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```Analysis of Optical Beam Propagation Through Turbulence
Heba M. El-Erian, Walid Ali Atia* , Linda Wasiczko, and Christopher C. Davis
Department of Electrical and Computer Engineering, University of Maryland, College Park
*now with Axsun, Inc. Billerica, Mass.
The Random Interface Geometric
Optics Model
Models the random medium as random
curved interfaces with random refractive index
discontinuities across them. Snell’s law
applied to evaluate the trajectories of the rays
crossing the interfaces. Various ray behaviors
are Monte-Carlo simulated.
The model contains two simulation routines
that:
 Calculate the beam wander of a single
pencil thin ray traveling through
turbulence
 Calculate the phase wander between two
parallel rays traveling through turbulence
Beam Wander Simulation Model
The Maryland Optics Group
Single Pass Aperture Averaging
Collecting lens
Filter
PD
Amplifier
Calculates the 3-D trajectory for a single ray traveling at distance L through a
simulated random medium. The mean square beam wander is averaged over
each run.
To
Labview
PD
1cm collecting
telescope with
narrowband filter
Incident HeNe
Phase Wander Simulation Model
Turbulence Equations
For a plane wave propating in the z-direction, E  E0 e j t  kz 
2
2
For a Laser Beam, E r   E0 e  r w
Beam Wander
Beam centroid deviates as it propagates through turbulence.
Mean square beam wander for a Gaussian beam (by Ishimaru),
W02
2
1 z 2  1   2 z 2  2.2Cn2 l01 / 3 z 3
l 
2
where l 0 is the inner scale,  2  1 R0, where R0 is the radius of
2
equivalent Gaussian wave, W0 is the spot-size, and  1   W0.
For a plane wave, it simplifies to,




The trajectories of two parallel rays propagating through the simulated random
medium are computed simultaneously. The difference in the path length
traveled l will yield the phase difference  between the rays    2  l
Beam Wander Simulation Results
 l2  2.2C n2 l 01 / 3 z 3
0.030
Phase Wander
D p r   0.32Cn2 k 2 Lr 5 / 3
Aperture Averaging
The intensity variance decreases with the area of the receiver.
Aperture averaging (by Tatarski),

16 bI  
G D  
K  d ,
D 2 0 bI 0
where, K    arccos D    D1   D 
GD  is the intensity variance of the actual receiver relative to
a point receiver D  L .
2
2
12
0.025
2
Conclusions
-12
Nominal Cn = 9.95*10
2
MS BEAM WANDER (m )
Mean square phase difference for a plane wave propagating
through weak turbulence found using the phase screen method.
For two narrow collimated beams, D p r  becomes,
Mean squre beam wander
Cubic fit
0.020
 Model to evaluate beam wander,
aperture averaging, and phase
decorrelations for a Gaussian beamwave input have been developed
0.015
0.010
0.005
0.000
0
100
200
300
400
500
TARGET LENGTH (m)
Parameters used: L = 500m with step size = 50m,  l  100m ,  l  90m,
 n  1.00001 ,  n  0.000001, path length threshold = 2cm, N = 1000.
Results show excellent agreement with the cubic fit described in
theory  l2  2.2C n2 l 01 / 3 z 3
Model will be verified with
measurements
```
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