# Towards rigorous simulations of Kerr non-linear photonic components in frequency domain.ppt by malj

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

y   z

NL
n changes
in z
Zatím nevyřešené problémy

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

Eigenmode expansion in each (linear) section                  x

y   z

NL
n changes
in z
One-way technique

x

y   z

NL
n changes
in 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
NL-BEP

Critical power
NL-BEP

Critical power
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

y

metal

Pin      w     dielectric                             P1

t

w                                            P2

z
Example 1: Nonlinear plasmonic coupler

No loss                                   Loss

Calculation parameters:                           Coupling length decreases with loss
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 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

•   NL-EME does not seem to converge for     •   Computational efficiency (memory
(absurdly) high values of nonlinearity       requirement, speed) is one of the main
•   For moderate nonlinearities good
convergence and agreement with FEM       •   Computational time does not
(COMSOL, RF module)                          significantly increase with nonlinearity
strength
•   Reasonable approximate results even
with low number of modes used in the     •   Approximate calculations (low mode
expansion                                    numbers) are extremely fast
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
D                                       created
(soliton position)
PSPP
Example 2: Soliton-plasmon interaction
•   Conversion efficiency and coupling             Coupling length
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

Soliton

Power at the
coupling length

SPP
Example 3: Nonlinear cavity

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