Transcript NL-BEP
Comments about BEP Towards rigorous simulations of Kerr non-linear photonic components in frequency domain
Eigenmode expansion (BEP) for
nonlinear
Didn’t you learn math?
structures? No problem for Kerr-nonlinearity, it’s just an iterative loop :) Define refractive index profile Linear calculation Update refractive index profile
Eigenmode expansion (BEP) for
nonlinear
structures? Eigenmode expansion in each (linear) section
x y z
NL
n
changes in
z
Nonlinear sections are divided Eigenmode expansion in each (linear) section
x y z
NL
n
changes in
z
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.
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)
Coupled Mode Theory Eigenmode expansion in each (linear) section NL
n
changes in
z x y z
Zatím nevyřešené problémy • • • 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) NL
n
changes in
z x y z
One-way technique Eigenmode expansion in each (linear) section NL
n
changes in
z x y z
One-way technique NL
n
changes in
z x y z
- Simple solution using Runge-Kutta technique - No iteration needed :D - Reminiscent of BPM
Example: Nonlinear directional coupler FEM - Comsol RF module
Example: Nonlinear directional coupler FEM - Comsol RF module
Linear coupler Nonlinear coupler FEM - Comsol RF module
Critical power NL-BEP
Critical power NL-BEP
Example:
NL-BEP
Conclusions :-) Principle of NL-BEP proposed.
:-) One-way technique successfully tested.
:-) Bidirectional technique under development.
Example 1: Nonlinear plasmonic coupler • two nonlinear dielectric slot waveguides with metallic claddings (silver at 480 nm)
y P
in metal dielectric
P
1
w t w P
2
z
Example 1: Nonlinear plasmonic coupler No loss Loss Calculation parameters:
w
= 0.06λ (λ = wavelength in vacuum),
t P
in = 0.1, = λ/10,
P
in = normalized input power = maximum of nonlinear index change at the input Coupling length decreases with loss
Coupling length
L
c • the loss significantly affects coupler behaviour • structure does not exhibit critical power • the nonlinear functionality (switching) is still possible Power at
L
c of the linear device (
L
c are different for each
w
/λ)
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
Example 2: Soliton-plasmon interaction nonlinear dielectric metal (silver at 1500 nm) The structure is excited with fundamental spatial soliton
P
0
D
(soliton position)
P
soliton SPP may be created
P
SPP
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
Example 3: Nonlinear cavity