Several circuits, the following two are denoted by (A) and (B), are needed for decoupling of lightwave circuits (<i>LWCs</i>) and for a future transfer function matrix (<i>TFM</i>) synthesis of <i>LWC</i>s. (A) Controller matrix of a feedback (<i>FB</i>) which is an optical network composed of (i) variable weighted distributed <i>FBs</i>, implemented by electro-optically tunable gratings at waveguides (<i>WGs</i>) and (ii) optical isolators (<i>OIs</i>) or (iii) switches, respectively, for the correct flow of the optical field in the network. The controller matrix may be generalized to include space-variable elements according to a step function by step-wise arranged electrodes at the substrate of the gratings.Another generalization is the introduction of several wavelengths by subsequently arranged gratings of different spacing at the <i>WGs</i> of the controller matrix. The inclusion of such controller matrices at several wavelengths into a <i>WDM</i> switch requires various frequency converters (<i>FCs</i>) for the non-blocking operation. (B) Optical differentiator with regard to the longitudinal space coordinate of a <i>WG</i> which is composed of (i) reflection-type Mach-Zehnder interferometers (<i>RMZIs</i>) arranged at a (ii) fish-bone <i>WG</i> pattern for the implementation of the parallel <i>RMZIs</i>. The interleaved organization of the arms in (ii) provides the approximate differentiation of the optical field in a <i>WG</i>. The application of (A) and (B) to the advanced control of couplers is proposed (derivative <i>FB</i>). The paper presents circuits and their principles rather than their evaluation. The final goal is the photonic integration of the presented decoupling circuits.
The application of transfer function matrix (TFM) synthesis concepts of linear space-invariant systems to lightwave circuits (LWCs) is reviewed and briefly discussed. The applied decoupling concepts are 1-D though the systems are 2-D where the application of 2-D decoupling concepts is left for forthcoming papers. Although the approach is applicable to nonlinear (NL) systems we restrict presently to linear systems. Distributed feedbacks (FBs) are established by means of a flexible grating concept which includes the electro-optical generation of reflections as well as their tuning. In this way reflections may be switched off and on, enabling novel devices and operations (presented in forthcoming papers) , and the gain of FB controller matrices may be adjusted according to the algebraic results of control theory. The discussion of further components (optical isolators and frequency converters), needed throughout the proposed concept, is left for forthcoming papers.
For recently proposed 2-D lightwave circuits (LWCs) the architectural implications of introducing several wavelengths are discussed. Three types of 3-D architectures are considered (1) reconfigurable router (2) straight-forward extension of the 2-D LWCs of any geometry N 3 to 3-D by introducing several wavelengths at every waveguide (WG) and (3) mapping a generated network topology (which starts with the 2-D LWC) onto the 3-D LWC in (2). The architectures in (2) require the generation of the total number of permutations at every switch for non-blocking networks whereas the architectures in (3) allow (amongst others) for some switches the reduction of the number of permutations. The computation of the total number of permutations requires (i) a photonic feedback (FB) controller matrix at several wavelengths which provide rn! x k! permutations and additionally (ii) several frequency conversions (FCs) which complete the total number (m x k)! permutations where k is the size of switches and rn is the number of wavelengths.
A major drawback of lightwave circuits (LWCs) is the nearest- neighbor (NN) interconnection scheme. An attempt to overcome within the technological restrictions is the repetitive triangulation (RTR) of the proposed N-gon cell complexes. (Higher-order RTR is aimed to be done in the frequency domain.) The 2-D LWCs are analyzed by (1) 2-D models (projection onto the plane) and (2) 3-D models. The 2-D models are (a) orthogonal 2-D grids where faulty edges comes in and (b) double triangulated 2-D grids for the embedding of the N- gon cell complexes subject to RTR. The 3-D models are (i) orthogonal 3-D grids and (ii) orthogonal 3-D grids with triangulated plane facets as spatial triangulation causes a topology which is difficult to realize by LWCs. The random walks within these architectures are considered. Random walks in orthogonal grids are known to exhibit different properties dependent on the dimension. These properties have to do with the propagation in all 2d directions (d is the dimension). The question arises whether these properties are obtainable also within the proposed feed-forward (FF) networks where backward couplings are excluded. As an approach to control these random walk characteristics (synthesis) the biased random walk is proposed.
Active fiber bundles (FBs) are aimed to model photonic switching and processing in 3-D without the restrictions of the photonic technology. The 2-D photonic architectures are assumed to be implemented by networks of directional couplers (DCs) and Mach-Zehnder interferometers (MZIs), respectively. For the implementation several crucial problems are expected: (1) proper operation of the spatial couplers/switches (nonblocking interconnections) and (2) coupling in the interstage interconnection section mainly caused by parallel and crossing fibers/waveguides (WGs). For the design of proper operating switches (refinement of couplers) the application of decoupling concepts of modern control theory is proposed. The final goal is to translate the refined couplers into integrated photonic architectures rather than into additional lightwave circuits (LWCs) which simply would increase the coupling. The decoupling concepts are reviewed. The paper is an attempt to prepare for applying well-known system engineering concepts to the upcoming technology of photonics.
Repetitive triangulation (RTR) of a N-gon cell complex is applied to generate a variety of active fiber bundles which, if integrated, become 2D lightwave circuits (LWCs). By means of RTR the size of existing couplers increases and new couplers are generated according to some principles. The RTR shows different results for the geometries N >= 3 and the result also depend on the degree of RTR. For the purpose of simplifying photonic integration, the large-sized couplers arising (i) at the center of the N-gon cells and (ii) interconnecting N-gon cells to cell complexes, are aimed to be reduced by the introduction of several wavelengths. The operation of the various couplers at several wavelengths represent a serious restrictions of the permutation capabilities of the switches but the applicability to optical logic is assumed. RTR may be applied to generate scalable networks of 2D LWCs which overcome the restriction of nearest-neighbor interconnections schemes and extend them to 3D LWCs. Two different space filling problems arise dependent on the chosen RTR procedure.
The proposed 3D lightwave circuits (LWCs) with square-shaped intersection geometry are transformed into 3D LWCs of another geometry (rectangles) with a lower number of layers (n<SUB>1</SUB>) by grid embeddings (GEs). In the limit, 2-layer sandwich architectures are obtained by GEs and by space filling curves, respectively. The purpose of applying GEs is to control the dilation (number of edges between two modes of the rectangle which are adjacent in the square-shaped grid) of the binary topology, related to the 3D LWCs and introduced by embeddings into cubes.
Optical number representations (NRs) beyond the common binary radix-2 concept and based on multiple-valued logic (MVL) and redundant NRs (RNRs) are presently restricted to optical free-space techniques. The present paper is an attempt to apply these concepts to lightwave circuits (LWCs). Starting with planar waveguide (WG) devices, capable to provide logic, their extension to 3D and large coupler size is assumed. The attempt to RNR by Wgs is based on two tuning parameters (i) the geometry N >= 3 of the optical architectures and (ii) the degree of repetitive triangularization where (ii) determine the increase of the parallelism dependent on (i). This parallelism is applied for RNR and (i) and (ii) determine (1) the number of couplers (2) the number of additional interconnections and (3) the distribution of the additional interconnections at the couplers. The proposed concept includes the common binary radix-2 logic and the higher radix WG logic is by the organization of the cores of the couplers. The reason for establishing RNR by LWCs is its necessary for minor depth optical WG logic. The large sized couplers, arising y repetitive triangularization, are aimed to be implemented by fibers and the logic circuits by active fiber bundles but the goal of the ongoing work is the photonic integration of the architectures.
Planar switching matrices of parallel waveguides (WGs) have reduced loss due to the absence of tapering but require some confinement of wave propagation reported from Kerr nonlinearities (NL). Parallel switching matrices are fed by the multiple splitting of the input WGs, an appropriate network model is the parallel version of the Spanke-Benes (PSB) network and the reduction of the number of stages (NSs) below N (for N i/o) is analyzed. However, in the parallel case, regarding WGs and SB networks, the location of switches can no longer be fixed but must be a moving location (ML). From the several parallel paths through the PSB network the shortest path is chosen either at the end by path selection switches (PS-SWs) or at the beginning of the switching matrix, respectively. It turns out that the reduction of NS of the switching matrix and in turn the saving of the number of switches (NSWs) is compensated by the number of PS-SWs at the end or at the beginning of the matrix. The replacement of the PS-SWs by combiners at the output (i) restores the energy balance but (ii) causes phase mismatch (iii) provides redundant paths (iv) restricts the overall NS to the NS of the SB network for each copy but (v) improves the nonblocking (NB) characteristic. The routing of the switching matrices and their optical implementation is also briefly discussed.
The proposed all-optical 2D networks represent machines in the 3D physical space for switching, processing and logic of light and the operation principles of the machines depend on the propagation and interaction of electromagnetic waves in waveguides. Thus at least three kinds of state-space problems arise: (1) The state-space of physical quantities expanded by the propagation and the coupling of modes through the waveguides (2) the cycle-state-space of these machines if subject to switching, processing and all-optical logic (3) the state-space of these machines for reliability and performability modeling and analysis.
The spatial arrangement of several waveguides (WGs) in close proximity is considered for (1) passive coupler with no external electrical fields (2) active coupler (external electric fields) and in both cases for (3) circular cylindric WGs as well as for (4) rectangular WGs. The mixture of cross and bar states in (2) is the next step and proposed to be obtained by the electro-optic effect. Two concepts of planar WGs-coupler are proposed to be applied to spatial coupler: ((alpha) ) The center WG of 3 X 3- and a 5 X 5-couplers and ((beta) ) passive parasitic WGs. For integrated architectures of rectangular WGs the restriction of the number of layers causes difficulties for the implementation of ((alpha) ) and ((beta) ). The extension of the reversed (Delta) (beta) -coupler principle to spatial switching is proposed as a first step towards (2). The step from coupler to switches for N >= 3 by increasing the optical parallelism is also briefly discussed.
The principles of the photonic integration of proposed 2D switching networks are presented which is mainly based on the utilization of planar integration techniques and their development. This `sandwich' technique composes layers of planar integration with the major goal of minimizing (1) the number of interconnections between adjacent layers with slopes (2) the number of crossing waveguides within single layers and between adjacent layers and of optimizing (3) the number of layers. This is a multi-level optimization problem whose first step can be solved by utilizing the symmetry of the 3D architectures. The presented work explains concepts and prepares their geometry for the ongoing physical analysis. The presented architectures are restricted to passive switches, capable to provide e.g. all-optical logic circuits. Hints for an extension towards active switches are also briefly presented.
Planar and spatial lightwave (LW) switches (synonym: all- optical switches) are reviewed and compared with regard to their (1) architectural principles (2) possible implementations and (3) performance. The first step towards spatial LW switches are spatial LW coupler which are aimed to utilize symmetries and circular geometries covering several planar LW couplers. Then spatial LW switches, capable to generate simultaneously several rearrangeable nonblocking interconnections, are the next more involved step. The goal are switch architectures with a low number of subsequent switching sections/stages. The spatial LW switches may be applied as (1) single components and (2) switches within large 3D LW circuits. For integrated switch architectures principle problems arise and the restriction of the number of layers causes additional difficulties.
Routing of all-optical 2D switching networks (which connect 2D data whereas the networks span the 3D physical space) is more complex than routing of (planar) 1D networks mainly caused by the (1) combinatorial `explosion' and (2) the arising spatial all-optical >= 3 X 3-switches which are difficult to implement. The routing problem may be subdivided into ((alpha) ) the combinatorial problem of determining the k to generate arbitrary permutations of the inputs at the output of a (rearrangeable nonblocking) network and ((beta) ) the realization of the states of the all-optical >= 3 X 3- switches by the search for waveguide-electrode configurations and voltage adjustments where the paper concentrates on ((alpha) ). The 2D switching networks are (1) projected into plane graphs which--in turn--are (2) mapped onto hypercubes (N equals 4 and 6) and (3) routed by means of the algorithms of (2). Several routing concepts are reviewed and the introduction of several wavelengths is discussed.
Assuming advantages of all-optical 2D switching networks, the question arises which advantages are preserved by the planarization and what are the costs. The planarization of all-optical 2D switching networks show different results depending on (1) the geometry N >= 3 and the (2) implementation by directional couplers and Mach-Zehnder interferometers, respectively. The feed-forward graphs (FFGs), being the complete graph models of planar 2D switching networks, are the starting point for the planarization which is classified into (1) the direct realization where the 2D network is transformed into its isomorphic FFG according to a certain scheme (2) the recurrent realization of 2D networks (a network is represented by smaller networks) and (3) multi-layer realizations. The FFGs are optimized with regard to (4) a minimum number of crossings and (5) minimum skews.
For the proposed all-optical 3D grids of directional couplers and Mach-Zehnder interferometers, network models (1) and (6) and their routing algorithms are reviewed and some are presented and discussed in more details. (1) Planar cellular arrays superposed by by-passing networks (2) 3D cellular arrays described by their projection onto Cayley graphs (3) shuffle networks (4) Clos networks (5) Projection onto 2D grids mapped onto hypercubes and onto star and pancake graphs and (6) Sorting in all-optical 3D grids. The extension to geometries N >= 5 and the introduction of wavelengths is briefly discussed.
All-optical signal processing by >= 2D lightwave circuits (LCs) is (i) aimed to allow the (later) inclusion of the frequency domain and is (ii) subject to photonic integration and thus the architectural and algorithmic framework has to be prepared carefully. Much work has been done in >= 2D algebraic system theory/modern control theory which has been applied in the electronic field of signal and image processing. For the application to modeling, analysis and design of the proposed 3D lightwave circuits (LCs) some elements are needed to describe and evalute the system efficiency as the number of system states of 3D LCs increases dramatically with regard to the number of i/o. Several problems, arising throughput such an attempt, are made transparent and solutions are proposed.
Problems arising throughout the design of 3D lightwave circuits are briefly addressed in the following. (1) The implementation of M X N-gon switches. (2) The routing of M X N-gon prism switches and (3) the routing of all- optical 3D girds and (5) dual switches. (6) The improvement of the switches towards the crossbar and (7) the generalization of all-optical grids to any N >= 3.
2D networks of directional couplers (DCs) and Mach-Zehnder interferometers for of all optical switching in the 3D physical space are presented and arising problems and their solutions are made transparent. The result also applies to optical signal processing. The all-optical 2D switching networks, based on 4-gon switches, are compared of 2 by 2, 3 by 3 and 4 by 4 couplers and nearest-neighbor interconnections where the crucial problem is the implementation of >= 3 by 3 couplers. However, various advantages are expected (1) combination of the massive parallelism of optics and the parallelism in the frequency domain (2) considerable reduction of the number of stages (3) novel properties which are caused by the increase of freedom contributed by the additional space coordinate and (4) an increase of the range of possible applications.
The design of the planar multi-layer switches, obtained by the Euler path through the switch intersection graph (IG), is discussed with regard to the (1) minimum crossings (2) minimum number of stages (NS) and (3) minimum crosstalk (of 1st order). The main part is devoted to crossing waveguides between the switching matrix (equivalent parallel waveguides, PWs) and the path-selection switches (PS - SWs).
Transformation paths (TPs) allow to transform 1-D shuffle interstage interconnections into their topologically equivalent 2-D shuffles and vice versa. By the assumption of different TPs in source arrays (SAs) and destination arrays (DAs), respectively, TPs exist through switching arrays independent of form (square or rectangle) and size. The ongoing work is devoted to scalable algorithms for arbitrary large photonics systems.
Massive parallel optical interconnection systems in the 3-D physical space offer advantages but they also represent serious difficulties caused by the increase of the degree of freedom. These difficulties are mainly (1) packaging and precise mounting but also (2) problems of combinatorial nature. Throughout the paper, the presentation of routing information in greater than or equal to 2-D MINs is discussed and the results are aimed to avoid delay and additional hardware. The results apply to distributed switching and processing under the optical interconnection regime.
Planar lightwave circuits (PLCs) of all-optical multi-layer switches are presented where the cycle structure of the spatial switches is modeled by rearrangeable nonblocking pruned rectangular cellular arrays (PRCAs). The chosen network model is the coupled PRCA. However, refined modeling of the proposed M X N-gon prism switches causes, beside coupling, additionally, the embedding of small PRCAs into larger ones and their intersection. Dependent on the implementation of PLCs of PRCAs (ordinary, reversed and combinations) the (1) moving location of switches is needed and the (2) multiple use of directional couplers arises, respectively. The PLCs of wide-sense nonblocking dilated multi-layer switches with simple switch setting are also presented.
The decomposition of 2D shuffle multistage interconnection networks without loss of functionality is presented and analyzed. The major costs for the decomposition arise by the interconnection of the subdivided switches. The attempts considered differ with regard to the implementation of the dimension-dependent switches (hybrid and all-optical) and with regard to the organization of the interconnection scheme (hierarchical and direct). These attempts are described, evaluated and compared. The application of the decomposition to distributed switching/processing is briefly illustrated.
Multi-layer switching architectures, where the layers and the parallel waveguides form N-gon prisms (N is the number of waveguides), represent a novel class of all-optical switches which (1) approach shuffle results through their 2 X 2- switches are interconnected in a nearest-neighbor manner and moreover (2) improve shuffle results by increasing the connectivity (equalsVnumber of 2 X 2-switches) between the prisms. However, as multi-layer switches may be difficult to implement by the current technology of photonics, the paper provides a first step towards the planar realization of this class of multi-layer switches.
Throughout the paper, the concept of the planar 1-D lay-out (one layer) of kXk-switches and the concept of its compact double-layer/multi-layer counterpart will be presented and experimental results analysed for k equals 4. The paper presents the construction and treats of the working principles of switching system that can operate with minimum number of stages. A reduction of the number of stages is obtained due to combination of the electro-optic (EO) polymer films on the isotropic substrate. The optical switches are collected in one optical layer and each layer composes of at most two 2X2- switches simultaneously active. Poly (methyl methacrylate) (PMMA) film doped with azo dye and para-nitro- aniline/polyvinyl alcohol (p-N-An/PVA) were used for the EO films preparation. Thereby switching is applied horizontally (use one layer only) and vertically (between the two layers). The scheme of such a 4X4-switch in double-layer technique has been presented.
The optical N-gon prism switch is the simplest multi-layer architecture and a starting point for novel all-optical switching architectures. The N-gon prism switch is composed of parallel waveguides at each layer; switching between the waveguides at each layer and switching between the waveguides at adjacent layers is assumed, and the optics comes in by the transmission of light through the waveguides. Throughout the paper, switch setting algorithms for the N-gon prism switch are discussed where (1) an approach to an algebraic and scalable algorithm is presented for its fast implementation and (2) the further development of the multi-layer switching concept is taken into account.
Throughout the paper, novel all-optical planar 1-stage k multiplied by k-switches and compact minimum-stage k multiplied by k-switches in double-layer and multi-layer technique, are presented and analyzed. In the first case, the number of k(k - 1)/2 switches of size 2 multiplied by 2 (equivalent minimum of the Spanke-Benes network) are arranged in parallel instead of the number of k (equivalent maximum) cascaded 2 multiplied by 2-switches of the Spanke- Benes network. In the second case, the number of 2 multiplied by 2-switches depends on the geometry of the 'pipes' of the switches formed by the layers and waveguides [for a square it is 3k/2(k/2 - 1) for rearrangeable nonblocking and 3(k - 1)k/2(k/2 - 1) for circuit switching networks]. The number of stages (NS) (horizontal cascaded) of the proposed compact switches for the nonblocking interconnection is NS equals n - 1 if the waveguides form an n-gon (n greater than or equal to 3) for any size of the k multiplied by k-switch. In this way, the attenuation of optical signals passing through a photonic network may be minimized. In particular, for any size of a k multiplied by k-switch, dependent on the n-gon, the minimum NS is n-1 equals 2 (triangle) or n - 1 equals 3 (square) etc. Thus the proposed switch concept is of complexity O(1), i.e. the NS is independent of the number of inputs/outputs. Additionally, the proposed switches are capable to operate in the circuit switching mode if and only if (iff) the parallelism increases by the factor k-1.
The WDM/OFDM-networks of the paper are multigraphs. Transformations of these graph models are presented and applied to WDM/OFDM-networks for their later analysis and optimization. The paper concentrates on two transformations, the complement and the dual of WDM/OFDM-networks. However, the complement exists only for simple graphs whereas WDM/OFDM-networks are multigraphs with multiple edges between two nodes. The computation of the dual is simple only for planar graphs and for regular graphs in terms of hypergraphs. Throughout the paper, the dual and the complement of nonregular simple graphs/networks are computed by means of computer programs. However, the aim of the ongoing work is to compute the dual and the complement of multigraphs. By means of the dual of multigraphs the topologically equivalent simple graph and in turn the complement may be obtained which both are aimed to extend the range of analysis and design concepts of WDM/OFDM- networks. Various examples are presented which explain the applied concepts.
Architectural and algorithmic principles of the common planar multistage interconnection network are reviewed. For the purpose of its proper operation in the circuit switching mode the enlarged planar network and the multiple planar network are introduced. Additionally, a novel planar 4 X 4-switch is presented and applied to two planar architectures with switches of size >= 4 X 4.
Kautz and de Bruijn networks are applied to model the free-space optica' interconnection of data arrays. For this purpose, the Kronecker sum (KS) and the Kronecker product (KP) of these networks is mapped into the 3-D physicaI space. The properties of the KS and KP networks are analysed and discussed. A switch and graph preserving transformation of l-D de Bruijn networks into their 2-D networks (and vice versa) is presented. The realization/refinement of the de Bruijn graphs by optical interconnections generates shuffle networks and thus the KP of 1st order of de Bruijn networks equals 2-D shuffle networks. The hardware requirement for the generation of permutations is analysed.
Keywords: Data arrays, de Bruijn, Kautz, d-dimensional shuffle, Kronecker sum, Kronecker product, switch and graph preserving transformation
Kautz directed graphs (digraphs) arise from larger de Bruijn digraphs by the deletion of nodes or from smaller de Bruijn digraphs by additional nodes, respectively. Thus all-optical self- routing networks with Kautz topologies have different properties compared with de Bruijn. Kautz digraphs may be extended by their sum and product graphs and contain planar (crossover-free) embeddings.
Proc. SPIE. 2537, Novel Optical Systems Design and Optimization
KEYWORDS: Matrices, Multiplexing, Colorimetry, Fluorescence correlation spectroscopy, Niobium, Orthogonal frequency division multiplexing, Optimization (mathematics), Frequency division multiplexing, Signal detection, Lead
The assignment of wavelengths to the paths of wavelength division multiplexing/optical frequency division multiplexing networks (wavelength allocation) is analyzed and the applicability of mathematical methods (optimization) is discussed.
Two basic architectures are deduced from 1D multistage architectures with nearest-neighbor (NN) interconnection of switches and extended into 2D architectures by a mathematical transformation. These two architectures are the Spanke-Benes (SB) network and the NN multistage interconnection network of switches (NN-MIN). The properties and applications of the two 2D architectures are described.
Selfrouting in multistage interconnection networks (MINs) with nearest-neighbor (NN) interstage interconnects is applied by mapping the system onto an equivalent shuffle net. For this purpose, additional interconnects between the switches are introduced which provide a 1:1 mapping of shuffle MINs onto NN-MINs (and vice versa). Both homogeneous and inhomogeneous lay-outs of the interconnection system are applied. Examples are presented and discussed for the interconnection of 8 X 8 and 16 X 16 data arrays, respectively.
The permutation of switches is applied to generate (1) butterfly, (2) crossover and (3) global/local interconnection patterns. Both classes, the shuffle-based interconnections (1) and (2) and the global/local interconnections (3) are analyzed and compared.
The length of multistage architectures is a crucial design parameter. The compact architectures overcome this problem but raise several others: additional interconnects because of the separation of the switching arrays, and search for the best interconnection scheme.
For OFDM systems based on multistage interconnection networks with space-frequency interstage patterns the number of crossed channels and the interconnection length in the frequency domain is aimed to be a minimum. A solution which only requires nearest-neighbor interconnections in the frequency domain is proposed and discussed.