Towards rigorous simulations of Kerr non-linear photonic components in frequency domain
Comments about BEP
Eigenmode expansion (BEP) for nonlinear structures? Didn’t you learn math?
Linear calculation
Define refractive index profile
No problem for Kerr-nonlinearity,
Update refractive index profile
it’s just an iterative loop :)
Eigenmode expansion (BEP) for nonlinear structures?
Eigenmode expansion in each (linear) section
y
x
NL
n changes in z
z
Nonlinear sections are divided
Eigenmode expansion in each (linear) section
y
x
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
y
x
NL
n changes in z
z
The change is small (<1e-4).
Couldn’t we use CMT?
(Coupled Mode Theory)
Coupled Mode Theory
Eigenmode expansion in each (linear) section
y
x
NL
n changes in z
z
• 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)
Zatím nevyřešené problémy
y
x
NL
n changes in z
z
One-way technique
Eigenmode expansion in each (linear) section
y
x
NL
n changes in z
z
One-way technique
y
x
NL
n changes in z
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
metal
• two nonlinear dielectric slot waveguides with metallic claddings (silver at 480 nm)
dielectric
t
w
w
Pin P1
P2
y
z
Example 1: Nonlinear plasmonic coupler
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
No loss Loss
Coupling length decreases with loss
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
Computational efficiency and comparison with FEM
• 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
• 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
Example 2: Soliton-plasmon interaction
metal
nonlineardielectric
(silver at 1500 nm)
Psoliton
PSPP
P0
The structure is excited with fundamental spatial soliton
D (soliton position)
SPP may be created
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
Power at the coupling length
Soliton
SPP
Example 3: Nonlinear cavity
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