Transcript No Slide Title

LPHY 2000
Bordeaux France
July 2000
The split operator numerical solution of
Maxwell’s equations
Q. Su
Intense Laser Physics Theory Unit
Illinois State University
S. Mandel
H. Wanare
R. Grobe
G. Rutherford
Acknowledgements: E. Gratton, M. Wolf, V. Toronov
NSF, Research Co, NCSA
Electromagnetic wave
Maxwell’s eqns
Light
scattering in
random media
Photon density wave
Photon diffusion
Boltzmann eqn
Diffusion eqn
Outline
• Split operator solution of Maxwell’s eqns
• Applications
• simple optics
• Fresnel coefficients
• transmission for FTIR
• random medium scattering
• Photon density wave
• solution of Boltzmann eqn
• diffusion and P1 approximations
• Outlook
Numerical algorithms for Maxwell’s eqns
Frequency domain methods
Time domain methods
Finite difference
U(t->t+dt)
A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995)
Split operator
J. Braun, Q. Su, R. Grobe, Phys. Rev. A 59, 604 (1999)
U. W. Rathe, P. Sanders, P.L. Knight, Parallel Computing 25, 525 (1999)
Exact numerical simulation of
Maxwell’s Equations
Initial pulse satisfies :
 r E  0
 B  0
Time evolution given by :
E
c

 B
t r 
2
B
  E
t
Split-Operator Technique

0



E

 


ct cB


Effect of vacuum
Effect of medium
1

E 
v
m E 
r 
 H  H  


cB
cB
0 

 0

H   


 0 

v

 1
0  
H , r  
1

 r  
0 0 

m
 
Er ,t  t  
E r, t  

 U 



cBr ,t  t 

cBr ,t 

U e
H H , r t
v
m
 U 1 2 U1 U1 2  Ot
m
v
m
3

Numerical implementation of evolution in Fourier space
Er ,t  t  
E r ,t  
m ˜ v ˜ m
˜
F 
 U 12 U1 U1 2 F 



cBr, t  t 

cBr ,t 

where
˜U v  e tF H F
1
v
-1
and
˜ m1  e
U
2
 
1
tF H m , r F -1
2
Reference:
“Numerical solution of the time-dependent Maxwell’s equations for random dielectric media”
- W. Harshawardhan, Q.Su and R.Grobe, submitted to Physical Review E
First tests : Snell’s law and Fresnel coefficients
Refraction at air-glass interface
10
5
y/l
n1
l
n2
0
q2
-5
-10
-10
-5
0
z/l
5
10
Fresnel Coefficient
fig2(n1=1,n2=2).d
0.7
0.6
Et / E i
0.5
0.4
0.3
0.2
0
20
40
q1
60
80
Second test
Tunneling due to frustrated total internal reflection
d
s
q
n1
n2
n1
Amplitude Transmission Coefficient vs Barrier Thickness
1
0.8
Et/Ei
0.6
0.4
0.2
0
0
0.5
1
d/l
1.5
2
Light interaction with random dielectric spheroids
• Microscopic realization
• Time resolved treatment
• Obtain field distribution at every point in space
One specific realization
•
•
•
•
400 ellipsoidal dielectric scatterers
Random radii range [0.3 l, 0.7 l]
Random refractive indices [1.1,1.5]
Input - Gaussian pulse
20
T=8
T = 16
T = 24
T = 40
10
0
-10
y/l
10
0
-10
-20
z/l
Summary - 1
• Developed a new algorithm to produce exact spatiotemporal solutions of the Maxwell’s equations
• Technique can be applied to obtain real-time
evolution of the fields in any complicated
inhomogeneous medium
» All near field effects arising due to phase are included
• Tool to test the validity of the Boltzmann equation
and the traditional diffusion approximation
Photon density wave
Infrared carrier
penetration but incoherent due to diffusion
Modulated wave 100 MHz ~ GHz
maintain coherence
Input light
Output light
D.A. Boas, M.A. O’Leary, B. Chance, A.G. Yodh, Phys. Rev. E 47, R2999, (1993)
tumor
Boltzmann Equation
for photon density wave
1 

   Ir,,t   s  d' p,'  I r,' ,t    s  a  Ir,,t 
c t

J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)
Studied diffusion approximation and P1 approximation
Q: How do diffusion and Boltzmann theories compare?
Bi-directional scattering phase function
1
1
p, '   1 g dcos   1  1  g dcos  1
2
2
 1 
 

  Rx, t   r   a  Rx, t  r Lx, t 
 c t x 
 1 
 

  Lx, t  r Rx,t   r   a  Lx, t
 c t x 
Diffusion approximation
r
1
  cos   1
2 s

(R  L)  0
t
Other phase functions
Mie cross-section: L. Reynolds, C. Johnson, A. Ishimaru, Appl. Opt. 15, 2059 (1976)
Henyey Greenstein: L.G. Henyey, J.L. Greenstein, Astrophys. J. 93, 70 (1941)
Eddington: J.H. Joseph, W.J. Wiscombe, J.A. Weinman, J. Atomos. Sci. 33, 2452 (1976)
Solution of Boltzmann equation
0.20
Transmitted intensity
Incident intensity
2
1.5
1
0.5
0
-30
-20
-10
0
10
20
30
0.15
0.10
0.05
0.00
-30
-20
-10
0
10
Position (cm)
Position (cm)
Incident: —
Transmitted: —
Diffusion: —
20
30
Frequency responses
-0.8
-0.5
Log Transmission
Log Reflection
0
-1
-1.5
reflected
-2
-2.5
-1.2
-1.6
-2
transmitted
-2.4
1
10
100
1
10
 (GHz)
 (GHz)
100
Exact Boltzmann: —
Diffusion approximation: —
Confirmed behavior obtained in P1 approx
J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)
Photon density wave
2
0.8
Right going
Left going
0.6
L (x)
1.5
R (x)
0.7
1
0.5
0.4
0.3
0.5
0.2
0.1
0
0
0.5
1
1.5
2
0
0
0.5
x (cm)
1
x (cm)
Exact Boltzmann: —
Diffusion approximation: —
1.5
2
Resonances
at w = n l/2 (n = integer)
-0.098
Log Transmission
Log Transmission
-0.09
-0.092
-0.094
-0.096
-0.098
-0.0985
-0.099
-0.0995
-0.1
1
10
100
 (GHz)
1000
10
4
-0.1
0
0.5
Exact Boltzmann: —
Diffusion approximation: —
1
1.5
l (mm)
2
2.5
3
Summary
Numerical Maxwell, Boltzmann equations obtained
Near field solution for random medium scattering
Direct comparison: Boltzmann and diffusion theories
Outlook
Maxwell to Boltzmann / Diffusion?
Inverse problem?
www.phy.ilstu.edu/ILP