The split operator numerical solution of Maxwell’s equations Q. Su Intense Laser Physics Theory Unit Illinois State University LPHY 2000 Bordeaux France July 2000 Acknowledgements: E. Gratton, M. Wolf, V. Toronov NSF, Research Co, NCSA S. Mandel R. Grobe H. Wanare G. Rutherford
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The split operator numerical solution of Maxwell’s equations Q. Su Intense Laser Physics Theory Unit Illinois State University LPHY 2000Bordeaux FranceJuly.
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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
Electromagnetic wave
Maxwell’s eqns
Lightscattering in
random media
Photon density wave
Boltzmann eqn
Photon diffusion
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 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)
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
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
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
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
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
d
n1
s
n2
n1
Second testTunneling due to frustrated total internal reflection
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
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
20
0
10
-10
0
10
-10
y/
-20 z/
T = 8 T = 16
T = 24 T = 40
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
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)
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
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
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)
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)
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)
-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: —
Summary
Numerical Maxwell, Boltzmann equations obtainedNear field solution for random medium scatteringDirect comparison: Boltzmann and diffusion theories