Light Propagation, Discrete Diffraction, Discrete ...

Nonlinear Optic.', anil Quantum Optics. Viil. 43. pp. 269-279 ReprinLs available dircclly from Ihc publisher Pholocopying permilled by liecnse iiniy

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Light Propagation, Discrete Diffraction, Discrete Solitons and Discrete Beats in Periodic and Non-Periodic POLICRYPS Structures L. PEZZI, A . D E LUCA, A. VELTRI AND C . UMETON

LICRYL (Liquid Ciystals Laboraloty), IPCF-CNR, Cetiter of Excellence CEMIF.CAL and Departtnent of Physics, University of Calabria 87036 Arcavacata di Rende (CS), Italy Received: Novettiher 15, 2010. Accepted: November 23, 2010.

We present a simple model which enables the derivation of light propagation in a POLICRYPS structure, utilized as an array of planar waveguides realized in liquid crystalline composite materials. Discrete diffraction and soliton solutions are lirst numerically derived, without the need of any approximation, for a sinusoidal periodic profile; results are in good agreement with theoretical data and experimental results already reported in literature for similar structures. Furthermore, the direct derivation performed by means of a general approach allows to extend the analysis to a generic structure with no periodic conditions. As particular cases, light propagation in a double gaussian profile and in an alternated, non-periodic refractive index profile are reported. Keywords: nonlinear optics; discrete diffraction; polymeric gratings; optical discrete solitons; periodic and aperiodic waveguide arrays PACS Numbers: 42., 42.25.-p. 42.25.Fx. 42.65.Tg. 42.82.Et

1 INTRODUCTION In recent decades, great attention has been devoted to the investigation of electrically switchable holographic devices which exploit liquid crystalline composite materials. In particular, it has been shown that diffraction gratings based on Holographic Polymer Dispersed Liquid Crystals (HPDLCs)

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can exhibit a good diffraction efficiency [1, 2]. However, their application oriented utilization is limited by the strong scattering of light, which occurs when the droplet size of the Nematic Liquid Crystal (NLC) component inside the polymeric matrix is comparable with the wavelength of the impinging light. We have recently proposed a new kind of holographic grating called POLICRYPS, made of polymer slices alternated to films of regularly aligned NLC. These structures do not present optical inhomogeneities and can exhibit good optical characteristics, with values of the diffraction efficiency as high as 98% [3, 4]. Soon we have realized that low scattering losses and good switchability of POLICRYPS gratings open a wide range of possible applications; in particular, we believe that their polymeric rigid frame can not only be exploited to realize a good 'physical confinement' (stabilization) of the NLC molecules, but can also be exploited to fabricate an array of regular 'channels', where light could be guided and, eventually, amplified. In the past, light propagation and non-linear optical effects in waveguide arrays have been intensively studied, both experimentally [5, 6, 7, 8] and theoretically [9, 10, 5, 7, 8, 11, 12] by exploiting models that refer to the "Fermi-Pasta-Ulam" approach to one-dimensional dynamical systems [13]. Numerical models have been implemented by exploiting the Coupled-Mode Theory (CMT) [9], which considers only nearest-neighbor interactions; in the continuum (or long-wavelength) approximation, this discrete process can be described by the non-linear Schrödinger equation [9, 14, 15, 16]. This approach enables to predict whether the light beam experiences normal diffraction (the beam spreads while propagating), anomalous diffraction (the beam spreads while propagating, but some anomaly is observed) or if diffraction disappears [5, 16, 6, 7]. Besides CMT, the Floquet-Bloch (FB) analysis, which is applicable also to lossy waveguides [17, 8, 18, 19, 20], has been used to extend concepts introduced previously. The FB approach, however, while well enabling investigation of light propagation in periodic structures, prevents extending the study to those cases where the physical system does not exhibit any periodicity. In investigating discrete diffraction, also another approach has been exploited, which is able to predict the beam evolution in non-periodic systems too: It is the (non-linear) beam propagator method (BPM), a simulation technique used to study the propagation of electromagnetic waves in inhomogeneous media [21, 22,23]; unfortunately, the BPM is valid only for small spatial variation of the refractive index. In this paper, we present a simple model that enables to investigate propagation of light in POLICRYPS structures exhibiting a variable transverse profile of the refractive index. The novelty and interest of our approach is in showing that, under suitable conditions, discrete diffraction and discrete solitons can be obtained by simply coupling the Maxwell equations with the POLICRYPS structures, without any necessity of the different approximations that have to be used in all the systems described in the cited references.

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2 THE POLICRYPS STRUCTURE Investigation of the POLICRYPS structur enabled us to obtain a deep knowledge of the basic phenomena that yield its formation, a good characterization of its morphology, a satisfactory understanding of the working principles and effects that can be exploited for applications, and a pretty clear idea of the wide range of possible utilization; at present, we have a general vision of POLICRYPS that puts together the different scientific aspects related to this new device. In agreement with a detailed experimental chatacterization, we have implemented a model which explains how realization of a POLICRYPS structure exploits the high diflusivity of NLC molecules in the isotropic state to avoid the formation and separation of the nematic phase (as NLC droplets) during the curing process [24]. Fabrication steps are here biiefiy recapitulated: By means of a hot stage, a syrup of NLC, monomer and photo-initiator is heated above the Nematic-Isotropic transition point of the NLC component. A typical syrup composition is 5CB NLC (by Merk, ^ 30 % wt.) in the prepolymer NOA61 (by Norland). The syrup is 'cured' with the interference pattern of a UV radiation (À=0.351 /xm) for about 10-''Í; then, it is brought below the Isotropic-Nematic transition point by means of a slow, linear, cooling down to room temperature. The experimental set-up exploits an active system for suppression of vibrations [25]. After the curing process has come to an end, the fabricated sample remains sandwiched between two glass slabs, whom the polymeric slices are "glued" to. This circumstance prevents the onset of any spatial modulation of the sample thickness, that could, eventually, be due to a "shrinkage" of the polymer. Depending on the required nano/micioscale dimensions of the structure, the spatial period of the interference pattern can be varied in the range A = 0.2 -^ \5ßm by adjusting the interference angle, in the range 1-60°. Samples can be fabricated with thickness in the range D = 5-^lOOjUni with amplitude of the index modulation in the range An = 0.01 -^ 0.2. Figure 1 shows a typical Scanning Electronic Microscope (SEM) image of a POLICRYPS structure with its sharp morphology.

3 LIGHT PROPAGATION IN PERIODIC POLICRYPS STRUCTURES (WAVEGUIDE ARRAYS) The propagation of a guided optical field E{x,z) is analyzed in the .xz-plane of a POLICRYPS, whose refractive index n{x) varies along the x-direction only. Our model is very simple, general and elegant, but implies that obtained equation can be solved only by using a numerical approach. We analyze the problem by starting directly from Maxwell's equations, without supposing any discreteness of the medium neither a periodicity of the solutions; we stress, indeed, that we perform a numerical analysis which makes use "only" of the

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FIGURE I SEM image of a POLICRYPS holographie diffraetion grating

Slowly Varying Envelope Approximation (S VEA) on the optical electric field. We hypothesizes a solution of the kind: f (Ç, ^, i) = e(Ç, f ) exp [ - /(k • r-tot)] where e = E/fo is the normalized electric field, £0 being the field amplitude; Ç and f are the normalized coordinates Ç = x/D, Ç = z/D, D being the sample thickness; f = (Ç, Ç); k = {k^,k^) is the normalized light propagation constant with modulus |k| = 2miD/X\ X indicates the wavelength of the optical field and n is the refractive index of the medium where light is coming from. Under these conditions, the Maxwell's wave equation becomes: 9e

= 0

(1)

where Ao = 2TTD/X and n(Ç) indicates the transverse profile of the refraetive index. In order to solve equation (1), we utilize a numerical technique based on a finite-difference scheme for both spatial derivatives (with respect to ^,t; spatial directions). We create a lattice of computational grid points {i,j) in the Ç-Ç plane, where lattice steps AC and A^' allow to go from (' to Í-|-1 and from j toy-I- 1 respectively. Evolution of the optical field in the f direction is then simulated by determining the value of the field at the (¿,7 -|-1 ) point by means of the field values at previous points. All simulations have been carried out

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by putting D = 100/iw, X — 532nni and n — 1, while AC and AC are chosen case by case in such a way that the numerical stability of solutions is ensured. The reliability of our model has been checked by comparing obtained numerical results with both experimental and theoretical data already reported in literature for similar structures. From literature [5] it is well known that, by indicating with kx the transverse component of the impinging light wavevector and with A the transverse periodicity of the refractive index of the medium, the light beam experiences anomalous discrete diffraction when diserete values of A (A = [2m -\- \]X/2, m integer) are chosen in the interval 7r/2 < \kxA\ ^ 7r. In fact, light distribution tends to broaden while propagating (like in normal diffraction) but most of the light concentrates into two distinct outermost lobes. Moreover, dilfraetion completely disappears around the two values kx — ±;r/2A. We have applied our model to the case of a POLICRYPS with a sinusoidal transverse modulation of the refractive index: An = AH,,,(U sin(2:7rÇ/A) where Ä indicates a noriTialized periodieity (À = A/D). Results for the case A«,,,,« = 0.04 and A = 0.02926 (m = 5) are reported in Fig. 2, where the beam intensity is represented in

(b)

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FIGURE 2 (a) Top view of the anomalous discrete diffraction of light occurring in a POLICRYPS waveguide array; (b) and (c) show the refractive index and the intensity profiles at the sample input and output respectively. All values are reported in normalized units (n.u.). Intensity is normalized to 1.

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the (Ç,0 plane. It is evident that, if only one waveguide is excited, most of the light is concentrated into two distinct outermost lobes, reproducing the solution obtained with CMT in the discrete diffraction problem (Green's function) [8, 7]. Besides these results, our model predicts also normal discrete diffraction and discrete soliton: An initially localized excitation (more than one waveguide is excited simultaneously), tends to spread over the whole array, as in continuous systems (Fig. 3) [7]. Appearance of discrete diffraction depends on the propagation direction and on the incidence angle; in particular cases (particular incidence angles), diffraction can be completely suppressed and a discrete soliton appears, which moves across the array. As shown in Fig. 4, also this particular case, numerically solved by using our approach, confirms the validity of the model which proves suitable to predict the onset, in our POLICRYPS waveguide array, of all kind of results reported in literature for similar structures.

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FIGURE 3 (a) Top view of the normal discrete diffraction of light occurring in a waveguide array constituted by a POLICRYPS with a sinusoidal transverse modulation of the refractive index; (b) and (c) show the refractive index and the intensity profiles at the sample input and output respectively.

LIGHT PROPAGATION IN PRRIODIC AND NON PERIODIC POLICRYPS STRUCTURES

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FIGURE 4 (a) Top view of a discrete soliton propagation in a sinusoidal POLICRYPS waveguide array for the case of a non zero incidence angle; (b) and (c) show the refractive index and the intensity profile at the sample input and output respectively.

4 LIGHT PROPAGATION IN A NON-PERIODIC POLICRYPS STRUCTURE Our approach becomes irreplaceable when we investigate a physical system which does not exhibit any periodicity. In the last years, realization of non-periodic, or even naif, holographic structures [26] has attracted a great attention for the interest they excite in application oriented research. Fabrication of samples, however, can hardly exploit multi-beam interferometric methods, which would require the design, realization and monitoring of an extremely cumbersome setup. A new technique has been therefore implemented, which exploits software driven Spatial Light Modulators (SLM) to locally modify the phase of a light beam and give rise to a precise, desired, distribution of the intensity impinging on the sample. SLMs enable utilization of the same, very stable, single beam setup to fabricate complex structures of different shapes, by exploiting the photo-polymerization induced phase

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separation of monomer- liquid crystalline composite materials [27]. A similar setup has been successfully utilized also to fabricate POLICRYPS - like structures, with different shapes and characteristics [28]. In this case, samples present a good phase separation of components, and are characterized by a sharp morphology and good optical and electro-optical properties, typical of those POLICRYPS gratings that are realized by utilizing the standard interferometric method [3]. The SLM technique can be therefore used to fabricate non periodic, or arbitrarily shaped, arrays of optical waveguides based on POLICRYPS. Let us consider, first, the particular case of a waveguide array formed by two waveguides, only which exhibit a gaussian transverse profile of the refractive index. We consider the behavior of a focused gaussian beam, injected in a single guide (right guide of Fig. 5): during propagation, the optical power is periodically exchanged between the two guides. This result is well known and is reported in [7, 29] for a similar system of couple of waveguides; in

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0.3 0.4 0.5 B, (n.u.)

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0.7

FIGURE 5 (a) Top view of the "switch-like" behavior of light in a non-periodic POLICRYPS structure representing two waveguides with a gaussian refractive index profile, (b) and (c) are the refractive index and the gaussian intensity profile at the sample input and output respectively. The inset represent the intensity profile along the f direction, in the central part of each guide.

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that system, however, the approach takes into account only the value of the refractive index in the waveguides, without allowing to manage a more realistic transverse modulation of material parameters. The "switch-like" behavior predicted by our model is confirmed by plotting the intensity profile in the central part of each guide as a function of the Ç coordinate (inset of Fig.5). Obtained functional curves behave as cos^ (Ç) and sin" (Ç) and a total energy transfer between the two waveguides is predicted. Very interesting is also the case of the propagation of a gaussian beam in an alternated, but non periodic, structure. In particular, we can design a POLICRYPS where the spacing between the different waveguides (in terms of gaussian profiles of the index of refraction) linearly increases by starting from the center of the sample. In this case, our model predicts the onset of anomalous discrete diffraction only until this spacing becomes so laige that there is no more coupling between the transverse modes; the optical field is then reflected by the sides of the structure and a sort of multi-guide "discrete beats" behavior starts, as shown in Fig. 6.

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^ (n.u.) FIGURE 6 (a) Top view of the "discrete beats" behavior of light in a non-periodic structure made of waveguides spaced by a distance that linearly increases by starting from the center of the sample; (b) and (c) show the refractive index and the gaussian intensity profile at the cell input and output respectively.

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We stress that, at our knowledge, only our model is able to reproduce this ease, where no periodicity is imposed and the index modulation An can reaeh values as high as 0.56, a case that is not reported in this paper, but has been used in one of our numerical simulations. 5 CONCLUSIONS In conclusion, we have presented a simple model that, by starting directly from Maxwell's equations, is able to predict all the kind of known discrete phenomena, like normal, anomalous discrete diffraction and discrete solitons, related to light propagation in a POLICRYPS structure utilized as an array of optical waveguides. The apptoaeh exhibits several advantages in comparison with different models utilized in literature for planar waveguide struetures. First of all, it is direetly derived from Maxwell's equations written in the medium of interest, with the only limitation of the "Slowly Varying Envelope Apptoximations". It is not necessary to introduce any "coupling constant", which determines the transverse propagation of the optical field, thus neglecting the existence of a different medium between two waveguides (CMT model). It is not limited to small modulations of the transverse pt ofile of the refractive index (BPM approach). Finally it is not necessary to hypothesize any periodicity of the transverse profile of the refractive index of the medium (FB model): In fact, our approach enables to predict also the propagation of light in non periodic systems. Two examples of this particular case are reported: the first one is related to the "switch-like" behavior obtained in a POLICRYPS which represents two adjacent waveguides, each of them exhibiting a gaussian transverse profile of the refractive index; the seeond one is related to an alternated, but non periodic system where the POLICRYPS strueture represents (gaussian) waveguides spaeed by a distance that linearly increases by starting from the center of the sample; in both cases, interesting light propagation behaviors are predicted. Design of experiments devoted to cheek in details our approach is under development.

ACKNOWLEDGMENTS The research has been supported by PRIN2006 - UMETON - Prot. 2006022132 001. REFERENCES [1] J. D. Margerum , A. M. Lackner, E. Ramos, G. W. Smith, N. A. Vaz, J. L. Köhler, and C. R. Allison. US Patent Specification, 5:096, 282 (1992). [2] R. L. Sutherland, V. R Tondiglia, L. V. Natarajan, T. J. Bunning, and W. W. Adams. / Nonlinear Opt. Phys. Mater 5: 89 (1996).

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[3] R. Caputo, L. D. Sio, A. V. Sukhov, A. Veltri, and C.Umeton. Opt. Lett., 29. 1261 (2004). [4] R. Caputo, A. V. Sukhov. C.Umeton. and A. Veltri. J. Opt. Soc. Am. B,2\. 1939. (2004). [5] H. S. Eisenberg, Y. Silberberg, R. Morandotti. and J. S. Aitchison. Phys. Rev. Lett., 85:1863-1866,(2000). [6] R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and i. S. Aitchison. Phys. Rev. Lett., 86(15):3296-3299, (2001). [7] F. Lederer and Y. Silberberg. Optics atid Phototiics News, pages 49-53, (2002). [8] D. N. Christodoulides, F. Lederer. and Y. Silberberg. Nature, 424:817-823. (2003). [9] D. N. Christodoulides and R. I. Joseph. Opt. Lett., 13:794-796, (1988). [10] M. J. Ablowitz and Z. H. Mussiimani. Phys. Rev. E, 65:056618, (2002). [Il] J. M. Bendickson. J. P. Dowling. and M. Scalora. Plm. Rev. E, 53:4107^121, (1996). [12] Falk Lederer, Sergey Darmanyan, and AndreyKobyakov. Spatial Solitons. Springer-Verlag Berlin Heidelberg New York, 2001. pag. 269-292. [13] E. Fermi. Pasta J, S. Ulam, and M. Tsingou. (Los Alamos Scientific Laboratory of the University of California) Tech. Rep., LA-1940 (Report written May 1955). Los alamos scientific laboratory of the university of california. [14] AmnonYariv. Optical Electronics. Saunders College Publishing, 1991. [15] S. Suntsov. K. G. Makris, D. N. Chri.stodoulide.s. and G.I. Stegeman. Phys. Rev. Lett., 96:063901.2006. [16] G. I. Stegeman and M. Segev. diversity. Science. 286:1518-1522, 1999. [17] D. Mandelik. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison. fioquetbloch solitons. Physical Review Utters, 90:053602, 2003. [18] Nai-Hsiang Sun, J. K. Butler, G. A. Evans, L. Pang, and P. Congdon. floquet-bloch theory. Jounial of Lightwave technology, 15:2301-2315, 1997. [19] A. Fratalocchi. G. Assanto, K.A. Brzdakiewicz, and M.A. Karpierz. 13:1808-1815,2005.

Optics Express,

[20] Charles Kittel. Introduction to solid state physics. New York, 1996. [21] M. D. Feit and J. A. Fleck. Applied Optics, 17:3990-3997. 1978. [22] Lars Thylèn and Chung Ming Lee. J Opt. Soc Am.A, 9:142-146. 1992. [23] M. Shalaby. Pure Applied Optics, 5:991)004. 1996. [24] A. Veltri, R. Caputo, C. Umeton. and A. Sukhov. Appl. Phys. Lett., 84. 3492 (2004) [25] L. De Sio, R. Caputo. A. De Luca, A. Veltri, A. V. Sukhov, and C. Umelon. Apll. Opt. 45. 3721,(2006) [26] J. Li, Y. Liu, X. Xie, P. Zhang. B. Liang, L. Yan. J. Zhou. G. Kurizki. D. Jacobs. K. S. Wong, and Y. Zhong Opt. Express 16, 13287 (2008). [27] G. Zito, B. Piccirillo. E. Santamato, A. Marino, V. Tkachenko. and G. Abbate Opt. Expre.is 16,5164(2008). [28] M. Infusino , PhD Thesis, University of Calabria 2010. [29] Tullio Rozzi and Andrea Di Donato. Componenti e circtiiti ottici. Tecnoprint S.n.c, 2005.

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Copyright of Nonlinear Optics, Quantum Optics: Concepts in Modern Optics is the property of Old City Publishing, Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.