beam propagation method devang parekh 3/2/04 ee290f
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Beam Propagation Method
Devang Parekh3/2/04EE290F
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Outline
What is it? FFT FDM Conclusion
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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
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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
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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
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BPM (cont.)
Substituting again
First two-linear; last-nonlinear
22
22
1 1''
2 2j j kn
dz dt
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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
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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
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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
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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
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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
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Fast Fourier Transform (FFTBPM)
Numerically solving
Discrete Fourier Transform
Fast Fourier Transform
Divide and conquer method
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Fast Fourier Transform (FFTBPM)
Cool Pictures
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Fast Fourier Transform (FFTBPM)
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Finite Difference Method (FDMBPM)
Represent as differential equation
Apply Finite Difference Method
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Finite Difference Method (FDMBPM)
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Finite Difference Method (FDMBPM)
Cool Pictures
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Finite Difference Method (FDMBPM)
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Conclusion Can be used for linear and nonlinear
propagation Either method depending on
computational complexity can be used
Generates nice graphs of light propagation
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Reference Okamoto K. 2000 Fundamentals of Optical Waveguides
(San Diego, CA: Academic)