Beam Propagation Method

GEOMETRY OF THE BEAM PROPAGATION METHOD

THE UNIVERSITY OF TEXAS AT EL PASO

Pioneering 21st Century

Electromagnetics and Photonics

The beam propagation method (BPM) is widely used in photonics and nonlinear optics. It propagates a beam through nonhomogeneous media and achieves its efficiency by handling the

propagation problem one grid slice at a time. The basis formulation assumes one-way propagation under the paraxial approximation. Bi-directional and wide-angle formulations exist.

BPM is primarily a “forward” propagating algorithm

where the dominant direction of propagation is

longitudinal.

The grid is computed and interpreted as it is in FDFD.

The algorithm and implementation looks more like the

method of lines.

BENEFITS

• Highly efficient method

• Can easily incorporate nonlinear materials properties. This is very unique

for a frequency-domain method.

• Simple to formulate and implement

FFT-BPM is simpler to formulate and implement.

BPM is commonly used to model nonlinear optical devices and waveguide

circuits.

DRAWBACKS

Not a rigorous method

Small angle approximation

Ignores backward reflections

FFT-BPM is slower, less stable, and less versatile than FD-BPM

FORMULATION OF THE BASIC BPM ALGORITHM

Step 1: Start with Maxwell’s equations.

yzxx x

x zyy y

y xzz z

EEH

y z

E EH

z x

E EH

x y

yzxx x

x zyy y

y xzz z

HHE

y z

H HE

z x

H HE

x y

0

0

0

x k x

y k y

z k z

y xxx xx yy yy zz zz x y

x y

y xxx xx yy yy zz zz x y

x y

s ss s

s s

s ss s

s s

Step 2: Reduce problem to two dimensions. 0y

y

xx x

x zyy y

y

zz z

EH

z

E EH

z x

EH

x

y

xx x

x zyy y

y

zz z

HE

z

H HE

z x

HE

x

x zyy y

y

xx x

y

zz z

H HE

z x

EH

z

EH

x

x zyy y

y

xx x

y

zz z

E EH

z x

HE

z

HE

x

E-Mode H-Mode

Step 3: Assume a solution using the slowly varying envelope approximation.

eff

eff

, ,

, ,

jn z

jn z

E x z x z e

H x z x z e

eff

eff

x zx yy y

y

y xx x

y

zz z

jnz x

jnz

x

eff

eff

x zx yy y

y

y xx x

y

zz z

jnz x

jnz

x

E-Mode H-Mode

Step 4: Write equations in matrix form.

eff

eff

hxx x z yy y

y

y xx x

e

x y zz z

jnz

jnz

hh D h ε e

ee μ h

D e μ h

eff

eff

exx x z yy y

y

y xx x

h

x y zz z

jnz

jnz

ee D e μ h

hh ε e

D h ε e

E-Mode H-Mode

Step 5: Derive matrix wave equation with small angle approximation.

2

2

y

z

e 1 2eff eff2

y H E

xx x zz x y xx yy yj n nz

eμ D μ D e μ ε I e 0E-Mode

H-Mode 2

2

y

z

h 1 2eff eff2

y e h

xx x zz x y xx yy yj n nz

hε D ε D h ε μ h 0

Step 6: Approximate the z-derivative

1 2effeff2

y H E

xx x zz x xx yy y

jn

z n

eμ D μ D μ ε I e

1 2effeff2

y e h

xx x zz x xx yy y

jn

z n

hε D ε D ε μ h

E-Mode

H-Mode

1 1 1

eff2 2

i i i i i i

y y e y e yj

z n

e e A e A e

1 1 1

eff2 2

i i i i i i

y y h y h yj

z n

h h A h A h

1

2

eff

i i H i E i i

e xx x zz x xx yy n

A μ D μ D μ ε I

1

2

eff

i i h i e i i

h xx x zz x xx yy n

A ε D ε D ε μ I

Step 7: Solve field at i+1

E-Mode

H-Mode

1

1 1

eff eff4 4

i i i i

y e e y

j z j z

n n

e I A I A e

1

1 1

eff eff4 4

i i i i

y h h y

j z j z

n n

h I A I A h

BLOCK DIAGRAM OF THE ALGORITHM

SNAPSHOTS FROM A TYPICAL MODEL