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