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Beam propagation method

for waveguide device simulation

Chrisada Sookdhis

Photonics Research Centre, Nanyang Technological University 

This is for III-V Group Internal Tutorial

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Beam propagation method (BPM)Beam propagation method is a mathematical procedure used to

study the evolution of electromagnetic fields in arbitrary inhomoge-nious medium. BPM yields the response of a given device to an exter-nal optical sigal, similar to in an experiment.

Applications

• Devices that defy eigenmode study, e.g. tapers

• Avoids difficult approximations

• Cases where radiation is important.• Behaviours of special devices.

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Starting from Maxwell’s Equations (CGS units)

∇× E = −1

c 

∂ H

∂ t −

4π

c 

∂ M

∂ t 

∇× H =   1c ∂ E∂ t 

+  4πc ∂ P∂ t 

+  4πc 

J

∇· E = −4π∇·P + 4πρ

∇·H = −4π∇·M

(1)

Assumptions: Non-magnetic materials M

= 0, no charge ρ = 0, zerocurrent  J = 0, linear isotropic media  P = χeE, eiωt  time dependence.

∇× E = −ik0H

∇× H = ik0E

∇· E = 0

∇·H = 0,

(2)

where  = 1 + 4πχe = n2

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E  formulationEquation (3) can be written as

∇2E + k2

0E = ∇(∇· E)

z -invariant media

∂

∂ z E z  ≈ 0

∇2Et + k2

0Et  = −∇t (∇t  ln · Et )   (5)

or in terms of  x  and y  components:

∇2E  x + k20E  x  = −∂ 

∂  x ∂  ln

∂  x  E  x −

  ∂ 

∂  x ∂  ln

∂  y  E  y 

∇2E  y + k20E  y  = −

∂ 

∂  y 

∂  ln

∂  x E  x 

−

  ∂ 

∂  y 

∂  ln

∂  y E  y 

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Slowly varying envelope approximation

Et  =  Et e−in0k0z ,

∂ 2Et 

∂ z 2

2n0k0

∂ Et 

∂ z 

With above approximations, the electric field’s evolution is given by

Vectorial BPM

i

∂ 

∂ z E  x E  y = A xx    A xy  A yx    A yy 

E  x E  y 

  (6)

where (for illustration)

 A xy E  y  =

  1

2n0k0 ∂ 

∂  x 1

∂ (E  y )

∂  y −   ∂ 2E  y 

∂  x ∂  y 

 A xx  = · · ·   ,   A yy  = · · ·   ,   A yx  = · · ·

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Semi-vector BPM

i∂ E  x 

∂ z =  A xx E  x 

i∂ E  y 

∂ z =  A yy E  y 

(7)

This taks into account the polarization ( A xx  =  A yy ), but neglects the

coupling between E  x  and E  y  ( A xy  =  A yx  = 0).When variation of refractive index is small in transverse dimen-sion, polarization dependency and coupling are weak and may beneglected.

It is safe to treat the two polarizations as decoupled, as long as theyare not synchronized with some mechanism within the device.

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Scalar BPM

i∂ E 

∂ z 

=  Ascalar E    (8)

where

 Ascalar  =  1

2n0k0

 ∂ 2

∂  x 2 +

  ∂ 2

∂  y 2 + (− n2

0)k20

(8) governs the conventional scalar beam propagation method. This isuseful where device is weakly-guiding and/or polarization dependencecan be neglected.

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Finite-difference scheme

In beam propagation

z

z+∆z

we apply the propagator  U  to calculate

Et (z +∆z ) =  U (∆z )Et 

Propagator  U  can take many forms, depending on the chosen BPMtechnique, e.g.

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• Paraxial/Wide angle

• Vectorial/Scalar BPM

• Boundary conditions

Discretization

n n+1n-1

n n+1n-1

m+1

m

m-1

 x 

 y 

2D FD-BPM

3D FD-BPM

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Finite-difference BPM

nn+1

n-1

m+1

mm-1

 x 

 y 

n n+1

n-1

m+1

m

m-1

 x 

 y 

z 

z+∆z 

Discretize the device volume and re-place differentiations in BPM propagator with difference operator.

∆ x  =  x m+1 − x m ∂  f m∂  x 

=   f m+1 − f m−1

2∆ x 

 f m( x ) =  f ( x m)  ∂ 2 f m

∂  x 2  =

  f m+1 − 2 f m + f m−1

∆ x 2

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So we replace

i∂ 

∂ z 

E  x E  y 

=

 A xx    A xy  A yx    A yy 

E  x E  y 

with

Et (z +∆z ) =  U FD-BPM(∆z )Et    (9)

This has been a simplest discussion, the method entails much morefiner points and best left to experts—or commercial software develop-ers.

Things we have not consider:

• Iterative algorithms for solving the coupled equations

• Speed

• Boundary conditions

• Stability

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Optiwave OptiBPM

Device definition

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Profile definition

This is a new implementation from version 5 of OptiBPM. The ma-terial and profile library can be called from many layout files.

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2D vs. 3D FD-BPM

Comparing the time complexity O of 2D- and 3D-BPM,

O3D

O2D

≥ n x 

  (10)

It is recommended to use 2D algorithm  whenever possible.

So when is it possible?

• Device is actually 2D. (slab waveguides)

• Using Effective index method (EIM) for – Weakly guiding structures with low level of radiation

– Full-vectorial BPM is not required

ε(x,y)

ε(y)

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After calculation, the electric field distribuion within the device is dis-played. We can choose to open the Analyzer module for data analysis.

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Examples of data presentation

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W id d l l ti

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Waveguide mode calculation

3 micron

3 micron

Waveguide modes are solved using Alternate Direct Implicit (ADI)method. Users can specify

• The solver engine (Vectorial, scalar)• Boundary conditions (Transparent, Neumann)

• Accuracy of the resultant mode field (e.g. to 1E-007)

• whether to start with the fundamental mode or any other mode.

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Waveguide modesummary table

St l

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Star couplers

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3dB couplers

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3dB couplers

Scanning script

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Scanning script 

Const NumIterations = 7d = 0For x = 1 to NumIterations

ParamMGR.SetParam "offset", CStr(d)ParamMgr.SimulateWGMgr.Sleep(5)d = d - 0 . 2

NextParamMGR.SetParam "offset", 0

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Scanning script results. Best performance is between iteration 3 and4. So we know the separation should be ≈ 2.5  µ m.

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The End

We have discussed

• What is BPM

• The technical backgrounds behind BPM

• Optiwave BPM and its features

– Device definition

– Path monitor and output extractions

– Waveguide Mode solver 

– Parameterization and scripting for optimization of device

Q & A