computational photonics beam propagation method (bpm) · 2019. 6. 7. · –beam propagation method...

Computational Photonics

– Beam Propagation Method (BPM) –

Dr. rer. nat. Thomas Kaiser

Introduction

07.06.2019 Computational Photonics SS2019 - Dr. Thomas Kaiser 2

Introduction

• so far: treated rigorous propagation:

– solving Maxwell’s equations directly without

approximations

(however, we were assuming material properties!)

– leapfrog iteration, 1st order PDE FDTD

– covers whole physics, includes all effects

(reflection, refraction, diffraction, field-enhancement,

polarization effects, … )

– but: numerically very expensive number of

treatable problems limited


Introduction

• We need less heavy numerical methods to

increase number of treatable problems

• Therefore, we need to get less rigorous

• We want to look at problems for approximated

cases now

• Which often-used approximations do we know?


Introduction

• no diffraction, interference or other wave

phenomena:

Ray Tracing

• no polarization effects,

paraxial approximation:

BPM


zeemax.com

https://pingo.coactum.de/421066

Which statement about the paraxial approximation is true?

The Slowly Varying Envelope Approximation (SVEA)


SVEA

• Recap from FOMO: paraxial approximation

• scalar Helmholtz equation:

• principal solutions without approximations:

– plane waves:

– spherical waves:


SVEA

• approximating the spherical wave by a parabolic wave:

• but what is the corresponding PDE?


spherical

parabolic

plane

SVEA

• starting with Helmholtz equation:

• introducing an ansatz which separates the fast

oscillations in z-direction with an average

• is a number which must be chosen depending on

the problem, e.g. with average refractive index


SVEA

• inserting into Helmholtz equation:


SVEA

• The equation for the envelop A(r) is still exact!

• Now comes the trick:

– we assume solutions where the envelop is just

slowly varying, i.e.


SVEA

• This is the slowly varying envelop approximation

• The implications are profound! The paraxial

Helmholtz equation is a parabolic PDE:

• In contrast to an elliptic PDE (boundary value

problem like mode calculation), the parabolic

Helmholtz equation can be numerically iterated

(initial value problem)


SVEA

• However, switching from an elliptic type of PDE to a parabolic type of PDE has consequences!

• Neglecting the 2nd derivative in z means that we do not have ±kz solutions any more (diffusion type problem)

in BPM, there are no backward waves!(and therefore also no reflections!)

• How can we iteratively solve the paraxial Helmholtz equation?


Discretization Schemes for parabolic PDEs


Finite Difference Schemes

• Let us rearrange the equation (assuming 2D for

simplicity)

• Task: Given A(x) in a plane z=0, calculate A(x,z)

in any other plane!

(initial value

problem)


z (n)

x (j)


• paraxial Helmholtz equation in operator notation:

• operator just acts on x-coordinate!

• Let’s discretize it with usual scheme:



• The operator L then has a tridiagonal

discretization:



• The RHS of the equation is now ready in a

discretized fashion!

• Only remaining question:

How do we discretize the 1st order derivative

on the LHS?



• Recap:

possibilities to discretize a 1st order derivative:

– forward differencing:

(1st order accurate)

– backward differencing:

(1st order accurate)

– central differencing:

(2nd order accurate)


BPM - explicit

• 1st choice: forward differencing (FTCS)

• We store the x-dependent field at each z-

position in a vector A

• Wow, it is trivial to iterate this vector explicitly

from step n to step (n+1):


BPM - explicit

• This explicit BPM scheme is very simple and

computationally effective (just multiplications of

sparse matrices)


• However, the method

is unstable under any

conditions (see next

lecture), errors grow

exponentially while

iterating

worthless!

BPM - implicit

• 2nd choice: backward differencing

• Oops, sth. goes wrong. We know and

want to know , not the other way around…

• Solving for requires to solve a linear system

of equations:


BPM - implicit

• This might look cumbersome and stupid, but the

system matrix is just tridiagonal and there are

very efficient numerical methods implemented

for solving such systems!


• Main argument for

this scheme: it’s

unconditionally

stable!

BPM – Crank-Nicolson

• 3rd choice: central differencing

• if things have to be implicit anyways, we could

go the extra mile and head for more accuracy

• central differences are 2nd order accurate!

• the equation is an explicit-implicit mixture:

Crank-Nicolson-Method


BPM – Crank-Nicolson

• Crank-Nicolson is the de-facto standard method

for iterating initial value problems (not just in the

BPM)

• it is unconditionally stable and 2nd order

accurate!

• It gives supreme results over the implicit BPM

scheme and the extra effort is rather minor (one

matrix multiplication)


BPM

• Comparison of the three schemes:


W.H. Press et al.,

“Numerical Recipes

in C++”, 2nd ed.

Cambridge

University Press

Next lecture

• Stability

• Boundaries

• Including polarization (vector BPM)

• Modifications for wide angles


computational photonics beam propagation method (bpm) · 2019. 6. 7. · –beam propagation method...

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