Abstract

Recent results for the atmospheric mode decomposition are applied to an idealized imaging problem in which the receiver has a priori knowledge of the channel impulse response and mode decomposition. It is shown that a channel-matched filter receiver is essentially optimum and, on the average, achieves diffraction-limited performance. Furthermore, when the transmitting aperture lies within a single isoplanatic patch, this system may be realized without a priori channel knowledge by transmitted reference techniques.

© 1974 Optical Society of America

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References

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  1. J. H. Shapiro, Appl. Opt. 13, 2714 (1974).
  2. The far-field case is not of interest here, because in the far field the transmitter appears to the receiver as an unresolved point source.
  3. C. K. Rushforth, R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
    [CrossRef]
  4. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [CrossRef] [PubMed]
  5. J. D. Gaskill, J. Opt. Soc. Am. 58, 600 (1968).
    [CrossRef]
  6. H. T. Yura, Appl. Opt. 12, 1188 (1973).
    [CrossRef] [PubMed]
  7. Because∑m=1∞ηm=∫R2dr¯′∫R1dr¯|h21(r¯′,r¯)|2,we have∑m=1∞ηm<∞,with probability 1, when both apertures have finite diameters.
  8. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), pp. 34–36.
  9. Ref. 8, pp. 68–69.
  10. We should note, however, that there are special circumstances under which we may reliably estimate em for m > Df for atmospheric propagation, or m > Dfo for free-space propagation, e.g., if we have a priori information that em is such that No/ηmem2 ≪ 1, despite ηm being very small. For free-space imaging, this leads to studies of analytic continuation and/or extrapolation using the prolate spheroidal wave functions. In this paper we assume no a priori knowledge of Ei, and hence we discount the possibility of reliably estimating mode-amplitudes associated with very small eigenvalues.
  11. C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).
    [CrossRef]
  12. J. L. Horner, J. Opt. Soc. Am. 59, 553 (1969).
    [CrossRef]
  13. J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 239–244.
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 80.
  15. B. L. McGlammery, J. Opt. Soc. Am. 57, 293 (1967).
    [CrossRef]

1974 (1)

J. H. Shapiro, Appl. Opt. 13, 2714 (1974).

1973 (1)

1969 (2)

1968 (2)

1967 (2)

Gaskill, J. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 80.

Harris, R. W.

Helstrom, C. W.

Horner, J. L.

Jacobs, I. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 239–244.

McGlammery, B. L.

Rushforth, C. K.

Shapiro, J. H.

J. H. Shapiro, Appl. Opt. 13, 2714 (1974).

Toraldo di Francia, G.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), pp. 34–36.

Wozencraft, J. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 239–244.

Yura, H. T.

Appl. Opt. (2)

J. H. Shapiro, Appl. Opt. 13, 2714 (1974).

H. T. Yura, Appl. Opt. 12, 1188 (1973).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (6)

Other (7)

The far-field case is not of interest here, because in the far field the transmitter appears to the receiver as an unresolved point source.

Because∑m=1∞ηm=∫R2dr¯′∫R1dr¯|h21(r¯′,r¯)|2,we have∑m=1∞ηm<∞,with probability 1, when both apertures have finite diameters.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), pp. 34–36.

Ref. 8, pp. 68–69.

We should note, however, that there are special circumstances under which we may reliably estimate em for m > Df for atmospheric propagation, or m > Dfo for free-space propagation, e.g., if we have a priori information that em is such that No/ηmem2 ≪ 1, despite ηm being very small. For free-space imaging, this leads to studies of analytic continuation and/or extrapolation using the prolate spheroidal wave functions. In this paper we assume no a priori knowledge of Ei, and hence we discount the possibility of reliably estimating mode-amplitudes associated with very small eigenvalues.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 239–244.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 80.

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Figures (2)

Fig. 1
Fig. 1

H CF ( f ¯ ) vs f′ for free-space propagation (solid line) and worst-case atmospheric propagation (broken line). Dfo = 100 for both curves.

Fig. 2
Fig. 2

H CF ( f ¯ ) vs f′ free-space propagation (solid line) and worst-case atmospheric propagation (broken line). Dfo = 500 for both curves.

Equations (37)

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E ( r ¯ ) = E o ( r ¯ ) + E n ( r ¯ ) , for r ¯ R 2 ,
E o ( r ¯ ) = R 1 d r ¯ E i ( r ¯ ) h 21 ( r ¯ , r ¯ )
E n ( r ¯ 1 ) E n * ( r ¯ 2 ) n = N o δ ( r ¯ 1 r ¯ 2 )
m = 1 η m < ) ( See Ref . , 7 )
E i ( r ¯ ) = m = 1 e m Φ m ( r ¯ ) , r ¯ R 1 ,
E o ( r ¯ ) = m = 1 e m η m 1 / 2 ϕ m ( r ¯ ) , r ¯ R 2 ,
E n ( r ¯ ) = m = 1 n m ϕ m ( r ¯ ) , r ¯ R 2 ,
E ˆ i ( r ¯ ) = m = 1 e ˆ m Φ m ( r ¯ ) , r ¯ R 1 ,
e m = R 1 d r ¯ E i ( r ¯ ) Φ m * ( r ¯ ) ,
n m = R 2 d r ¯ E n ( r ¯ ) ϕ m * ( r ¯ ) ,
e ˆ m = R 1 d r ¯ E ˆ i ( r ¯ ) Φ m * ( r ¯ ) ,
e ˆ mML = e m + n m η m 1 / 2 = η m 1 / 2 R 2 d r ¯ E ( r ¯ ) ϕ m * ( r ¯ ) ;
Var ( e ˆ mML ) n = | e ˆ mML e m | 2 n = N o / η m .
η m 1 , for 1 m D f ,
η m 0 , for m > D f ,
D f = m = 1 η m .
m = 1 η m < ,
E ˆ iML ( r ¯ ) = m = 1 N e ˆ mML Φ m ( r ¯ ) , r ¯ R 1
E ˆ iML ( r ¯ ) = R 2 d r ¯ E ( r ¯ ) g N ( r ¯ , r ¯ ) , r ¯ R 1 ,
g N ( r ¯ , r ¯ ) = m = 1 N η m 1 / 2 Φ m ( r ¯ ) ϕ m * ( r ¯ )
E ˆ iCF ( r ¯ ) = R 2 d r ¯ E ( r ¯ ) h 21 * ( r ¯ , r ¯ ) , r ¯ R 1
= m = 1 ( e m η m + n m η m 1 / 2 ) Φ m ( r ¯ ) , r ¯ R 1 .
e ˆ mCF = e m η m + n m η m 1 / 2 , 1 m <
e ˆ mCF n = e m η m , 1 m <
Var e ˆ mCF n = N o η m , 1 m < .
R 1 d r ¯ Var [ E ˆ iCF ( r ¯ ) ] n = m = 1 N o η m = N o D f < .
η m 1 , for 1 m D f
η m 0 , for m > D f ,
t ( r ¯ ) = exp [ i 2 π ( n Δ o | r ¯ | 2 / z ) / λ ]
exp ( i π | r ¯ 2 | 2 / λ z ) E ˆ i ( r ¯ 2 ) n = { exp [ i 2 π ( 2 z + n Δ o + | r ¯ 2 | 2 / z ) / λ ] / ( λ z ) 2 } · R 2 d r ¯ g * ( r ¯ , r ¯ 2 ) { R 1 d r ¯ 1 [ exp ( i π | r ¯ 1 | 2 / λ z ) E i ( r ¯ 1 ) ] · g ( r ¯ , r ¯ 1 ) } ,
D f t = ( π d 1 d 2 / 4 λ z ) 2 = D fo ,
exp ( i π | r ¯ 2 | 2 / λ z ) E ˆ iCF ( r ¯ 2 ) = R 1 d r ¯ 1 [ exp ( i π | r ¯ 1 | 2 / λ z ) E i ( r ¯ 1 ) ] exp [ 1 2 D ( | r ¯ 1 r ¯ 2 | ) ] · d 2 J 1 ( π | r ¯ 1 r ¯ 2 | d 2 / λ z ) / 2 λ z | r ¯ 1 r ¯ 2 | ,
h CF ( r ¯ ) = exp [ 1 2 D ( | r ¯ | ) ] d 2 J 1 ( π | r ¯ | d 2 / λ z ) / 2 λ z | r ¯ | .
H CF ( f ¯ ) = 0 d x exp [ 1 2 D ( π d 1 D f o x / 2 ) ] 2 π J 1 ( 2 π x ) J o ( 2 π f x ) ,
R 1 d r ¯ E i ( r ¯ ) h 21 ( r ¯ , r ¯ ) h 21 * ( r ¯ , r ¯ o ) t = exp [ i π ( 2 r ¯ · r ¯ o | r ¯ o | 2 ) / λ z ] R 1 d r ¯ [ exp ( i π | r ¯ | 2 / λ z ) · E i ( r ¯ ) ] exp ( i 2 π r ¯ · r ¯ / λ z ) exp [ 1 2 D ( | r ¯ r ¯ o | ) ] / ( λ z ) 2 .
m=1ηm=R2dr¯R1dr¯|h21(r¯,r¯)|2,
m=1ηm<,

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