Abstract

The average power received at a space craft from a reciprocity-tracking transmitter is shown to be the free-space diffraction-limited result times a gain-reduction factor that is due to the point-ahead requirement. For a constant-power transmitter, the gain-reduction factor is approximately equal to the appropriate spherical-wave mutual-coherence function. For a constant-average-power transmitter, an exact expression is obtained for the gain-reduction factor.

© 1975 Optical Society of America

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References

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  1. J. H. Shapiro, IEEE Trans. COM19, 410 (1971).
    [CrossRef]
  2. D. L. Fried and H. T. Yura, J. Opt. Soc. Am. 62, 600 (1972).
    [CrossRef]
  3. J. H. Shapiro, J. Opt. Soc. Am. 61, 492 (1971).
    [CrossRef]
  4. R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [CrossRef] [PubMed]
  5. P. J. Titterton, J. Opt. Soc. Am. 63, 439 (1973).
    [CrossRef]
  6. H. T. Yura, Appl. Opt. 11, 1399 (1972).
    [CrossRef] [PubMed]
  7. J. H. Shapiro, Appl. Opt. 13, 2614 (1974).
    [CrossRef] [PubMed]
  8. In terms of the appropriate vector space of functions, what we are assuming is that |a¯−b¯|is small enough to ensure that |a¯·b¯|2/|a¯|2|b¯|2≈1−|a¯−b¯|2/2|b¯|2.
  9. R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1967).
  10. D. Korff, J. Opt. Soc. Am. 63, 971 (1973).
    [CrossRef]
  11. V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, J. Opt. Soc. Am. 64, 516 (1974).
    [CrossRef]
  12. This implies that the 〈Pr〉/Pt curve shown in Fig. 2 lies below the true result when d1is less than the phase-coherence length.

1974 (2)

1973 (2)

1972 (2)

1971 (3)

1967 (1)

R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1967).

Banakh, V. A.

Fried, D. L.

Khmelevtsov, S. S.

Korff, D.

Krekov, G. M.

Lutomirski, R. F.

Mironov, V. L.

Schmeltzer, R. A.

R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1967).

Shapiro, J. H.

Titterton, P. J.

Tsvik, R. Sh.

Yura, H. T.

Appl. Opt. (3)

IEEE Trans. (1)

J. H. Shapiro, IEEE Trans. COM19, 410 (1971).
[CrossRef]

J. Opt. Soc. Am. (5)

Q. Appl. Math. (1)

R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1967).

Other (2)

This implies that the 〈Pr〉/Pt curve shown in Fig. 2 lies below the true result when d1is less than the phase-coherence length.

In terms of the appropriate vector space of functions, what we are assuming is that |a¯−b¯|is small enough to ensure that |a¯·b¯|2/|a¯|2|b¯|2≈1−|a¯−b¯|2/2|b¯|2.

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Figures (2)

FIG. 1
FIG. 1

Planar propagation geometry. The vectors r ¯ and r ¯ are two-dimensional vectors in stationary coordinate systems whose planes are determined, respectively, by the R1 and R2 apertures.

FIG. 2
FIG. 2

Average fractional earth-to-space power transfer. Curve (a), diffraction-limited free-space system. Curve (b), constant-power reciprocity-tracking system. Curve (c), constant-average-power reciprocity-tracking system. Curve (d), nonadaptive transmitter.

Tables (1)

Tables Icon

TABLE I Point-ahead angle for 1/e gain reduction.

Equations (19)

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E p ( r ¯ ) = d r ¯ circ ( 2 | r ¯ r ¯ 0 | / d 2 ) × exp ( i π | r ¯ | 2 / λ L ) · h 21 ( r ¯ , r ¯ ) .
E t ( r ¯ ) = d r ¯ circ ( 2 | r ¯ r ¯ 0 2 υ ¯ r L / c | / d 2 ) × exp ( i π | r ¯ | 2 / λ L ) · h 21 * ( r ¯ , r ¯ ) .
P r = [ | R 1 d r ¯ E p ( r ¯ ) E t ( r ¯ ) exp ( i 4 π υ ¯ r · r ¯ / c λ ) | 2 × ( R 1 d r ¯ | E p ( r ¯ ) | 2 R 1 d r ¯ | E t ( r ¯ ) | 2 ) 1 ] × P t R 1 d r ¯ d r ¯ circ ( 2 | r ¯ r ¯ 0 2 υ ¯ r L / c | / d 2 ) × | h 21 ( r ¯ , r ¯ ) | 2 .
[ | R 1 d r ¯ E p ( r ¯ ) E t ( r ¯ ) exp ( i 4 π υ ¯ r · r ¯ / c λ ) | 2 × ( R 1 d r ¯ | E p ( r ¯ ) | 2 R 1 d r ¯ | E t ( r ¯ ) | 2 ) 1 ] 1 ( R 1 d r ¯ | E p * ( r ¯ ) exp ( i 4 π υ ¯ r · r ¯ / c λ ) E t ( r ¯ ) | 2 ) × ( 2 R 1 d r | E t ( r ¯ ) | 2 ) 1 .
P r [ P t ( π d 1 d 2 / 4 λ L ) 2 ] exp ( D ( 2 υ ¯ r L / c , 0 ¯ ) / 2 ) ,
D ( ρ ¯ , r ¯ ) = 2.91 k 2 L 5 / 3 0 L d s C n 2 ( s cos β ) × | ρ ¯ s + r ¯ ( L s ) | 5 / 3 ,
P r P t ( π d 1 d 2 / 4 λ L ) 2
θ p = ( 1.45 k 2 0 L C n 2 ( s cos β ) s 5 / 3 d s ) 3 / 5 .
θ p = ( 1.45 k 2 ( sec β ) 8 / 3 0 C n 2 ( h ) h 5 / 3 d h ) 3 / 5 .
P r = [ | R 1 d r ¯ E p ( r ¯ ) E t ( r ¯ ) exp ( i 4 π υ ¯ r · r ¯ / c λ ) | 2 × ( R 1 d r ¯ | E p ( r ¯ ) | 2 R 1 d r ¯ | E t ( r ¯ ) | 2 ) 1 ] · P t R 1 d r ¯ d r ¯ circ ( 2 | r ¯ r ¯ 0 2 υ ¯ r L / c | / d 2 ) × | h 21 ( r ¯ , r ¯ ) | 2 ,
P r = P t ( π d 2 2 / 4 ( λ L ) 2 ) d r ¯ circ ( | r ¯ | / d 1 ) · exp { D ( 0 ¯ , r ¯ ) D ( 2 υ ¯ r L / c , 0 ¯ ) + D ( 2 υ ¯ r L / c , r ¯ ) / 2 + D ( 2 υ ¯ r L / c , r ¯ ) / 2 + 2 [ C χ ( 2 υ ¯ r L / c , r ¯ ) + i C χ , ϕ ( 2 υ ¯ r L / c , r ¯ ) ] + 2 [ C χ ( 2 υ ¯ r L / c , r ¯ ) i C χ , ϕ ( 2 υ ¯ r L / c , r ¯ ) ] } · ( 2 / π ) [ cos 1 ( | r ¯ | / d 1 ) ( 1 | r ¯ | 2 / d 1 2 ) 1 / 2 | r ¯ | / d 1 ] .
C χ ( ρ ¯ , r ¯ ) + i C χ , ϕ ( ρ ¯ , r ¯ ) = 4 π 2 k 2 i 0 L d s 0 d u u C n 2 ( s cos β ) 0.033 u 11 / 3 · J 0 [ | ρ ¯ ( s / L ) + r ¯ ( L s ) / L | u ] sin [ u 2 s ( L s ) / 2 k L ] · exp [ i u 2 s ( L s ) / 2 k L ] ,
| R 1 d r ¯ E p ( r ¯ ) E t ( r ¯ ) exp ( i 4 π υ ¯ r · r ¯ / c λ ) | 2 | R 1 d r ¯ E p ( r ¯ ) E t ( r ¯ ) exp ( i 4 π υ ¯ r · r ¯ / c λ ) | 2 ,
P r P t ( π d 1 d 2 / 4 λ L ) 2 exp [ D ( 2 υ ¯ r L / c , 0 ¯ ) ] .
lim d 1 P r P t ( π d 1 d 2 / 4 λ L ) 2 = exp [ D ( 2 υ ¯ r L / c , 0 ¯ ) ] .
lim d 1 0 P r P t ( π d 1 d 2 / 4 λ L ) 2 = exp [ 4 C χ ( 2 υ ¯ r L / c , 0 ¯ ) ] ,
lim d 1 0 P r P t ( π d 1 d 2 / 4 λ L ) 2 = 1 ,
C n 2 ( h ) = π 1 / 2 10 13 δ ( h 1.5 × 10 4 ) m 2 / 3 .
P r = P t [ π d 2 2 / 4 ( λ L ) 2 ] · d r ¯ circ ( | r ¯ | / d 1 ) exp [ D ( 0 ¯ , r ¯ ) / 2 ] · ( 2 / π ) [ cos 1 ( | r ¯ | / d 1 ) ( 1 | r ¯ | 2 / d 1 2 ) 1 / 2 | r ¯ | / d 1 ] .