Fig. 1. Simplified schematic of a TBU. It is made up of two 50%:50% DCs on the left and right, and two optical PSs parameterized by {θ,ϕ} in the middle. {θ,ϕ} can be adjusted freely in [0,2π) by thermo- or electro-optic control of the two PSs.

Fig. 2. Schematic of an N×M square-mesh PPIC. For derivation simplicity, we disable the TBUs at the right-most column marked by dashed lines and assume that the top and bottom connections (yellow lines) are ideal.

Fig. 3. Naming conventions for port and direction. Capitalized “A” and “B” should be regarded as port names, and lowercases “a” and “b” are the complex magnitudes ahead of ejωt. For conciseness, we have omitted the ejωt dependence.

Fig. 4. Illustration of the forward input at the left and the backward input at the right. Note that the directions I and O in the parenthesized superscripts are defined according to going into or coming out of the associated vertical TBU, as defined in Fig. 3.

Fig. 5. Schematic of a 1×M square PPIC. The PSs in green horizontal TBUs are fixed to θ=0 and ϕ=π.

Fig. 6. Case 1, routing. (a) and (b) show the target response U(ω) with magnitude normalized to input, and phase, respectively, used in the cost function. (c) shows a heat map of all optimized PS values (see Appendix C for colored cell ordering details). (d) shows the resulting optimized configuration (π omitted). Red lines are those PS changes larger than 0.2π before and after optimization, while blue lines are those with changes smaller than 0.2π. (e) shows the port magnitude at frequency fcenter—orange for inward direction and purple for outward direction (refer to Fig. 3 for definition). Port magnitudes less than 0.2 are not drawn. The light path is plotted in black. (f) shows the power coupling ratio (i.e., cos2ϕ−θ2) of each TBU with a percentage in a shaded bounding box. The edge color of the TBU shows common PS π−ϕ−θ2; see Appendix D; (g) synthesized magnitude response; (h) synthesized phase response; (g) and (h) show the frequency response that the optimized configuration is able to achieve. The square meshes in (e) and (f) share the same color bar, shown in (c).

Fig. 7. Case 2, splitting. (a) Target equal three-way split magnitude response (normalized to the input); (b) heat map of all optimized PS values; (c) optimized configuration (π omitted); (d) port magnitude at frequency fcenter; (e) power coupling ratio and common PS; (f) synthesized magnitude reponse; (g) synthesized phase response; (f) and (g) show the frequency response that the optimized configuration is able to achieve. There are three lines colored in red, blue, and green in (a), and they overlap here and in (f).

Fig. 8. Case 3, coherent two-way splitting. (a) and (b), respectively, show the target equal magnitude split with equal phase response. (c) Heat map of all optimized PS values; (d) optimized configuration (π omitted); (e) port magnitude at frequency fcenter; (f) power coupling ratio and common PS; (g) synthesized magnitude response; (h) synthesized phase reponse; (g) and (h) show the frequency response that the optimized configuration is able to achieve. There are two lines colored in red and blue in (a), and they overlap here and in (b), (g), and (h).

Fig. 9. Case 4, coherent four-way splitting. (a) and (b), respectively, show the target equal four-way magnitude split and phase response. (c) Heat map of all optimized PS values; (d) optimized configuration (π omitted); (e) port magnitude at frequency fcenter; (f) power coupling ratio and common PS; (g) synthesized magnitude response; (h) synthesized phase response; (g) and (h) show the frequency response that the optimized configuration is able to achieve. There are four lines colored in red, blue, green, and cyan in (a), and they overlap here and in (b).

Fig. 10. Case 5, optical filtering. (a) Target magnitude response; (b) heat map of all optimized PS values; (c) optimized configuration (π omitted); (d) port magnitude at frequency fcenter; (e) power coupling ratio and common PS; (f) synthesized magnitude response; (g) synthesized phase response; (f) and (g) show the frequency response that the optimized configuration is able to achieve. Note that in this case, only port magnitudes over 0.3 are plotted in (d) for clarity.

Fig. 11. Case 6, WDM. (a) Target magnitude response; (b) heat map of all optimized PS values; (c) optimized configuration (π omitted); (d) port magnitude at frequency fcenter; (e) power coupling ratio and common PS; (f) synthesized magnitude response; (g) synthesized phase response; (f) and (g) show the frequency response that the optimized configuration is able to achieve. Note that in this case, only port magnitudes over 0.3 are plotted in (d) for clarity.

Fig. 12. Case 7, simultaneously synthesizing two light-processing functions for two in-phase inputs. Figure caption is similar to that of Fig. 6, except that in this case, only port magnitudes over 0.3 are plotted in (d) for clarity.

Fig. 13. Three different configurations are obtained under three random initializations; all satisfy the goal of routing in Case 1. Each row represents one synthesized configuration. Figure 6 is not included here. Left column, the optimized configuration (π omitted); right column, power coupling ratio (%) and common PS; all synthesized magnitude responses are identical to those in Fig. 6(g), hence not shown.

Fig. 14. Three different configurations are obtained under three random initializations; all satisfy the goal of filtering in Case 5. Each row represents one synthesized configuration. Figure 10 is not included here. Left column, synthesized magnitude response; right column, optimized configuration (π omitted).

Fig. 15. Demonstration of how we plot the heat map, using Figs. 6(c) and 6(e) as an example.