Abstract

The narrow beam widths associated with intersatellite optical communication links make such links susceptible to signal fading because of pointing jitter. Such fading can be aggravated by stationary offsets in pointing. We calculated the fade rates for the case of two spaceborne telescopes having Gaussian beam profiles, a pointing offset, and pointing jitter that can be described by Gaussian statistics. An integral solution is derived for the general case of a nonsymmetrical system, with and without pointing bias, and closed-form solutions are presented for the case of a symmetrical system (identical platforms and optics). These results show that, for a system with 3-dB margin, the rms pointing jitter must be held to less than 7% of the full beam width to keep the fade rate below once per year.

© 1997 Optical Society of America

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References

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  1. D. L. Begley, ed., Free-Space Laser Communication. SPIE Milestone Series Vol. 30 (SPIE, Bellingham, Wash., 1991).
  2. E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N. J., 1968).
  3. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944).
    [CrossRef]
  4. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–156 (1945).
    [CrossRef]
  5. S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed., (Dover, New York, 1954), p. 189.
  6. D. Middleton, An Introduction to Statistical Communication Theory, International Series in Pure and Applied Physics, L. I. Schiff, ed. (McGraw-Hill, New York, 1960).
  7. N. M. Blachman, Noise and its Effect on Communication (McGraw-Hill, New York, 1966).
  8. N. Coburn, Vector and Tensor Analysis (Macmillan, New York, 1955), p. 198.
  9. K. S. Miller, Multidimensional Gaussian Distributions (Wiley, New York, 1964).
  10. S. G. Lambert, W. L. Casey, Laser Communications in Space (Artech, Boston, 1995).

1945 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–156 (1945).
[CrossRef]

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944).
[CrossRef]

Balmain, K. G.

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N. J., 1968).

Blachman, N. M.

N. M. Blachman, Noise and its Effect on Communication (McGraw-Hill, New York, 1966).

Casey, W. L.

S. G. Lambert, W. L. Casey, Laser Communications in Space (Artech, Boston, 1995).

Coburn, N.

N. Coburn, Vector and Tensor Analysis (Macmillan, New York, 1955), p. 198.

Jordan, E. C.

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N. J., 1968).

Lambert, S. G.

S. G. Lambert, W. L. Casey, Laser Communications in Space (Artech, Boston, 1995).

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory, International Series in Pure and Applied Physics, L. I. Schiff, ed. (McGraw-Hill, New York, 1960).

Miller, K. S.

K. S. Miller, Multidimensional Gaussian Distributions (Wiley, New York, 1964).

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–156 (1945).
[CrossRef]

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944).
[CrossRef]

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed., (Dover, New York, 1954), p. 189.

Bell Syst. Tech. J. (2)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944).
[CrossRef]

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 24, 46–156 (1945).
[CrossRef]

Other (8)

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed., (Dover, New York, 1954), p. 189.

D. Middleton, An Introduction to Statistical Communication Theory, International Series in Pure and Applied Physics, L. I. Schiff, ed. (McGraw-Hill, New York, 1960).

N. M. Blachman, Noise and its Effect on Communication (McGraw-Hill, New York, 1966).

N. Coburn, Vector and Tensor Analysis (Macmillan, New York, 1955), p. 198.

K. S. Miller, Multidimensional Gaussian Distributions (Wiley, New York, 1964).

S. G. Lambert, W. L. Casey, Laser Communications in Space (Artech, Boston, 1995).

D. L. Begley, ed., Free-Space Laser Communication. SPIE Milestone Series Vol. 30 (SPIE, Bellingham, Wash., 1991).

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N. J., 1968).

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Figures (2)

Fig. 1
Fig. 1

Mean duration of fade versus the fade rate for an f0 of 100 Hz and an s0 of 0.5. Note the very weak dependence of τ on the fade rate.

Fig. 2
Fig. 2

Double-ended fade rates for the case of (a) no pointing bias and (b) a pointing bias that gives a 10% reduction in jitter-free transmittance. The horizontal dashed line corresponds to one fade per year. m = 40.6, f0 = 100 Hz.

Equations (35)

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Δθ=cλD,
rate=0 s˙Ws0, s˙ds˙.
s=sx1, x2, , xn.
s=exp-x1-x022χ12-x222χ22-x322χ32  xn22χn2,
Wx, x˙=12πσρexp-x2σ2+x˙2ρ2,
ρ=2πf0σ
Fk1, k2=expik1s+ik2s˙,
s˙=sx1x˙1+sx2x˙2+ sxnx˙n=-sx1-x0x˙1χ12+x2x˙2χ22+ xnx˙nχn2.
Fk1, k2=12πnΠi σiρi-dx1  -dx˙n×exp-xi22σi2+x˙i22ρi2×expik1s+ik2s˙.
Fk1, k2=12πn/2Πiσi-dx1  -dxn×exp-xi22σi2+ik1s-k22s2ρ12x1-x022χ14+ρ22x222χ24+.
rate=0 s˙Ws0, s˙ds˙=0 s˙ds˙ 12π2-dk1-dk2×exp-ik1s0-ik2s˙Fk1, k2.
rate=-12π2-dk1 exp-ik1s0-dk2Fk1, k2k22= -dx1  -dxn exp-xi22σi2×sρ12x1-x022χ14+ ρn2xn22χn41/2×-dk1 expik1s-s0.
χ1=χ2=χa,χ3=χ4=χb,σ1=σ2=σa,σ3=σ4=σb,
r2=x1-x022χ12+x222χ22+x322χ32+x422χ42
x1-x02 χa=r cos ϕ sin θ cos ξ, x22 χa=r sin ϕ sin θ cos ξ, x32 χb=r cos θ cos ξ, x42 χb=sin ξ.
-expiks-s0=2πδs-s0, ds=-2rsdr,
rate= r03ρaχaexp-x02+χa2r022σa20πsin θ dθ×-π/2π/2cos2 ξ dξ×I02 x0χar0 sin θ cos ξσa2×exp-ηr021-sin θ cos ξ2×1+ε1-sin θ cos ξ21/2,
η=χb2σb2-χa2σa2, ε=ρbχaρaχb2-1.
φ=ξ-π2, cos ξsin φ, u=sin θ sin φ, ν=sin φ, cos ξdξdθ=νdudνν2-u21-ν2.
rate=fa2π2ma ln1/s03/2×exp-ma2θ022χa2s0mambma×01duuI0θ0χa2ma ln1/s01/2u×exp-η ln1/s01-u2×1+ε1-u21/2,
mi=χi2σi2, x0θ0.
rate=f2π 2m ln1/s03/2 exp-m2θ02χ2×s0mI1mθ0χ2 ln1/s01/2m θ0χ2 ln1/s01/2.
rate=f2π 2m ln1/s01/2exp-m2θ02χ2×s0mI0mθ0χ2 ln1/s01/2.
rate=fa2π 2ma ln1/s03/2×exp-ma2θoa22χa2-mb2θob22χb2s0mambma×01duuI0θoaχa2ma ln1/s01/2u×I0θobχa2ma ln1/s01/21-u2×exp-η ln1/s01-u2×1+ε1-u2.
θ02=θoa2+θob2.
rate=2f0π m ln1/s01/2s0m.
rate=2f0π m ln1/s03/2s0m.
rate=1τ0s0 Wsds=integrated probabilitymean duration of fade.
Ws=msm-1.
Ws= ds1 ds2Ws1Ws2δs-s1s2=s1ds1s1Ws1Ws/s1,
Ws=m2 ln1/ssm-1,
Ws=mnsn-1m-n+sm-1n-m.
Ws=lmnsl-1m-ln-l+sm-1l-mn-m+sn-1l-nm-n,
1τ=2f0πm ln1/s0,
1τ=2f0πm ln1/s03/21+m ln1/s02f0πm ln1/s0,

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