Towards rigorous simulations of Kerr non-linear photonic components in frequency domain

1. Towards rigorous simulations of Kerr non-linear photonic components in frequency domain Comments about BEP

2. Eigenmode expansion (BEP) for nonlinear structures? Didn’t you learn math? No problem for Kerr-nonlinearity, it’s just an iterative loop :) Define refractive index profile Linear calculation Update refractive index profile

3. Eigenmode expansion (BEP) for nonlinear structures? Eigenmode expansion in each (linear) section x y z NL n changes in z

4. Nonlinear sections are divided Eigenmode expansion in each (linear) section x y z NL n changes in z

5. Nonlinear sections are divided • The main advantage of BEP, simple propagation in z-invariant sections, is lost • Suitable for complex structures with small number of short nonlinear sections. • For longer nonlinear structures it is probably better to use FEM (in the frequency domain) which is optimized for this task.

6. Eigenmode expansion (BEP) for nonlinear structures? Eigenmode expansion in each (linear) section x y z NL n changes in z The change is small (<1e-4). Couldn’t we use CMT? (Coupled Mode Theory)

7. Coupled Mode Theory Eigenmode expansion in each (linear) section x y z NL n changes in z

8. Zatím nevyřešené problémy x • Jak určit S-matici nelineárního úseku? • přeformulovat vázané rovnice v rovnice pro jednotlivé složky matice S, je více možností • ... ? • Pozor na součet velkých a malých čísel • Jak se změní S-matice na rozhraní? (zatím změnu zanedbávám) y z NL n changes in z

9. One-way technique Eigenmode expansion in each (linear) section x y z NL n changes in z

10. One-way technique x y z NL n changes in z - Simple solution using Runge-Kutta technique - No iteration needed :D - Reminiscent of BPM

11. Example: Nonlinear directional coupler FEM - Comsol RF module

12. Example: Nonlinear directional coupler FEM - Comsol RF module

13. Linear coupler Nonlinear coupler FEM - Comsol RF module

14. NL-BEP Critical power

15. NL-BEP Critical power

16. Example:

17. NL-BEP

18. Conclusions :-) Principle of NL-BEP proposed. :-) One-way technique successfully tested. :-) Bidirectional technique under development.

19. Example 1: Nonlinear plasmonic coupler • two nonlinear dielectric slot waveguides with metallic claddings (silver at 480 nm) y metal dielectric Pin P1 w t P2 w z

20. Example 1: Nonlinear plasmonic coupler No loss Loss Calculation parameters: w = 0.06λ (λ = wavelength in vacuum), t = λ/10, Pin = 0.1, Pin = normalized input power = maximum of nonlinear index change at the input Coupling length decreases with loss

21. Coupling length Lc Power at Lc of the linear device (Lc are different for each w/λ) • the loss significantly affects coupler behaviour • structure does not exhibit critical power • the nonlinear functionality (switching) is still possible

22. Computational efficiency and comparison with FEM • NL-EME does not seem to converge for (absurdly) high values of nonlinearity • For moderate nonlinearities good convergence and agreement with FEM (COMSOL, RF module) • Reasonable approximate results even with low number of modes used in the expansion • Computational efficiency (memory requirement, speed) is one of the main NL-EME advantages • Computational time does not significantly increase with nonlinearity strength • Approximate calculations (low mode numbers) are extremely fast

23. Example 2: Soliton-plasmon interaction nonlinear dielectric metal (silver at 1500 nm) The structure is excited with fundamental spatial soliton Psoliton P0 SPP may be created D(soliton position) PSPP

24. Example 2: Soliton-plasmon interaction • Conversion efficiency and coupling length strongly depend on soliton position D • The results do not appear to depend significantly on soliton amplitude nor propagation constant provided these parameters are near the resonance Coupling length Soliton Power at the coupling length SPP

25. Example 3: Nonlinear cavity