Abstract

We describe a concept for a phase conjugate adaptive optics system which avoids the use of a shared aperture and the attendant difficulties when such a system is used with a high-power laser. The concept consists of measuring the phase of a reference wave around the edge of the aperture and interpolating with a suitable formula to find the phase inside the aperture. We assess the potential of this technique to compensate for atmospheric turbulence when the transmitting aperture is circular or annular. Good correction is indicated, particularly for the case of an annular aperture where perimeter phase measurements can give an estimate of the amount of turbulence-induced misfocus.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. E. Pearson, “Atmospheric turbulence compensation using coherent optical adaptive techniques,” Appl. Opt. 15, 622–631 (1976).
    [Crossref] [PubMed]
  2. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [Crossref]
  3. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 44.
  4. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [Crossref]
  5. C. B. Hogge and R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propagat. AP-24, 144–154 (1976).
    [Crossref]
  6. C. W. Clenshaw, “Chebyshev series for mathematical functions,” from National Physical Laboratory Mathematical Tables, Vol. 5 (Her Majesty’s Stationery Office, London, 1962).

1976 (2)

J. E. Pearson, “Atmospheric turbulence compensation using coherent optical adaptive techniques,” Appl. Opt. 15, 622–631 (1976).
[Crossref] [PubMed]

C. B. Hogge and R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propagat. AP-24, 144–154 (1976).
[Crossref]

1970 (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

1965 (1)

Butts, R. R.

C. B. Hogge and R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propagat. AP-24, 144–154 (1976).
[Crossref]

Clenshaw, C. W.

C. W. Clenshaw, “Chebyshev series for mathematical functions,” from National Physical Laboratory Mathematical Tables, Vol. 5 (Her Majesty’s Stationery Office, London, 1962).

Fried, D. L.

Hogge, C. B.

C. B. Hogge and R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propagat. AP-24, 144–154 (1976).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 44.

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

Pearson, J. E.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Antennas Propagat. (1)

C. B. Hogge and R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propagat. AP-24, 144–154 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[Crossref]

Other (2)

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 44.

C. W. Clenshaw, “Chebyshev series for mathematical functions,” from National Physical Laboratory Mathematical Tables, Vol. 5 (Her Majesty’s Stationery Office, London, 1962).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

FIG. 1
FIG. 1

C(s) vs s = a/b, where E(ϕ) = C(s) (2b/r0)5/3 and ϕ is the phase of the uncompensated wave from an annular aperture of inner radius a and outer radius b.

FIG. 2
FIG. 2

D(s) vs s = a/b, where E(ψ) = D(s) (2b/r0)5/3 and ψ is the phase of the compensated wave from an annular aperture of inner radius a and outer radius b.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

T 2 u - 2 i K 0 n 0 u z + K 0 2 [ n 2 ( r ) - n 0 2 ] u = 0.
u 0 = ( C / z ) exp [ - i K 0 n 0 r 2 / 2 z ] = A 0 exp ( i ϕ 0 ) .
2 ( T A 1 ) · ( T ϕ 1 ) + A 1 T 2 ϕ 1 - 2 K 0 n 0 A 1 z = 0.
T 2 ϕ = 0.
2 G ( r , r ) = δ ( r - r ) ,             r , r R
G ( r , r ) = 0             r C .
P f ( r ) = C f ( r ) G ( r , r ) n d s ,
Φ ( r ) = P Φ ( r )             r R
ψ ( r ) = ϕ ( r ) - P ϕ ( r ) .
E ( ψ ) = ( ψ - ψ ¯ ) 2 ¯ = ψ 2 ¯ - ( ψ ¯ ) 2 ,
f ¯ = 1 A A f ( r ) d a .
D ϕ ( r 1 , r 2 ) = [ ϕ ( r 1 ) - ϕ ( r 2 ) ] 2 .
E ( ϕ ) = 1 2 A 2 A d 2 r A d 2 r D ϕ ( r , r ) .
Ψ ( r ) = ϕ ( r ) - C ϕ ( r ) H ( r , r ) d s .
C ϕ ( r 1 , r 2 ) = ϕ ( r 1 ) ϕ ( r 2 ) ,
C ϕ ( r 1 , r 2 ) = 1 2 [ σ ϕ 2 ( r 1 ) + σ ϕ 2 ( r 2 ) - D ϕ ( r 1 , r 2 ) ] ,
Ψ 2 ( r ) = σ ϕ 2 ( r ) - C [ σ ϕ 2 ( r ) + σ ϕ 2 ( r ) - D ϕ ( r , r ) ] H ( r , r ) d s + 1 2 C d s C d s [ σ ϕ 2 ( r ) + σ ϕ 2 ( r ) - D ϕ ( r , r ) ] × H ( r , r ) H ( r , r ) .
1 = C d s H ( r , r ) ,
Ψ 2 ( r ) = C D ϕ ( r , r ) H ( r , r ) d s - 1 2 C d s C d s D ϕ ( r , r ) H ( r , r ) H ( r , r ) .
Ψ ¯ = ϕ ¯ - C d s ϕ ( r ) H ¯ ( r ) ,
( Ψ ¯ ) 2 = ( ϕ ¯ ) 2 - 2 C d s ϕ ¯ ϕ ( r ) H ¯ ( r ) + C d s σ ϕ 2 ( r ) H ¯ ( r ) - 1 2 C d s C d s D ϕ ( r , r ) H ¯ ( r ) H ¯ ( r ) .
( Ψ ¯ ) 2 = - E ( ϕ ) - 1 2 C d s C d s D ϕ ( r , r ) H ¯ ( r ) H ¯ ( r ) + 1 A A d 2 r C d s D ϕ ( r , r ) H ¯ ( r ) .
D ϕ ( r 1 , r 2 ) = 1.089 K 0 2 C n 2 L r 5 / 3 - 2 [ B χ ( 0 ) - B χ ( r ) ] ,
D ϕ ( r ) = 6.88 ( r / r 0 ) 5 / 3 ,
P ϕ ( r , θ ) = b 2 - r 2 2 π 0 2 π ϕ ( b , θ ) d θ b 2 + r 2 - 2 b r cos ( θ - θ ) .
E ( Ψ ) = 0.088 ( 2 b / r 0 ) 5 / 3 .
E ( ϕ ) = 1.032 ( 2 b / r 0 ) 5 / 3 .
F = n = 1 10 a n F n .
G = F - P F = n = 1 10 b n F n ,
E ( ϕ ) = 3 ( 6.88 ) 11 π ( q 2 - 1 ) 2 [ 8 q 4 ( 4 3 ! ) ( 1 2 ! ) ( 17 6 ! ) ( 1 + s ) 17 / 3 - 2 - 2 / 3 π q 2 ( 1 + s ) 5 / 3 F ( - 5 6 , 3 2 ; 3 ; 4 s ( 1 + s ) 2 ) ] ( 2 b r 0 ) 5 / 3 ,
E ( ϕ ) = C ( s ) ( 2 b / r 0 ) 5 / 3 .
E ( Ψ ) = D ( s ) ( 2 b / r 0 ) 5 / 3 .