Method of configuration and optimisation of programmable photonic devices

Abstract

The method object of the invention enables the scalable configuration and performance optimisation to be carried out for programmable optical circuits based on meshed structures, in such a way that they can perform optical/quantum signal processing functions. The object of the invention can be applied in circuits with arbitrary degrees of complexity implemented by means of programming a waveguide mesh. The method object of the invention enables not only the analysis and evaluation of performance to be carried out, but also the subsequent programming and optimisation of programmable optical devices.

Claims

1. A method of configuration and optimisation of programmable optical devices based on meshed optical structures, a meshed optical structure comprising a highly coupled structure defined by at least three or more tunable basic units implemented with two coupled waveguides providing independent values of power and phase division; wherein the method comprises the steps of: a. segmenting an entire mesh into tunable basic units or subsets of tunable basic units in the initial configuration, b. determining a complete frequency response with the tunable basic units or subsets of tunable basic units in the initial configuration, wherein said complete response comprises amplitude and phase of input/output ports of a 2D waveguide mesh, c. calculating at least one parameter of the 2D waveguide mesh from a result of the preceding step (b), and d. modifying the initial configuration of at least one tunable basic unit or subsets of tunable basic units based on the at least one parameter calculated in the preceding step.

2. The method according to claim 1 wherein the complete frequency response of the entire mesh is obtained by applying an inductive method wherein a resulting matrix is obtained with a matrix that defines a mesh formed by n?1 subsets of tunable basic units and a matrix that defines an additional subset that is connected to the mesh formed by n?1 subsets of tunable basic units.

3. The method according to claim 2 wherein a number of input/output ports to connect and a number of new cavities originated after the interconnection of each new subset of tunable basic units defines a different interconnection scenario selected from: a. a scenario 0 defined by the interconnection in a single port, b. a scenario 1 defined by the interconnection of two ports no giving rise to one new cavity, c. a scenario 2 defined by the interconnection of two ports giving rise to one new cavity, and d. a scenario 3 defined by the interconnection of three ports giving rise to one new cavity.

4. The method according to claim 1 wherein an evaluation and a modification of the tunable basic units are carried out using recursive algorithms.

5. The method according to claim 4 wherein the recursive algorithms comprise: a. selecting elements that make up a main circuit to be programmed, b. selecting a subset of tunable basic units adjacent to the main circuit to be used and modifying a configuration of the tunable basic units, c. performing an evaluation of the entire mesh of the system that defines a 2D programmable optical mesh, d. checking a status of the at least one parameter to be optimised, e. calculating a change in configuration of each tunable basic units not present in the main circuit, and f. repeating steps b-e recursively until a desired optimisation is reached.

6. The method according to claim 1 wherein the evaluation stage of the programmable circuit combines an analytical evaluation with an experimental monitoring of an optical signal in a subset of the output ports or in internal points of the circuit.

7. The method according to claim 1 wherein at least one of the tunable basic units is a non-resonant interferometer of Mach-Zehnder type.

8. The method according to claim 7 wherein the Mach-Zehnder interferometer is balanced, wherein both arms that make up the interferometer are equal with 3 dB losses.

9. The method according to claim 1 wherein at least one of the tunable basic units is a double actuation directional coupler.

10. The method according to claim 1 wherein at least one of the tunable basic units is a resonant interferometer.

11. The method according to claim 1 wherein at least one of the tunable basic units has an arbitrary number of ports.

12. The method according to claim 1 wherein at least one of the tunable basic units is configured with tuning elements based on: MEMS, thermo-optic tuning, electro-optical tuning, optomechanical or electro-capacitive tuning.

13. The method according to claim 1 wherein the subsets of tunable basic units form uniform topologies of 2D programmable optical circuits.

14. The method according to claim 1 wherein the subsets of tunable basic units form non-uniform topologies of 2D programmable optical circuits.

15. The method according to claim 1 wherein the at least one parameter to be calculated and optimised is related to programming of the programmable optical device.

16. The method according to claim 15 wherein the at least one parameter to be calculated and optimised is selected from a set consisting of: total power consumption, loss reduction, interference and crosstalk reduction, isolation between circuits and reduction of an area used.

Description

DESCRIPTION OF THE DRAWINGS

(1) To complement the description that is being made and for the purpose of helping to better understand the features of the invention according to a preferred practical exemplary embodiment thereof, a set of drawings is attached as an integral part of said description in which the following is depicted in an illustrative and non-limiting manner:

(2) FIG. 1 shows different meshed circuits and segmentation options in TBUs or subset of TBUs. All of them and any circuit that can be discretised in identical tuning units are eligible the application of the disclosed method. (a) Square mesh application, (b) hexagonal mesh application, (c) triangular mesh application.

(3) FIG. 2 shows the discretisation in TBUs for different meshed circuit topologies (a) Hexagonal uniform, (b) Square uniform, (c) Triangular uniform, (d) Unidirectional propagation uniform interferometer and (e) non-uniform wherein each TBU can have a different orientation and size.

(4) FIG. 3 shows the tri-TBUs building block for 2D hexagonal waveguide meshes and the magnification ratio between the number of optical nodes and optical ports with the number of cells. (a), tri-TBU made up of three TBUs and associated symbol, (b), two tri-TBUs interconnected by the optical node P1, (c) Three tri-TBUs that create a closed hexagonal cell, (d), eight tri-TBUs interconnected to obtain a waveguide mesh formed by four cells. (e), number of optical nodes (ON) and optical ports versus number of closed cells (C) in a meshed waveguide IC integrated photonic circuit.

(5) FIG. 4 depicts the inductive method to obtain the dispersion matrix H(n) of a 2D hexagonal waveguide mesh made up of n basic tri-TBU units by adding one tri-TBU unit H(1) to a 2D hexagonal waveguide mesh composed of n?1 basic tri-TBU units H(n?1) and a general signal flow diagram to derive H(n) as a function of h(n?1) and H(1).a, Interconnection Scenario 0.b, Interconnection Scenario 1.c, Interconnection Scenario 2.d, Interconnection Scenario 3.

(6) FIG. 5 depicts scenario 0. (a) Connection diagram with mesh n?1, (b) interconnection diagram with the label contributions, (c) resulting sections of the matrix. S1:x=P?1. The direct contribution within the ports of network N is not included in the graph.

(7) FIG. 6 depicts scenario 1. (a) Connection diagram with mesh n?1, (b) interconnection diagram with the label contributions, (c) resulting sections of the matrix. S1:x=P?1,y=P. The direct contribution within the ports of network N is not included in the graph.

(8) For graphs showing the signal flow, the connections N, M, X, Y, F, D E, F, Q, R, C, D, A, B, S, U, I, J, B, F, hyy, hzz, hxx represent signal flow pathways with transfer functions given by the coefficients of the dispersion matrix H(n?1). The connections K, L, O, P, A, H, C, E, T, G, V, W represent the additional signal flow pathways resulting from the additional tri-TBU.

(9) FIG. 7 depicts scenario 2. (a) Connection diagram with n?1 mesh, (b) interconnection diagram with the contributions label, (c) resulting sections of the matrix. x=P?1, y=P. Note that the direct contribution within the ports of network N is not included in the graph.

(10) FIG. 8 depicts scenario 3. (a) Connection diagram with mesh n?1, (b) interconnection diagram with labeled contributions, (c) resulting sections of the matrix. x=P?2, y=P?1, z=P. The direct contribution within network ports is not included in the graph.

(11) FIGS. 9-11 depict practical examples of use of the method and the technical advantages obtained. In the first case, a structure has been configured that implements an optical filter based on interferometric cavities and the response thereof has been evaluated for each combination of explored TBU configurations. For the second case (FIG. 10), the mesh is programmed to perform a complex optical circuit formed by 4 resonant cavities loaded in a balanced MZI interferometer. The optimisation is carried out to evaluate the performance related to the filtering (extinction range, losses and ripple in the passband). For the third case (FIG. 11), the mesh implements two independent circuits. The first is based on three coupled cavities and the second is an unbalanced MZI type two-sample filter. The figure shows that the application of the proposed method returns an improvement in the reduction of optical interferences between circuits, improving the performance of both.

PREFERRED EMBODIMENT OF THE INVENTION

(12) In a preferred exemplary embodiment of the object of the invention, the starting point is a 2D waveguide mesh formed from the replication of a basic tuning element implemented by means of two waveguides coupled by an independent (in power and phase division) tunable basic unit (TBU), tunable basic unit (TBU) that is configured by means of tuning elements based on: MEMS, thermo-optical tuning, electro-optical tuning, or optomechanical or electro-capacitive tuning.

(13) This tunable basic unit (TBU) can be preferably implemented by means of balanced, tunable Mach-Zehnder interferometers (MZI), or by means of a double actuation directional coupler and representable by means of a H.sub.TBU 2?2 transmission matrix. Depending on the orientation and the interconnection of the TBUs, uniform (square, hexagonal, triangular etc.) or non-uniform topologies are originated if each TBU has an arbitrary length and orientation. Next, a theoretical segmentation in TBUs or subset of TBUs of the target mesh is performed to apply the implementation of mathematical induction (MI). In the case of hexagonal waveguide meshes, an option for the basic or tri-TBU building block is made up of three TBUs (A, B, and C) connected in a Y-configuration as shown in FIG. 3.a. The tri-TBU set is described by means of a 6?6 dispersion matrix calculated from the three H.sub.TBU dispersion matrices that describe the respective internal TBUs thereof. To aid in the graphical illustration of the method, we will use a triangle symbol to represent the tri-TBU, wherein each port has, in principle, internal connections to the four opposite ports (that is, port 1 to ports 3, 4, 5, 6, etc.). The tri-TBU can be replicated and distributed N times to generate any desired hexagonal mesh arrangement of any size. For example, FIGS. 3b and 3c show the process that leads to the construction of a single hexagonal cell made up of three tri-TBUs (we will use the notation Ai, Bi, Ci to identify the TBUs that make up the tri-TBU i).

(14) Even for the simplest structure represented by the unit cell, there are already twelve input/output ports and six intermediate auxiliary nodes required for the calculation of the 12?12 transfer matrix (that is, 144 elements). With an increasing number of cells, the above figures show a drastic increase. For example, the four-cell structure shown in FIG. 3.d, which is still a low-complexity structure, has twenty input/output ports, thirty-eight internal nodes, and a 20?20 dispersion matrix (i.e., 400 elements). FIG. 3.e provides the exact number of input/output ports and internal nodes according to the number of hexagonal cells, and clearly shows that the analytical derivation of dispersion matrices for 2D meshes becomes seemingly unapproachable even for a very low cell count.

(15) Also, the numerical methods to analyse the responses of the circuits, such as the FDTD (finite-difference time domain) and eigen-mode based solutions, do not scale well as the number of components in the photonic circuit increases.

(16) Formally, the method object of the invention is expressed as follows, a 2D structure formed by a tri-TBU is described by a unitary dispersion matrix H (1) with known coefficients. Then, if a 2D structure formed by n?1?1 tri-TBUs is described by means of a unitary dispersion matrix H (n?1) with known coefficients, the structure made up of n tri-TBUs obtained by adding an additional H (1) tri-TBU to the first is described by means of a unitary dispersion matrix H (n) with known coefficients.

(17) This method enables the sequential derivation of the dispersion matrix of an arbitrary n-order hexagonal waveguide mesh using the above lower-order mesh dispersion matrix H (n?1) and that of the newly added H (1) tri-TBU. The final calculation thereof will depend on how the additional tri-TBU connects to the above lower order mesh. Four different interconnection scenarios can be identified, as shown in FIG. 2a, 4.a to 4.d, depending on the number of ports that are interconnected and the number of new complete hexagonal cells that appear after incorporating the new tri-TBU.

(18) In a first scenario, scenario 0, referring to the simplest case that represents the starting point of the design of a new mesh, only one of the 6 ports that define the triple frame is connected to the ports of the previous mesh. Adding the new tri-TBU increases the number of mesh ports by 4, correspondingly increasing the number of rows and columns in the dispersion matrix.

(19) In a second scenario, Scenario 1, adding the new tri-TBU increases the number of mesh ports by 2, but the number of complete hexagonal cells does not increase.

(20) In a third scenario, scenario 2, adding the new tri-TBU increases the number of ports by 2 and the number of complete cells by 1.

(21) In a fourth scenario, scenario 3, adding the new three-lattice network does not increase the number of ports, as it connects 3 ports to the previous mesh and the number of complete cells increases by 1.

(22) FIGS. 5-8 depict for each scenario the more general signal flow diagram that must be taken into account to derive the overall dispersion matrix H(n) according to H(n?1) and H(1). The nodes s, r shown on the left side represent any pair of input and output ports respectively (the ranges of variation allowed for s, r are also displayed depending on the scenario, wherein P is the input/output port count of H(n?1) before connecting the additional tri-TBU). The nodes x,y,z identify the input/output ports of H(n?1) that are used to connect this mesh to the newly added tri-TBU (the allowed values for x,y,z are also shown according to the scenario). In FIGS. 5-8 the connections

(23) N, M, X, Y, F, D E, F, Q, R, C, D, A, B, S, U, I, J, B, F, hyy, hzz, hxx represent signal flow pathways with transfer functions given by the coefficients of the dispersion matrix H(n?1). While the connections K, L, O, P, A, H, C, E, T, G, V, W represent the additional signal flow pathways that result from the additional tri-TBU. The transfer functions (additional matrix coefficients) for these connections must be calculated to obtain the overall dispersion matrix H(n).

(24) In order to carry out the aforementioned derivatives, the four scenarios described above are used, in this way we have:

(25) In scenario 0 only one of the 6 ports of the new tri-TBU (Latt N) which is added to H(n?1) is connected to the order mesh n?1. As shown in FIG. 4.a, adding one tri-TBU (Latt N) increases the number of mesh ports by 4, and correspondingly, the number of rows and columns in the dispersion matrix H(n). The interconnection diagram, shown in FIG. 5b, depicts the possibilities of signal flow within the order mesh n?1 and between this mesh and the newly added tri-TBU through the interface node x=P. This interconnection diagram defines a system of equations associated to the node x which can be solved, giving rise to the following equations (Eq. 1) that provide the matrix coefficients that characterise the new waveguide mesh ports:
Submatrix 1 coefficients:h.sub.s,r=X=h.sub.s,r.sup.N-1.
Submatrix 2 coefficients:h.sub.s,(P, . . . ,P+4).sup.N=GB
Submatrix 3 coefficients:h.sub.(P, . . . ,P+4),r.sup.N=TS,
Submatrix 4 coefficients:h.sub.(P, . . . ,P+4),(P, . . . ,P+4).sup.N=Th.sub.XXG+IntCon,(1)
wherein IntCon represents the internal connections given by the dispersion matrix of the triple-framed additional unitary cell latt n.

(26) Scenario 1: here, adding the new tri-TBU latt n increases the number of mesh ports by two but the number of complete hexagonal cells does not increase, as shown in FIG. 6.a. FIGS. 6.b. and 6.c wherein the associated interconnection diagram to be solved and the resulting matrix for the order mesh are n respectively depicted. In this case, the resulting equations are more complex since two interface nodes are required (x=P?1ey=P). Solving the system of equations related to the nodes x=P?1 and y=P equations (Eq. 2) are obtained that provide the matrix coefficients that characterise the new waveguide mesh ports and the four sub-matrices:
SM1 h.sub.s,r=X=h.sub.s,r.sup.N-1,
SM2 h.sub.s,(P?1, . . . ,P+2)=BG+DP,
SM3 h.sub.(P?1, . . . ,P+2),r=OE+TS,
SM4 h.sub.(P?1, . . . ,P+2),(P?1, . . . ,P+2)=(h.sub.xxG+PM)+O(h.sub.yyP+GN).(2)

(27) In scenario 2 adding the new tri-TBU increases the number of ports by two and the number of complete hexagonal cells by one, as shown in FIG. 7.a. In this case, the signal flow diagram is shown in FIG. 7.b, wherein the possibility is included of recirculation between the interface nodes x=P?1 and y=P and the newly added tri-TBU unit latt n as shown in the connections V, W. The procedure is similar to the two above scenarios 0 and 1, solving the system of equations associated to the nodes y, x; in this way, solving the system of equations related to the nodes x=P?1 and y=P equations (Eq. 3) are obtained that provide the dispersion matrix coefficients that characterise the new waveguide mesh ports and the four sub-matrices:

(28) $\begin{array}{cc}& \left(3\right)\end{array}$$\begin{array}{cc}\mathrm{SM1}& \end{array}$$\begin{array}{cc}{h}_{s,r}=X+\left[\frac{\begin{array}{c}DW\left[h_{\mathrm{xx}}^{\left(N-1\right)}{\mathrm{VE}}^{}+\left(1-\mathrm{VN}\right)S\right]+\\ B^{}V\left[h_{\mathrm{yy}}^{\left(N-1\right)}WS+\left(1-\mathrm{MW}\right)E^{}\right]\end{array}}{\left(1-\mathrm{VN}\right)\left(1-\mathrm{MW}\right)-h_{\mathrm{xx}}^{\left(N-1\right)}{h}_{\mathrm{yy}}^{\left(N-1\right)}\mathrm{VW}}\right],& \end{array}$$\begin{array}{cc}\mathrm{SM2}& \end{array}$$\begin{array}{cc}{h}_{s,\left(P-1,.Math.,P-2\right)}=\mathrm{UG}+F^{}P+\frac{\left(\begin{array}{c}{F}^{}W\left[h_{\mathrm{xx}}^{\left(N-1\right)}\left(G+{\mathrm{VPh}}_{\mathrm{yy}}^{\left(N-1\right)}\right)+\left(1-\mathrm{VN}\right)\mathrm{MP}\right]+\\ \mathrm{UV}\left[h_{\mathrm{yy}}^{\left(N-1\right)}\left(P+{\mathrm{WGh}}_{\mathrm{xx}}^{\left(N-1\right)}\right)+\left(1-\mathrm{MW}\right)\mathrm{NG}\right]\end{array}\right)}{\left(1-\mathrm{VN}\right)\left(1-\mathrm{MW}\right)-h_{\mathrm{xx}}^{\left(N-1\right)}{h}_{\mathrm{yy}}^{\left(N-1\right)}\mathrm{VW}},& \end{array}$$\begin{array}{cc}\mathrm{SM3}& \end{array}$$\begin{array}{cc}{h}_{\left(P-1,.Math.,P+2\right),r}=\frac{\left(\begin{array}{c}O\left[h_{\mathrm{yy}}^{\left(N-1\right)}\mathrm{WS}+\left(1-\mathrm{MW}\right)E^{}\right]+\\ T\left[h_{\mathrm{xx}}^{\left(N-1\right)}\mathrm{VF}+\left(1-\mathrm{NV}\right)S\right]\end{array}\right)}{\left(1-\mathrm{VN}\right)\left(1-\mathrm{MW}\right)-h_{\mathrm{xx}}^{\left(N-1\right)}{h}_{\mathrm{yy}}^{\left(N-1\right)}\mathrm{VW}}& \end{array}$$\begin{array}{cc}\mathrm{SM4}& \end{array}$$\begin{array}{cc}{h}_{\left(P-1,.Math.,P+2\right),\left(P-1,.Math.,P+2\right)}=\frac{\left(\begin{array}{c}O\left[h_{yy}^{\left(N-1\right)}P+h_{\mathrm{yy}}^{\left(N-1\right)}W{h}_{\mathrm{xx}}^{\left(N-1\right)}G+\left(1-MW\right)\mathrm{NG}\right]+\\ T\left[h_{\mathrm{xx}}^{\left(N-1\right)}G+h_{\mathrm{yy}}^{\left(N-1\right)}V{h}_{\mathrm{xx}}^{\left(N-1\right)}P+\left(1-VN\right)MP\right]\end{array}\right)}{\left(1-VN\right)\left(1-MW\right)-h_{\mathrm{xx}}^{\left(N-1\right)}{h}_{yy}^{\left(N-1\right)}\mathrm{VW}}+\mathrm{IntCont}.& \end{array}$

(29) In the third scenario, as shown in FIG. 8.a, adding the new tri-TBU does not increase the number of ports, as it connects three ports to the previous mesh and the number of complete cells is increased by one. Here, the interconnection diagram involves three interface nodes x,y,z, (represented in FIG. 8b). The procedure to obtain the coefficients of the different sub-matrices is similar to the three previous scenarios, but with more complexity given the complexity of the result of the addition, which leads to:

(30) $\begin{array}{cc}\mathrm{SM1}{?}_2=C\left(\mathrm{MK}+H_{\mathrm{xx}}E\right)+L\left(\mathrm{NE}+H_{\mathrm{yy}}K\right),{?}_1=1-\mathrm{FE}-\mathrm{IK}-H_{\mathrm{zz}}{?}_2,z_1=\frac{\left(C^{}\left(1-\mathrm{BC}-\mathrm{JL}\right)+\left(\mathrm{SC}+E^{}L\right)\mathrm{Hzz}\right)}{\left({?}_1\left(1-\mathrm{BC}-\mathrm{JL}\right)-\left(\mathrm{BC}+\mathrm{JL}\right)\mathrm{Hzz}{?}_2\right)},z_4=\frac{\left(\mathrm{CS}+{\mathrm{LE}}^{}+{?}_2{z}_1\right)}{\left(1-\mathrm{CB}-\mathrm{LJ}\right)},h_{s,r}=X+\left(\mathrm{DK}+B^{}E\right)z_1+{\mathrm{Rz}}_4.& \end{array}$$\begin{array}{cc}\mathrm{SM2}{?}_2=C\left(\mathrm{MK}+H_{\mathrm{xx}}E\right)+L\left(\mathrm{NE}+H_{\mathrm{yy}}K\right),{?}_1=1-\mathrm{FE}-\mathrm{IK}-H_{\mathrm{zz}}{?}_2,z_1=\frac{\begin{array}{c}(\left(1-\mathrm{BC}-\mathrm{JL}\right)+\left(\mathrm{IP}+\mathrm{FG}\right)+\\ \mathrm{Hzz}\left(H+\mathrm{CHxxG}+\mathrm{CMP}+\mathrm{LHyyP}+\mathrm{LNG}\right))\end{array}}{\left({?}_1\left(1-\mathrm{BC}-\mathrm{JL}\right)-\left(\mathrm{BC}+\mathrm{JL}\right)H_{\mathrm{zz}}{?}_2\right)},z_4=\frac{\left(H+{\mathrm{CH}}_{\mathrm{xx}}G+\mathrm{CMP}+{\mathrm{LH}}_{\mathrm{yy}}P+\mathrm{LNG}+z_1{?}_2\right)}{\left(1-\mathrm{BC}-\mathrm{JL}\right)}{h}_{s,\left(P-1,.Math.,P+2\right)}=D\left(P+{\mathrm{Kz}}_1\right)+{\mathrm{Rz}}_4+B^{}\left(G+{\mathrm{Ez}}_1\right).& \end{array}$$\begin{array}{cc}\mathrm{SM3}{?}_1=\mathrm{NE}+H_{\mathrm{yy}}K,{?}_2=\mathrm{MK}+H_{\mathrm{xx}}E,{?}_3=1-\mathrm{IK}-\mathrm{FE},{?}_1=\left(1-\mathrm{BC}\right){?}_1+\mathrm{JC}{?}_2,{?}_2=\left(1-\mathrm{BC}\right){?}_3-{\mathrm{CH}}_{\mathrm{zz}}{X}_2,{?}_1\left(1-\mathrm{BC}\right)E^{}+\mathrm{JCS},{?}_2=-\left(1-\mathrm{BC}\right)C^{}-{\mathrm{CH}}_{\mathrm{zz}}S,z_5=\frac{\left(1-\mathrm{BC}-\mathrm{JL}\right){?}_2-H_{\mathrm{zz}}L?1}{\left(H_{\mathrm{zz}}L{?}_1-\left(1-\mathrm{BC}-\mathrm{JL}\right){?}_2\right)},y_3=\frac{?1+{?}_1{z}_5}{\left(1-\mathrm{BC}-\mathrm{JL}\right)},x_3=\left(S+{\mathrm{BLy}}_3+{?}_2{z}_5\right)/\left(1-\mathrm{BC}\right),h_{\left(P-1,.Math.,P+2\right),r}={\mathrm{Oy}}_3+{\mathrm{Az}}_5+{\mathrm{Tx}}_3.& \end{array}$$\begin{array}{cc}\mathrm{SM4}{?}_1=\mathrm{NE}+H_{\mathrm{yy}}K,{?}_2=\mathrm{MK}+H_{\mathrm{xx}}E,{?}_3=1-\mathrm{IK}-\mathrm{FE},{?}_1=-{?}_2\left(H_{\mathrm{yy}}P+\mathrm{JH}+\mathrm{NG}\right)+.Math.{?}_1\left(H_{\mathrm{xx}}G+\mathrm{BH}+\mathrm{MP}\right),{?}_2=\left(1-\mathrm{BC}\right){?}_1+\mathrm{JC}{?}_2,{?}_3=-{?}_3\left(H_{\mathrm{xx}}G+\mathrm{BH}+\mathrm{MP}\right)-.Math.{?}_2\left(H_{\mathrm{zz}}H+\mathrm{IP}+\mathrm{FG}\right),{?}_4={?}_2{\mathrm{CH}}_{\mathrm{zz}}-{?}_3\left(1-\mathrm{BC}\right),{?}_1=\left(1-\mathrm{JL}\right){?}_2+\mathrm{BL}{?}_1,{?}_2={?}_3\mathrm{BL}+{?}_2{H}_{\mathrm{zz}}L,y_3=\left({?}_3{?}_2-{?}_1{?}_4\right)/\left({?}_4{?}_1+{?}_2{?}_2\right),x_3={\left(-y_{3^{}}*\mathrm{?2}+{?}_3\right)}_{}/{?}_4,z_5=\left(y_3\left(1-\mathrm{JL}\right)-H_{\mathrm{yy}}P-\mathrm{JH}-{\mathrm{JCx}}_3-\mathrm{NG}\right)/{?}_1,\mathrm{or}{z}_5=\left(H_{\mathrm{zz}}H+\mathrm{IP}+\mathrm{FG}+H_{\mathrm{zz}}{\mathrm{Cx}}_3+H_{\mathrm{zz}}{\mathrm{Ly}}_3\right)/{?}_3,h_{\left(P-1,\mathrm{.Math.P}+2\right),\left(P-1,\mathrm{.Math.P}+2\right)}={\mathrm{Oy}}_3+{\mathrm{Az}}_5+{\mathrm{Tx}}_3+\mathrm{IntCont}.& \end{array}$

(31) This completes the complete set of analytical expressions that enable the core of the algorithm responsible for evaluating the dispersion matrix that defines the system given the values of each TBU to be implemented. The core of the method is then recursively used to configure and optimise mesh performance.

(32) By way of example of implementation, a series of experimental results are provided in this document that reinforce the previous assertions regarding the flexibility and the advantages of the object of the invention.

(33) In this way, the method of the invention is applied to configure, optimise and evaluate circuits of different degrees of complexity implemented by programming a 40 input/40 output waveguide mesh. This involves calculating 40?40=1600 matrix coefficients subject to varying conditions imposed by the large number of possible combinations of individual configuration of the parameters of each TBU. Furthermore, for each wavelength, the method object of the invention makes it possible to evaluate the 40?40 matrix in a few seconds for each iteration of the optimisation/configuration process.