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![Page 1: The split operator numerical solution of Maxwell’s equations Q. Su Intense Laser Physics Theory Unit Illinois State University LPHY 2000Bordeaux FranceJuly.](https://reader030.fdocuments.net/reader030/viewer/2022032702/56649cb75503460f9497c7d9/html5/thumbnails/1.jpg)
The split operator numerical solution of Maxwell’s equations
Q. Su
Intense Laser Physics Theory UnitIllinois State University
LPHY 2000 Bordeaux France July 2000
Acknowledgements: E. Gratton, M. Wolf, V. ToronovNSF, Research Co, NCSA
S. Mandel R. Grobe H. Wanare G. Rutherford
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Electromagnetic wave
Maxwell’s eqns
Lightscattering in
random media
Photon density wave
Boltzmann eqn
Photon diffusion
Diffusion eqn
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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
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Numerical algorithms for Maxwell’s eqns
Frequency domain methods
Time domain methods U(t->t+t)Finite difference
A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995)
Split operatorJ. Braun, Q. Su, R. Grobe, Phys. Rev. A 59, 604 (1999)
U. W. Rathe, P. Sanders, P.L. Knight, Parallel Computing 25, 525 (1999)
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Exact numerical simulation of Maxwell’s Equations
Initial pulse satisfies :
Time evolution given by :
r E 0
B 0
E
t
c2
r
B
B
t
E
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H v
0
0
Split-Operator Technique
H m
,r 1
r 1
0
0 0
E r , t t
cB r , t t
U
E r , t
cB r , t
Effect of vacuum
Effect of medium
ct
E
cB
01
r
0
E
cB
H v H m
E
cB
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U eH v
H m
,r t
U 12
m U1v U1
2
m O t3
F
E r , t t
cB r , t t
˜ U 1
2
m ˜ U 1v ˜ U 1
2
m F
E r , t
cB r , t
˜ U 1
2
m e1
2tF H m
,r F -1
˜ U 1
v etF H v
F - 1
and
Numerical implementation of evolution in Fourier space
where
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
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n1
n2
-10 100 5-5 0
10
-10
0
-5
5
z/
y/
First tests : Snell’s law and Fresnel coefficientsRefraction at air-glass interface
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0.2
0.3
0.4
0.5
0.6
0.7
0 20 40 60 80
fig2(n1=1,n2=2).d
1
Et / E
i
Fresnel Coefficient
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d
n1
s
n2
n1
Second testTunneling due to frustrated total internal reflection
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0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2d/
Et/E
i
Amplitude Transmission Coefficient vs Barrier Thickness
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Light interaction with random dielectric spheroids
• Microscopic realization• Time resolved treatment• Obtain field distribution at every point in space
• 400 ellipsoidal dielectric scatterers• Random radii range [0.3 , 0.7 ]• Random refractive indices [1.1,1.5]• Input - Gaussian pulse
One specific realization
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20
0
10
-10
0
10
-10
y/
-20 z/
T = 8 T = 16
T = 24 T = 40
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Summary - 1
• Developed a new algorithm to produce exact spatio-temporal 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
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Photon density wave
Infrared carrier
penetration but incoherent due to diffusion
Modulated wave 100 MHz ~ GHz
maintain coherencetumor
Input light
Output light
D.A. Boas, M.A. O’Leary, B. Chance, A.G. Yodh, Phys. Rev. E 47, R2999, (1993)
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1
c
t
I r,, t s d' p ,' I r,' , t s a I r,, t
Boltzmann Equation for photon density wave
J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)
Q: How do diffusion and Boltzmann theories compare?
Studied diffusion approximation and P1 approximation
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Bi-directional scattering phase function
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)
Other phase functions
p ,' 1
21 g cos 1 1
21 g cos 1
1
c
tx
R x, t r a R x, t r L x, t
1
c
tx
L x, t r R x,t r a L x, t
t
(R L) 0
r 1
2 s cos 1
Diffusion approximation
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Incident: —Transmitted: —
Diffusion: —
Solution of Boltzmann equation
0
0.5
1
1.5
2
-30 -20 -10 0 10 20 30
Inci
dent
inte
nsit
y
Position (cm)
0.00
0.05
0.10
0.15
0.20
-30 -20 -10 0 10 20 30
Tra
nsm
itte
d in
tens
ity
Position (cm)
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J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)
Confirmed behavior obtained in P1 approx
Exact Boltzmann: —Diffusion approximation: —
Frequency responses
-2.4
-2
-1.6
-1.2
-0.8
1 10 100
Log
Tra
nsm
issi
on
(GHz)
reflected transmitted-2.5
-2
-1.5
-1
-0.5
0
1 10 100
Log
Ref
lect
ion
(GHz)
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Photon density wave
Right going Left going
0
0.5
1
1.5
2
0 0.5 1 1.5 2
R (
x)
x (cm)
Exact Boltzmann: —Diffusion approximation: —
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.5 1 1.5 2
L (
x)
x (cm)
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-0.1
-0.098
-0.096
-0.094
-0.092
-0.09
1 10 100 1000 10 4
Log
Tra
nsm
issi
on
(GHz)
-0.1
-0.0995
-0.099
-0.0985
-0.098
0 0.5 1 1.5 2 2.5 3
Log
Tra
nsm
issi
on (mm)
Resonancesat w = n /2 (n = integer)
Exact Boltzmann: —Diffusion approximation: —
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Summary
Numerical Maxwell, Boltzmann equations obtainedNear field solution for random medium scatteringDirect comparison: Boltzmann and diffusion theories
Outlook
Maxwell to Boltzmann / Diffusion?Inverse problem?
www.phy.ilstu.edu/ILP