Beam Propagation Method

Devang Parekh3/2/04EE290F

Outline

What is it? FFT FDM Conclusion

Beam Propagation Method Used to investigate linear and

nonlinear phenomena in lightwave propagation

Helmholtz’s Equation2 2 2

2 22 2 2

( , , ) 0E E E

k n x y z Edx dy dz

BPM (cont.)

Separating variables

( , , ) ( , , ) ojkn zE x y z x y z e

Substituting back in

2 2 2 22 ( ) 0o oj kn k n ndz

BPM (cont.)

Nonlinear Schrödinger Equation

22

22

1 1( ' ) ''

2 2

A A Aj A kn A Adz dt dt

Optical pulse envelope

Switch to moving reference frame

1'gv

( , ) ( , )A z t z t't z

BPM (cont.)

Substituting again

First two-linear; last-nonlinear

22

22

1 1''

2 2j j kn

dz dt

Fast Fourier Transform (FFTBPM)

Use operators to simplify

(A B)dz

2

2

1A= ''

2j

dt 2

2

1B=

2j kn

Solution

A AB2 2( , ) ( , )h hhz h t e e e z t

Fast Fourier Transform (FFTBPM)

A represents linear propagation

Switch to frequency domain

2

2

1= ''2j

dz dt

2''=- (2 ) ( , )

2j f z f

dz

Fast Fourier Transform (FFTBPM)

Solving back for the time domain( )( , ) ( , ) oj z z

oz f z f e 2''(2 )

2f

( )) ( )1 1 1( , ) [ ( , )] [ ( , ) ] [ [ ( , )] ]o oj z z j z zo oz F z f F z f e F F z f e

1 2( , ) [ [ ( , )] ]2

hj

o

hz t F F z f e

Plug in at h/2

Fast Fourier Transform (FFTBPM)

Similarly for B(nonlinear)

[B( ) B( )]2*( , ) ( , )

2 2

hz z hh h

z t e z t

Using this we can find the envelope at z+h

Fast Fourier Transform (FFTBPM)

Three step process

1. Linear propagation through h/2

1 2( , ) [ [ ( , )] ]2

hj

o

hz t F F z f e

2. Nonlinear over h1 2( , ) [ [ *( , )] ]

2

hjh

z h t F F z t e

3. Linear propagation through h/21 2( , ) [ [ *( , )] ]

2

hjh

z h t F F z t e

Fast Fourier Transform (FFTBPM)

Numerically solving

Discrete Fourier Transform

Fast Fourier Transform

Divide and conquer method

Fast Fourier Transform (FFTBPM)

Cool Pictures

Fast Fourier Transform (FFTBPM)

Finite Difference Method (FDMBPM)

Represent as differential equation

Apply Finite Difference Method

Finite Difference Method (FDMBPM)

Finite Difference Method (FDMBPM)

Cool Pictures

Finite Difference Method (FDMBPM)

Conclusion Can be used for linear and nonlinear

propagation Either method depending on

computational complexity can be used

Generates nice graphs of light propagation

Reference Okamoto K. 2000 Fundamentals of Optical Waveguides

(San Diego, CA: Academic)