Abstract

Measuring the Fried parameter r0 (atmospheric optical coherence length) in optical link scenarios is crucial to estimate a receiver’s telescope performance or to dimension atmospheric mitigation techniques, such as in adaptive optics. The task of measuring r0 is aggravated in mobile scenarios, when the receiver itself is prone to mechanical vibrations (e.g., when mounted on a moving platform) or when the receiver telescope has to track a fast-moving signal source, such as, in our case, a laser transmitter on board a satellite or aircraft. We have derived a method for estimating r0 that avoids the influence of angle-of-arrival errors by only using short-term tilt-removed focal intensity speckle patterns.

© 2011 Optical Society of America

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References

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  1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77(1967).
    [CrossRef]
  2. A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
    [CrossRef]
  3. A. Glindemann, “Beating the seeing limit—adaptive optics on large telescopes,” state doctorate thesis (Ruprecht-Karls-Universität, 1997).
  4. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998), p. 144.
  5. Proceedings of the International Workshop on Ground-to-OICETS Laser Communications Experiments 2010—GOLCE (Japanese National Institute of Information and Communications Technology, 2010).
    [PubMed]
  6. J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

2002 (1)

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
[CrossRef]

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77(1967).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998), p. 144.

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77(1967).
[CrossRef]

Glindemann, A.

A. Glindemann, “Beating the seeing limit—adaptive optics on large telescopes,” state doctorate thesis (Ruprecht-Karls-Universität, 1997).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998), p. 144.

Tokovinin, A.

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
[CrossRef]

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77(1967).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
[CrossRef]

Other (4)

A. Glindemann, “Beating the seeing limit—adaptive optics on large telescopes,” state doctorate thesis (Ruprecht-Karls-Universität, 1997).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998), p. 144.

Proceedings of the International Workshop on Ground-to-OICETS Laser Communications Experiments 2010—GOLCE (Japanese National Institute of Information and Communications Technology, 2010).
[PubMed]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

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Figures (6)

Fig. 1
Fig. 1

Typical tilt-removed (a) short-term FIS and (b) long-term tilt-removed FSD (right, overlay of 48 samples, equivalent to 1 s exposure) measured during KIODO–2009. Both images show 990 μm × 990 μm of the focal intensity image, signal wavelength λ = 847 nm , receiver aperture diameter D = 0.4 m , and focal length f = 8.3 m . These values imply a minimum diffraction-limited focal spot sigma radius of ρ dl = 0.61 * λ * f / D = 10.8 μm , which would only be seen under perfect seeing (without any atmospheric distortions). Note that all aAoA and all tracking AoA have been removed from these images.

Fig. 2
Fig. 2

Gaussian fits (dotted curves) to the (a) short- and (b) long-term FSD of Fig. 1. Instead of a simple horizontal cut through the focal intensity, a weighted circular integration of the FIS and FSD has been performed for these plots, providing a much better estimate for the sigma of the Gaussian fit. As a result of this processing, the curves are symmetric to the center axis; the ordinate is the intensity in arbitrary units.

Fig. 3
Fig. 3

Samples of the FIS sigma radius, plotted linearly over time (different satellite elevation angles are marked on the abscissa); the ordinate is the sigma radius in camera pixels (with a 9.9 μm pixel pitch). Dots represent the short-term tilt- removed radii ρ FIS - tr - st (sigma of its Gaussian fit), and the black curve is their 1 s sliding mean. A lower bound of the FIS size is given by the diffraction-limited focal spot size. The FIS reduces with increasing elevation as less disturbing atmosphere is passed by the laser beam from the satellite. A total of 7775 FIS samples was evaluated for this plot, representing 163 consecutive seconds of satellite downlink measurements.

Fig. 4
Fig. 4

Factor between tilt-included r 0 - st and tilt- removed r 0 - tr - st as a function of r 0 - st with the receiver aperture diameter D as the parameter. The relation becomes dubious for r 0 - st D and should only be applied for r 0 - st < D .

Fig. 5
Fig. 5

Relation between the measured r 0 - tr - st and the desired r 0 - st according to approximation (7), with the receiver aperture diameter D as the parameter. The exact numerical inversion of Eq. (5) is plotted with thin curves to show the quality of the approximation (7).

Fig. 6
Fig. 6

Comparison of different averaging times (1 and 10 s ) for r 0 estimation in the optical satellite downlink [5] according to the above formulas. The increase of r 0 10 with elevation during the satellite’s ascending path over the ground station coincides with the theory (less air mass in the link path at higher elevations); the unsteady behavior of this curve is caused by intermittent turbulent layers traversed by the link. The OGS aperture is D = 0.4 m , and the wavelength is λ = 847 nm .

Equations (8)

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r 0 0.98 · λ · f FWHM FSD = 0.59 · λ · f σ FSD ,
σ FSD 2 = ρ FIS 2 = ρ FIS - tr - st 2 + ρ aAoA 2 ,
β aAoA 2 = 0.182 · λ 2 D 1 / 3 · r 0 5 / 3 ; r 0 D .
( 0.59 · λ · f r 0 - st ) 2 = ( 0.59 · λ · f r 0 - tr - st ) 2 + 0.182 ( f · λ ) 2 D 1 / 3 · r 0 - st 5 / 3 ,
r 0 - tr - st = r 0 - st · [ 1 0.52 · ( r 0 st D ) 1 / 3 ] 1 2 .
r 0 - st r 0 - tr - st = f ( r 0 - st , D ) = 1 0.52 · ( r 0 - st D ) 1 / 3 .
r 0 - st 8 · D · r 0 - tr - st r 0 - tr - st + 10 · D ; r 0 - st D .
r 0 - st 8 1 D + ρ FIS - tr - st · 10 0.59 · λ · f ; r 0 - st D .

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