Abstract

According to the point-source point-receiver (PSPR) reciprocity, the received field remains equal when the positions of a point source and point receiver are interchanged. We extend the PSPR scenario to a finite receiver that spatially averages scintillation over its aperture. By use of weak-fluctuation theory, an analytical expression for the correlation coefficient between the received powers at both link ends is provided. The effects of turbulence profile, receiver aperture size, and central obscuration on the correlation are assessed. Because correlation is obtained to the detriment of antenna gain and aperture averaging, the net benefit of the channel reciprocity is highly scenario dependent.

© 2012 Optical Society of America

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References

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  1. J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am. 61, 492–495 (1971).
    [CrossRef]
  2. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef]
  3. D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. (to be published).
  4. R. R. Parenti, J. M. Roth, J. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments” in Applications of Lasers for Sensing and Free Space Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper LTuD3.
  5. V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
    [CrossRef]
  6. V. A. Banakh and V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, 1987).
  7. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media2nd ed. (SPIE, 2005).
  8. J. Goodman, Introduction to Fourier Optics3rd ed. (Roberts, 2005).
  9. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE, 55, 57–77 (1967).
    [CrossRef]
  10. S. Basu and D. Voelz, “Tracking in a ground-to-satellite optical link: effects due to lead-ahead and aperture mismatch, including temporal tracking response,” J. Opt. Soc. Am. A 25, 1594–1608 (2008).
    [CrossRef]
  11. J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford, 1998).
  12. V. A. Banakh and A. Z. Vagner, “Calculation of the variance of the strong intensity fluctuations for light beams propagating in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 24–28 (1992).
    [CrossRef]
  13. L. C. Andrews and R. L. Phillips, “Monostatic lidar in weak-to-strong turbulence,” Waves Random Media, 11, 233–245 (2001).
    [CrossRef]

2008 (1)

2001 (1)

L. C. Andrews and R. L. Phillips, “Monostatic lidar in weak-to-strong turbulence,” Waves Random Media, 11, 233–245 (2001).
[CrossRef]

1992 (1)

V. A. Banakh and A. Z. Vagner, “Calculation of the variance of the strong intensity fluctuations for light beams propagating in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 24–28 (1992).
[CrossRef]

1985 (1)

V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
[CrossRef]

1971 (2)

1967 (1)

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

Aksenov, V. P.

V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
[CrossRef]

Andrews, L.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media2nd ed. (SPIE, 2005).

Andrews, L. C.

L. C. Andrews and R. L. Phillips, “Monostatic lidar in weak-to-strong turbulence,” Waves Random Media, 11, 233–245 (2001).
[CrossRef]

Banakh, V. A.

V. A. Banakh and A. Z. Vagner, “Calculation of the variance of the strong intensity fluctuations for light beams propagating in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 24–28 (1992).
[CrossRef]

V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
[CrossRef]

V. A. Banakh and V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, 1987).

Basu, S.

Buldakov, V. M.

V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
[CrossRef]

Cowley, W.

D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. (to be published).

Fried, D. L.

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

Giggenbach, D.

D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. (to be published).

Goodman, J.

J. Goodman, Introduction to Fourier Optics3rd ed. (Roberts, 2005).

Grant, K.

D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. (to be published).

Hardy, J. W.

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

Lutomirski, R. F.

Mironov, V. L.

V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
[CrossRef]

V. A. Banakh and V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, 1987).

Parenti, R. R.

R. R. Parenti, J. M. Roth, J. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments” in Applications of Lasers for Sensing and Free Space Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper LTuD3.

Perlot, N.

D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. (to be published).

Phillips, R.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media2nd ed. (SPIE, 2005).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, “Monostatic lidar in weak-to-strong turbulence,” Waves Random Media, 11, 233–245 (2001).
[CrossRef]

Roth, J. M.

R. R. Parenti, J. M. Roth, J. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments” in Applications of Lasers for Sensing and Free Space Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper LTuD3.

Shapiro, J.

R. R. Parenti, J. M. Roth, J. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments” in Applications of Lasers for Sensing and Free Space Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper LTuD3.

Shapiro, J. H.

Tikhomirova, O. V.

V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
[CrossRef]

Vagner, A. Z.

V. A. Banakh and A. Z. Vagner, “Calculation of the variance of the strong intensity fluctuations for light beams propagating in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 24–28 (1992).
[CrossRef]

Voelz, D.

Walther, F. G.

R. R. Parenti, J. M. Roth, J. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments” in Applications of Lasers for Sensing and Free Space Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper LTuD3.

Yura, H. T.

Appl. Opt. (2)

D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. (to be published).

R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef]

Atmos. Oceanic Opt. (1)

V. A. Banakh and A. Z. Vagner, “Calculation of the variance of the strong intensity fluctuations for light beams propagating in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 24–28 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

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

Sov. J. Quantum Electron. (1)

V. P. Aksenov, V. A. Banakh, V. M. Buldakov, V. L. Mironov, and O. V. Tikhomirova, “Distribution of fluctuations of light intensity behind the objective of a telescope after reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 15, 1404–1406 (1985).
[CrossRef]

Waves Random Media (1)

L. C. Andrews and R. L. Phillips, “Monostatic lidar in weak-to-strong turbulence,” Waves Random Media, 11, 233–245 (2001).
[CrossRef]

Other (5)

V. A. Banakh and V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, 1987).

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media2nd ed. (SPIE, 2005).

J. Goodman, Introduction to Fourier Optics3rd ed. (Roberts, 2005).

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

R. R. Parenti, J. M. Roth, J. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments” in Applications of Lasers for Sensing and Free Space Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper LTuD3.

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Figures (4)

Fig. 1.
Fig. 1.

Propagation of a spherical wave in the forward and return directions. The turbulent screen Ψ is shown at a distance z1 from the point source, which is transmitting in the forward direction.

Fig. 2.
Fig. 2.

Correlation coefficient μFR between forward and return Rx powers for the case of a single turbulent screen along the path. Left: Rx apertures of the forward and return channels with the same geometry (m=1). Right: Rx aperture of the return channel 3 times larger than the forward-channel Rx aperture (m=3). The key and the y-axis are common to both graphs. Two values for the obscuration ratio q (ratio of inner to outer diameter) are considered.

Fig. 3.
Fig. 3.

Correlation coefficient μFR as a function of the normalized Rx diameter for the case of a constant-Cn2 path.

Fig. 4.
Fig. 4.

Correlation coefficient μFR as a function of the normalized Rx diameter on the ground for the case of one ground terminal and one on a spacecraft.

Tables (2)

Tables Icon

Table 1. Main Characteristics of Functions Involved in Eq. (22)

Tables Icon

Table 2. Correlation Coefficients μFR under Limit Conditions for a Single Turbulent Screen along the Path

Equations (25)

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UF(ρ)=exp(jkz2)jλz2exp(jkρ22z2)UF,0(s)exp(jks22z2)exp(jkz2s·ρ)ds;
UF,0(s)=AFz1exp[jk(s22z1+z1)]Ψ(s),
UR(ρ)=UF(z2z1ρ)exp[jkρ22z12(z1z2)],
IR(ρ)=IF(z2z1ρ).
σFR2=PFPRPFPR=IF(ρ1)WF(ρ1)dρ1IR(ρ2)WR(ρ2)dρ2IF(ρ1)WF(ρ1)dρ1IR(ρ2)WR(ρ2)dρ2.
WF(n˜)={1,forn˜inside aperture;0,otherwise.
σFR2=IF(n˜1)IR(n˜2)1WF(n˜1)WR(n˜2)dn˜1dn˜2.
IF(ρ)=0LiF(ρ;z1)dz1,
BF(r)=0LbF(r;z1)dz1.
bF(r;z)=0.033×8π2k2Cn2(z1)0κ8/3J0(κrz1L)(1cosκ2z1z2kL)dκ
IF(n˜1)IR(n˜2)1=0LiF(n˜1;z1)dz10LiR(n˜2;z1)dz1(0LiFdz1)2=0L0LiF(n˜1;z1*)iR(n˜2;z1)iF2dz1*dz1,
IF(n˜1)IR(n˜2)1=0L0LiF(n˜1;z1*)iF(z2z1n˜2;z1)iF2dz1*dz1.
0LiF(ρ1;z1*)iF(z2z1ρ2;z1)iF2dz1*0LiF(ρ1;z1)iF(z2z1ρ2;z1)iF2dz1*bF(|z2z1ρ2ρ1|;z1).
IF(n˜1)IR(n˜2)10LbF(|z2z1n˜2n˜1|;z1)dz1.
σFR2=R2R20LbF(|z2z1n˜2n˜1|;z1)WF(n˜1)WR(n˜2)dz1dn˜1dn˜2.
σFR2=0L(z1z2)2R2R2bF(|n˜2n˜1|;z1)WF(n˜1)WR(z1z2n˜2)dn˜1dn˜2dz1.
σFR2=2π0L0(z1/z2)2bF(r;z1)HFR(r;z1)rdrdz1,
HFR(r;z1)=WF(r)WR(z1z2(r+r))dr.
μFRσFR2/σFσR,
σF2=2π0BF(r)HF(r)rdr,
HF(r)=WF(r)WF(r+r)dr,
μFR=0L(z1/z2)20bF(r;z1)HFR(r;z1)rdrdz10BF(r)HF(r)rdr0BR(r)HR(r)rdr.
HFR(r;z1)aR(z2/z1)2WF(r).
2π0BR(r)HR(r)rdraR2BR(0).
μFR=0BF(r)WF(r)rdr(2π)1BR(0)0BF(r)HF(r)rdr.

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