978-1-4244-3559-3/09/$25.00 ©2009 IEEE IWSSC 2009

Impact of Various Flow-Fields on Laser Beam Propagation

Fotios Stathopoulos, Philip Constantinou, Athanasios D. Panagopoulos Institute of Communication and Computer Systems

Mobile Radiocommunications Laboratory Iroon Polytechniou 9, Zografou, Athens, GR-15780, Greece

fotis.stathopoulos@gmail.com, fkonst@mobile.ntua.gr, thpanag@ece.ntua.gr

Abstract — In this paper, we investigate how the aircraft

flow field affects the laser beam wavefront in an optical point-to-point link. We study various flow fields highlighting the different distortions that are created. The subject of the manuscript is twofold. Firstly, we study the steady laminar incompressible flow around an ellipse and secondly we assume a sphere as a flying object and we present simulated results for various types of flow, from laminar to turbulent, as the speed of the sphere is increasing. Furthermore we examine the performance of the distortions along the altitude, when density is getting smaller. Employing the above methology, we have presented one 'realistic' simulation scenario of an aircraft that communicates through an optical link.

Keywords — aircraft/satellite communications, optical

communication, wavefront distortions

I. INTRODUCTION In this paper we continue the research [1] of how the aircraft

flow field affects the laser beam wavefront, in optical point-to-point link. We extend the method, calculating the strehl ratio, and we simulate and study various cases of flow fields (for several velocities) around different types of aircrafts. Finally we set different air-conditions at the free stream area that affect the final results of the simulation.

II. WAVEFRONT DISTORTIONS CALCULATION The variation of the refractive index around an aircraft is

the reason of the wavefront distortions on a propagated laser beam. As the aircraft is moving, the value of each air property changes and affects the refractive index. Assuming that the air properties around the aircraft are known, we introduce a method in order to be able to calculate the wavefront distortions of the laser beam. From the Gladstone – Dale equation [2]:

1nG

ρ−= ορ 1n Gρ= + ⋅ , (1),(2)

where G is the Gladstone – Dale constant, we can calculate the refractive index (n). The Gladstone-Dale constant depends only on the beam wavelength, according to the equation [3], [4]:

( ) 240.017850762.192539 10G λ

λ−⎛ ⎞

= + ×⎜ ⎟⎜ ⎟⎝ ⎠

(3)

10-1 100 1010.2

0.22

0.24

0.26

0.28Gladstone - Dale constant over wavelength

wavelength [µm]

Gla

dsto

ne -

Dal

e co

nst.

( cm

3 / g

)

Figure 1. the Gladstone-Dale constant over the wavelength

From equations (2), (3) we get that in the situation of a laser beam we have a constant ratio between the density and the distance of refractive index from the unit, for two different points (A,B) of the medium:

11

A A

B B

nn

ρρ

−=− . (4)

As we have assumed that the air properties around the aircraft are known as well as the refractive index at the free stream area, it is easy to calculate the refractive index around the aircraft. Then we calculate the optical path length (OPL) of the laser beam:

( ) ( )0

, , , ,y

OPL x y z n x y z dy= ∫ , (5)

and the Optical Path Difference, OPD, that shows the configuration of the wavefront:

minOPD OPL OPL= − . (6)

Figure 2. Diagram of the wavefront error calculation method

Since we have obtained the OPD we are able to calculate and describe the wavefront distortions. Fig.2 shows a flow chart diagram of the method we have followed. First we calculate the boresight error and the mean tilt of the wavefront. Then we analyze the wavefront at the Zernike polynomials, and finally calculating the RMS value of the OPD, we are able to calculate the strehl ratio. In order to get a better aspect of the topic we present the results of extensive simulations.

III. SIMULATED RESULTS In this paragraph we observe the results of the simulation.

We can see various flow fields and then the applying our method we have the wavefront configuration of the laser beam, i.e. OPD. The wavelength of the laser beam is considered at 1550 nm.

A. Air conditions at the free stream area Being an ideal fluid, we consider the dry air to be the fluid

for our study. The velocity will be expressed trough the Mach number. The value of the speed of the air is related only to the temperature. In our research we will use two standard conditions for the value of temperature and pressure of the air at the free stream area. The first air properties model is the Standard Temperature and Pressure (STP). The values for the two mentioned properties are constant and equal with 1 bar (105 Pa) for the pressure and 273.15 K (0 °C) for the temperature. In this case we can calculate the density from which equals with ρSTP=1.2754 kg/m3 and the speed of the air equals with 331.31 m/s. The second model is the barometric formula related to the International Standard Atmosphere model (ISA) [5]. With this formula we can calculate the free stream conditions along the atmosphere.

B. Flow Fields In the first case (A) we assume a flying object as an

ellipse. The big radius of the ellipse is 2m and the small is 1m. The diameter of the concentric circular antenna is taken at 40cm. The velocity is 0.3 Mach (~99.5m/s). In this kind of flow there is symmetry over the y- and the z- axis, so we take in mind only the elevation angle (θ).

In case (B) we simulated several cases of flow around a sphere [6]. We present two examples of compressible flow for a flying sphere, with radius 1m, and a squere antenna of 40cm, for specific velocity and air conditions.

x- axis

y- a

xis

pressure coefficient over a sphere

-2 0 2

-3

-2

-1

0

1

2

3

-1

-0.5

0

0.5

Figure 3. Pressure coefficient around an eclipse, r/R=1m/2m (case A)

Figure 4. Density around a sphere, R=1m, u=37.5m/s, STP

Figure 5. Density around a sphere, R=1m, u=54m/s, air at 20km

Compressible flows around a sphere (case B1, B2)

C. Wavefront configuration – OPD For the cases above (A, B1, B2) we present the results of

the simulation. In this paragraph we can see the OPD, comparing the wavefront distortions for the different kind of flows, but also for the different propagating angles. We have to notice that for cases B1,B2 there is no symmetry only over xz-surface (because of the spherical shape) so we need to define the azimuthial angle. In case A the azimuthial angle equals to zero.

.

Figure 6. OPD for case A, θ= 0ο

Figure 7. OPD for case A, θ= 90ο

Figure 8. OPD for case B1, θ= 30ο, φ= 90ο

Figure 9. OPD for case B2, θ= 60ο, φ= 90ο

IV. MEAN TILT OF THE WAVEFRONT From the wavefront configuration, OPD(x,y), we are able to

calculate the boresight error, a(x,y), through the wavefront phase error, e(x,y) [7]:

( ) ( )0, ,e x y k OPD x y= ⋅ , (7)

where K0 is the wave number, and then:

( ) ( ) ( )a , , ,( , ) 2 ( , )

x y x y e x yx y x y

λφπ

∂ ∂= − = − ⋅∂ ∂ . (8)

Knowing the boresight error, we are able to calculate the mean wavefront tilt along the two coordinates of the wavefront surface (x,y). The mean tilt is the sum of the two coordinates:

,1

1a am

x y iim =

= ∑ and ( ) ( )2 2a a ax y= + (9),(10)

Comparing the results of case A to the same flow around a sphere [1], we notice that the maximum point of the mean is transferred close to elevation angle of 28o from the symmetric case of 45o.

Figure 10. Mean tilt vs the elevation angle, case A, air at STP.

Figure 11. Mean tilt vs height for several elevation angles, case A, θ=30°.

V. ZERNIKE POLYNOMIALS An analytical measure used to characterize the fluctuation

of the wavelength, are the Zernike polynomials. They show us the effect of the corresponding Zernike polynomial making an accurately describing of the wavefront aberrations. At the next figures (Fig.19-23) are presented the variation of the first six coefficients (each one for the corresponding polynomial) for elevation angles from 0° to 90°.

Figures 12-14. The value of the first six Zernike polynomials, case A, θ= 0ο, θ= 30ο, θ= 90ο.

Figures 15. The value of the first six Zernike polynomials for case B1, θ= 30ο, φ= 90ο

Figures 16. The value of the first six Zernike polynomials for Case B2, θ= 60ο, φ= 90ο

VI. RMS-OPD AND STREHL RATIO Another parameter we can research to define the wavefront

fluctuations is the difference-Root Mean Square (RMS) of the wavefront, defined as:

( )221RMSOPD OPD OPD dx dy

S= − ⋅∫ , (11)

where S is the surface of the aperture.

The ratio of the observed peak intensity to the theoretical maximum peak intensity of the beam, called strehl ratio, is another determinant. For the calculation of strehl ratio complex mathematic equations are required, thus we use the following equation, accurate for cases with small errors [8]:

( ) ( )2 2exp expS K OPD f⎡ ⎤= − ∗ =⎣ ⎦ , (12)

where S is the strehl ratio, K is the wave number, OPD is optical path length and f is the wavefront phase error.

Figure 17. OPDRMS vs height, case Α, θ=30° .

Figure 18. Strehl ratio vs height, case A, θ=30°.

VII. CONCLUSIONS AND FURTHER RESEARCH Our study gives many useful insights for further research.

We noticed how the wavefront distortions change when the flying object has more aerodynamic shape than a sphere. Furthermore we have seen the performance of the distortions when the aircraft accelerates and the air flow is more complicated. We succeeded in transferring and using data from a specialized for flows simulation S/W, Tecplot, in order to pass from the aircraft flow field to the laser beam distortions. With our analysis, we have understood how the distortions perform relatively to the flow field. For future work, we are going to simulate any flying conditions for any aircraft, reaching a realistic scenario. Our purpose is to simulate a High Altitude Platform model and observe the distortions in that case and then apply all these results to the research of pointing, acquisition and tracking capabilities of the receiver.

ACKNOWLEDGMENT This work is supported in part by the EC through the FP6 IST project “SatNEx II”. The authors would like to thank Dr. M. Gavaises and Dr. M.Giannadakes for their invaluable help with the flow simulations at Tecplot Software.

REFERENCES [1] Fotios Stathopoulos, Philip Constantinou, “Impact of aircraft boundary

layer on laser beam propagation”, in International Workshop on Satellite and Space Communications 2008, October 2008 Toulouse France.

[2] G. Havener, Optical Wave Front Variance, A study on Analytic Models in Use Today, 1992, AIAA 992-0654.

[3] K.G. Gilbert, L.J. Otten, W.C. Rose, Atmospheric Propagation of Radiation in The Infrared and Electro-optical Systems Handbook, chapter 3: Aerodynamic Effects, 1993.

[4] M.I. Jones, E. Bender, CFD – Based Computer Simulation of Optical Turbulence through Aircraft Flow Fields and Wakes, AIAA 2001-2798.

[5] U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.

[6] http://www.tecplot.com [7] M. Sarazin, F. Roddier, The ESO differential image motion monitor,

Astronomy and Astrophysics vol. 227, p.294-300, 1990. [8] A.J.Laderman, R. de Jonckheere, Subsonic Flow over Airborne

Optical Turrets, AIAA 82-4005.