501 (2005). 42. J. M. Moison and C. Minot, French Patent n° 07 54872 (2007). 43. E. Kapon, J. Katz, and A. Yariv, "Supermode analysis of phase-locked arrays of semiconductor lasers," Opt. Lett. 9, 125 (1984). 44. J. M. Moison and C. Minot, French Patent n° 08 05307 (2008). #103693 $15.00 USD Received 4 Nov 2008; revised 30 Dec 2008; accepted 4 Jan 2009; published 17 Feb 2009 (C) 2009 OSA 2 March 2009 / Vol. 17, No. 5 / OPTICS EXPRESS 3149


Introduction
Light propagation in homogeneous arrays of weakly coupled waveguides, i.e. identical waveguides identically coupled, has been studied extensively [1,2, and references therein]. In homogeneous arrays, light injected in a single guide or several waveguides couples to more and more waveguides as it propagates along the longitudinal direction Z. The widening distribution of lit waveguides thus exhibits a behavior analogous to diffraction [3]. Propagation of Floquet-Bloch waves in this periodic discrete 1D or 2D guided environment displays several interesting features. Characteristic discrete diffraction can be observed, and the light distribution within the array has a beam-like behavior and follows a specific diffraction relation [4]. In these weakly coupled waveguide arrays, the key parameters governing light propagation from one waveguide to the next and throughout the whole array are the coupling constant C between neighboring waveguides and the propagation constant β of individual waveguides.
From a fundamental point of view, these archetypal discrete systems enable exploration of universal discrete-like dynamics [5] and discrete non-linear behavior [6] via the study of the field evolution in each individual waveguide. They have been exploited both in the linear [7,8] and non linear [2,9,10] regimes to demonstrate Bloch oscillations [7,8], discrete diffraction engineering [11] and discrete optical solitons [4,12,13]. Beside the intrinsic fundamental interest of these elegant studies started in the 90s, all-optical signal processing in the efficient, sturdy and compact domain of guided optics has been pursued for several decades in couples of waveguides [e.g. 14,15] and waveguide arrays [e.g. [16][17][18]. For example, several schemes -phase [19] or power [20-23] controlled -have been proposed to steer a propagating discrete soliton in order to achieve multiport switching.
In practice though, experimental demonstrations of advances towards signal processing [10,24-26] remain scarce. One reason could be that the studied arrays constitute indeed for the Floquet-Bloch waves a homogeneous meta-material analogous to a bulk material in optics, acoustics or electronics, which is limiting in all domains when signal processing is at stake. Attempts of signal processing in homogeneous arrays, i.e. guiding, steering, routing, switching etc. of beams, thus relied on non-linear effects and self localization, which are fruitful routes but demand high optical power [21], as well as subtly modified structures [27] and intricate schemes, all the more so as the natural mode which carries information in a non linear homogeneous lattice is the discrete soliton, a challenging object of exploration [28] in itself.
Extending the free space approach of homogeneous arrays, the inspiring theoretical work of Syms [29], carried out in the context of supermode control in transverse-modelocked lasers, hinted that reflection, refraction and guiding of beam-like light distribution within the array could be obtained also by building boundaries between semi-infinite homogeneous arrays of different characteristics. This seminal functionalization of the discrete guided array space has remained implicit up to now to our knowledge, except for a recent experimental demonstration of refraction [30]. Other works involving array inhomogeneity [8,22,25,[31][32][33][34][35][36][37][38] followed a different path, well-grounded in the optics/nonlinear optics framework. They rely on the creation of defects and corrections to or gradients of the propagation constant β to mimic dc or ac electric force [8] In search of alternative routes for light control in waveguide arrays, we propose here to explore and implement explicitly another instance [42] of discrete refractive optics by: -fragmenting the infinite periodic discrete space of the homogeneous array into a structured array, i.e. a combination of finite or semi-infinite arrays where the key parameter governing the inter-guide couplingthe coupling constant C, and not β as in previous works -is different, thus building what we will call "C-patterns".
-introducing light processing functions in a built-in functionalized space, i.e. manipulation of Floquet-Bloch waves across interfaces between zones with different coupling constants.
In this paper, we present a first experimental demonstration of this concept, i.e. a demonstration that functional C-patterned arrays can indeed be designed, modeled, realized, and operated. Our test patterns are light-confining C-patterns which we call channels. In section 2, we validate a simple coupled-mode (tight-binding) model and discuss the modes that can be expected in channels. In section 3, we describe the sample, a first realization of a C-pattern, before demonstrating experimentally light confinement in designated areas in section 4. This finally shows that such C-patterns can be excellent realizations of discrete refractive optics schemes and designs.

Modes of C-pattern channels and validity of the coupled-mode-theory approach
Similar to [29], we propose creating a characteristic guided mode (i.e. light confinement within a specific zone of the structured array) in a composite waveguide array. Light confinement is achieved in a N-guide homogeneous array of type 1 (the channel), sandwiched between two semi-infinite confinement barriers made of homogeneous arrays of type 2. Homogeneous arrays of type 1 or 2 are identical except for their characteristic C: type 1 has a coupling constant C 1 and type 2 has a coupling constant C 2 < C 1 .
We will base our assessment of channel structures, with respect to both the modal structure and the actual propagation, on the widely-used coupled-mode theory (CMT) approximation which can be derived from Maxwell equations in the slowly varying envelope approximation. This approximation is generally taken as valid when the coupling is weak enough (C<<β). We propose here to test its relevance in a more refined manner for the case of channels, by checking that their propagation coefficients and eigenmodes shapes are correctly described by CMT; although the methods we use are already known separately, such a test has not been reported yet to our knowledge. Channel modes can be computed as direct numerical solutions of Maxwell equations, for instance via a finite element method (FEM) analysis. They can also be derived simply within CMT in the following way. The reduced CMT propagation equation along the waveguide axis Z is , where z is the reduced propagation coordinate =C 1 Z, c i,,j are reduced coupling constants C i,j /C 1 , and a m is the amplitude of the individual-waveguide mode carried by the m-th waveguide (~ electric field amplitude at its center). Solutions sought as Floquet-Bloch plane waves or eigenmodes can be computed by a method derived from the one used in laser arrays [43]: for any C-pattern, mode shapes and wavevectors are respectively the eigenvectors b (p) and eigenvalues k z CMT (p) (p=1,…,N) of the adjacency matrix, whose only non-zero elements are two diagonals of the reduced coupling constants. The relationship of the reduced wavevectors k z in CMT with the actual propagation constants β is given by where β 0 is the propagation coefficient of the isolated waveguide. Hence reduction of the propagation constants β to (β-β 0 )/C 1 wavevectors allows comparing them to k z CMT predictions. We test that CMT does describe correctly the channel modes in the extreme case of a perfectly confined channel (C 2 =0, i.e. no waveguides around those in the channel), for which CMT solutions are analytical and FEM calculations do not require too large structures. The FEM calculations -Comsol software package -are conducted on the actual experimental InGaAsP shallow-ridge layered structure described in the next section. The ladders of wavevectors and some corresponding mode shapes obtained in various perfectly confined channels are compared in Fig. 1. As expected, channels are naturally multimode structures with as many modes as waveguides. A very good agreement between FEM and CMT data is obtained, at least for the four or five first modes (highest k z or β), both on all mode shapes and all wavevector ladders -the various k z CMT and the various (β FEMβ 0 )/C 1 -for a single set of the adjustable parameters (C 1 =1.51 ± 0.07 mm -1 and β 0 =13.095µm -1 ). The agreement decreases for higher-order modes, due to the fact that FEM data for such modes become increasingly sensitive to boundary conditions; this shows on the increasingly large erroneous tail of the modes outside the channel, and the progressive deviation in wavevector ladders. The comparison between the analytical CMT approximation and the FEM data is indeed an encouraging validation for the very simple CMT approximation that we will use to model the experiments in section 4. It additionally provides us with an accurate determination of the coupling constant of homogeneous arrays required for designing our C-pattern. As checked against our experimental results (not shown) in the case of perfectly confined channels, C values obtained that way are indeed much more reliable than values usually obtained numerically from the beating between the symmetric and antisymmetric modes of two coupled waveguides (same method for channel N=2, k z =±1).
We therefore use CMT to design channels. By looking at the extension of the evanescent wave across a desired barrier, we estimate a useful and sufficient contrast C 1 / C 2 of 2 for Cpattern interfaces. We also estimate a desired value for the homogeneity of ∆C / C < 10 %.

Sample and experimental setup
In practice, we use 1D arrays of shallow-ridge InP-based waveguides operated at a wavelength of 1.55 µm. Following the method described in section 2, we design a layered structure and ridge geometry of appropriate and convenient coupling constant. The structure is carefully designed so that the dependence of C on the various structural parameters (layers thicknesses, ridge width, height and separation) is neither two steep -to ensure homogeneity and reproducibility of the structures -nor too gentle -to conveniently reach the desired ratios C 1 /C 2 by tuning one of these parameters. The quality of III-V waveguides enables fabrication of the C-pattern at the chosen working point. The sample thus consists of shallow-ridge waveguides etched 1.5µm deep on top of a stack composed by a 400 nm thick guiding layer of InGaAsP (refractive index 3.337 at 1.55 µm), sandwiched between 1.9 µm thick (upper) and 3.0 µm thick (lower) InP claddings. The ridge width is 1.5 µm. Typical samples are constituted both of homogeneous and structured regions (Fig. 2).
We first validate experimentally the simulation and technological choices in homogeneous systems homogeneous arrays. The source is a conventional 1 mW 1.55 µm cw fiber-output laser. We use a microlensed fiber to inject light at the input face of the sample. The output field is imaged with a x20 or a x8 microscope objective on an infrared camera. To measure the coupling constant of our homogeneous arrays, we compare the CMT simulation of the well-known discrete diffraction output and the experimental output image of the field intensity, when light is injected into the homogeneous arrays via a single waveguide. The samples are 3.4 mm long homogeneous arrays with a constant separation between the centers of the waveguides of s=5 µm (coupling constant C 1 ) or s=9 µm (chosen so that C 2 =C 1 /2). This propagation length corresponds to about 5 coupling lengths for the C 1 regions. For s=5 µm, we find C 1 =1.55±0.08 mm -1 . This value is in remarkable agreement with the value deduced from the left of Fig. 1 in section 2, C 1 =1.51±0.07 mm -1 . We also measure C 2 =0.67±0.08 mm -1 for s=9 µm. We validate the homogeneity of the sample by checking that the output image from the homogeneous array is not dependent on the guide chosen for the injection. We thus successfully fabricated the simulated homogeneous array of controlled coupling constant. We find the coupling is slightly (± 0.1 mm -1 ) dependent on the input beam polarization that we henceforward fix to be vertical. infinite regions of less coupled identical waveguides. For this first experimental demonstration, the coupling is simply structured by varying the spacing between the guides, s=5 µm in the channel (coupling constant C 1 ) and s= 9 µm for the confining barrier (coupling constant C 2 ). The C-pattern sample is shown on Fig. 2. A top view of the injection setup is given in Fig. 3.

Experimental results
The output images from channels composed of N guides (N=2, 4, 6, 8 and 10) after 3.4 mm propagation are shown in Fig. 3. In contrast with the typical discrete diffraction shown at the bottom of Fig. 3, these experimental data clearly exhibit the desired guiding effect in the more strongly coupled waveguides. The light distribution is indeed localized in the channel which successfully achieves guiding of the wave. Fig. 3. Left: Top view of the experimental setup: a microlensed fiber injects light into a 6guide-wide channel. Inside the channel, the ridge separation is s=5 µm. On both barrier sides, the less strongly coupled waveguides are separated by 9 µm. Right: Output intensity maps for various sizes {N} of channels (N more strongly coupled waveguides). Special care has been taken to keep the injection constant (w0=8.75 µm, see text) for all channels. Bottom: output of a homogeneous array with s=5 µm. Please note that a different microscope objective (MO) and different injection (w0=5 µm) have been used to image the homogeneous array sample {∞}.
In addition, comparison of the data of Fig. 3 with CMT results is given in Fig. 4. The size of the injection, that is set by the microlensed fiber position and identical for all channel sizes, is chosen experimentally large enough to smooth the discrete propagation patterns, and narrow enough for the light to be injected within the narrower channel. The size of Gaussian injection (waist =8.75 µm) and its position is deduced from the fits. As can be seen, the CMT efficiently predicts the intensity of light in the output field. One can note that the theoretical predictions agree very satisfactorily with experimental output for {N=2}, {N=4}, and {N=6}. The predicted and experimental results for {N=8} are comparable, though irregularities of the signal within the channel are not well caught. Further discrepancies arise when fit of the {N=10} structure is attempted. Several reasons can be invoked to explain this: inadequateness of CMT in which only the first band with cosine shape is considered, or more probably slight tilt in the injection or defects/lack of homogeneity in the structure.
It may be noted that the size of the injection beam is kept constant at a value which for most cases is quite smaller than the channel width. In other words, there is a large mismatch between the fundamental mode of the channels and the injection beam. This leads to the excitation of many modes of the structure, both guided modes -rather near to the ones of perfectly confined channels described in section 2 -and leaky modes whose k z lie in the barriers band, which can hence propagate in them, and which will be eliminated in the end by divergence -slower than in the channel -for long propagation paths, as in any conventional waveguide. In direct CMT simulations such as those reported in Fig. 4 (black vertical bars), all these effects are taken into account, but mode contributions are intermixed. Projection on the calculated modes show that guided propagation involves mostly the two highest even confined modes -those which have high amplitudes at the channel center -and that for our injection pattern and our propagation path they carry at least 80% of the injected power. The extension of evanescent waves is very small (~1 inter-guide spacing) and intensity outside the channel is mostly due to leaky modes.
Finally, Fig. 3 and Fig. 4 are the experimental demonstration of how patterning of the governing parameter C successfully leads to a discrete refractive optics function (here confinement of light) in structures that we can trust to be good realizations of the schemes elaborated with CMT.