Abstract

The modulation transfer function and the irradiance profile of a phase-compensated laser beam propagating through atmospheric turbulence are calculated for various values of the normalized transmitter diameter, D/r0. Both focused Gaussian beam and collimated beam with modal phase corrections of arbitrary orders are considered using Zernike polynomials as basic correcting modes. Examples are given for low-order corrections such as tilt, focus, and astigmatism as well as for higher order corrections. The beam quality criteria, defined in terms of Strehl ratio, spot radius, and energy in a bucket, are used to measure the beam performance with different compensating modes. The effects on the beam propagation of central obscuration of the transmitter are also examined, and numerical results are given.

© 1978 Optical Society of America

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References

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  1. M. J. Lavan, W. K. Cadwallender, T. F. Young, Opt. Eng. 15, 56 (1976).
  2. J. E. Pearson, S. Hansen, J. Opt. Soc. Am. 67, 325 (1977).
    [CrossRef]
  3. A. Buffington, F. S. Crawford, R. A. Muller, A. J. Schwemin, R. G. Smits, J. Opt. Soc. Am. 67, 298 (1977).
    [CrossRef]
  4. J. R. Dunphy, J. R. Kerr, J. Opt. Soc. Am. 64, 1015 (1974).
    [CrossRef]
  5. C. Neufeld, “Modal wavefront control system (MOWACS),” AD-A028 298, Perkin-Elmer Corporation (July1976).
  6. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [CrossRef]
  7. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  8. M. T. Tavis, H. T. Yura, Appl. Opt. 15, 2922 (1976).
    [CrossRef] [PubMed]
  9. R. F. Lutomirski, W. L. Woodie, R. G. Buser, Appl. Opt. 16, 665 (1977).
    [CrossRef] [PubMed]
  10. J. Y. Wang, J. Opt. Soc. Am. 67, 383 (1977).
    [CrossRef]
  11. J. Y. Wang, J. K. Markey, J. Opt. Soc. Am. 68, 78 (1978).
    [CrossRef]
  12. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [CrossRef]
  13. D. L. Fried, “How many pictures do you take to get a good one?” submitted to J. Opt. Soc. Am.
  14. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [CrossRef] [PubMed]

1978 (1)

1977 (4)

1976 (3)

1974 (1)

1971 (1)

1966 (1)

1965 (1)

Buffington, A.

Buser, R. G.

Cadwallender, W. K.

M. J. Lavan, W. K. Cadwallender, T. F. Young, Opt. Eng. 15, 56 (1976).

Crawford, F. S.

Dunphy, J. R.

Fried, D. L.

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
[CrossRef]

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
[CrossRef]

D. L. Fried, “How many pictures do you take to get a good one?” submitted to J. Opt. Soc. Am.

Hansen, S.

Kerr, J. R.

Lavan, M. J.

M. J. Lavan, W. K. Cadwallender, T. F. Young, Opt. Eng. 15, 56 (1976).

Lutomirski, R. F.

Markey, J. K.

Muller, R. A.

Neufeld, C.

C. Neufeld, “Modal wavefront control system (MOWACS),” AD-A028 298, Perkin-Elmer Corporation (July1976).

Noll, R. J.

Pearson, J. E.

Schwemin, A. J.

Smits, R. G.

Tavis, M. T.

Wang, J. Y.

Woodie, W. L.

Young, T. F.

M. J. Lavan, W. K. Cadwallender, T. F. Young, Opt. Eng. 15, 56 (1976).

Yura, H. T.

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Figures (12)

Fig. 1
Fig. 1

MTF vs spatial frequency for a focused Gaussian beam with (a) D/r0 = 2.0 and (b) D/r0 = 7.0. Dashed curves—phase correlation term is ignored.

Fig. 2
Fig. 2

MTF vs spatial frequency for a collimated beam with D/r0 = 2.0 and (a) without phase correction, (b) with tilt correction, and (c) with perfect phase corrections.

Fig. 3
Fig. 3

Tilt-corrected focused Gaussian beam irradiance profiles for several values of the normalized transmitter diameter D/r0.

Fig. 4
Fig. 4

Focused Gaussian beam irradiance profiles at the focal plane with phase compensations for D/r0 = 2.0.

Fig. 5
Fig. 5

Irradiance distribution vs far-field radius for a collimated beam at D/r0 = 2.0, NF = 30.

Fig. 6
Fig. 6

Strehl ratio vs normalized transmitter diameter for a focused Gaussian beam: curves A—no correction; curves B—tilt correction; curves C—tilt plus focus corrections; curves D—tilt plus focus plus astigmatism corrections; curves E—tilt plus focus plus astigmatism plus coma corrections; curves F—21 terms Zernike corrections. Solid curves—current results; dashed curves—results of Fried6 and Noll.7

Fig. 7
Fig. 7

Spot radii (αr) as functions of the normalized transmitter diameter: curve A—no correction; curve B—tilt correction; curve C—tilt plus focus corrections; curve D—tilt plus focus plus astigmatism corrections (focused Gaussian beam).

Fig. 8
Fig. 8

Spot radii ( α r ) as functions of the normalized transmitter diameter: curve A—no correction; curve B—tilt correction; curve C—tilt plus focus corrections; curve D—tilt plus focus plus astigmatism corrections (focused Gaussian beam).

Fig. 9
Fig. 9

Focal plane fractional energy vs far-field radius for a focused Gaussian beam with (a) D/r0 = 2.0 and (b) D/r0 = 5.0.

Fig. 10
Fig. 10

Geometric representation of aperture overlap area for the integration of 〈τ(ρ)〉 with central obscuration.

Fig. 11
Fig. 11

Tilt-corrected MTF vs spatial frequency for a focused Gaussian beam with central obscuration, D/r0 = 2.0.

Fig. 12
Fig. 12

Effect of obscuration on peak intensity of a focused Gaussian beam (a) without phase correction and (b) with tilt correction.

Equations (42)

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I ( p ) = ( k 2 π z ) 2 d 2 ρ exp ( i k z ρ · p ) × d 2 r G ( r + 1 2 ρ ) G * ( r 1 2 ρ ) × U ( r + 1 2 ρ ) U * ( r 1 2 ρ ) exp ( i k z ρ · r ) .
τ ( ρ ) = d 2 r G ( r + 1 2 ρ ) G * ( r 1 2 ρ ) U ( r + 1 2 ρ ) × U * ( r 1 2 ρ ) exp ( i k z ρ · r ) ,
G ( r + 1 2 ρ ) G * ( r 1 2 ρ ) = exp { l ( r + 1 2 ρ ) + i [ φ ( r + 1 2 ρ ) Φ ( r + 1 2 ρ ) ] + l ( r 1 2 ρ ) i [ φ ( r 1 2 ρ ) Φ ( r 1 2 ρ ) ] } ,
D ( ρ ) [ φ ( r + 1 2 ρ ) φ ( r 1 2 ρ ) ] 2 + [ l ( r + 1 2 ρ ) l ( r 1 2 ρ ) ] 2
Φ ( r ) = i = 1 N a i F i ( r ) ,
G ( r + 1 2 ρ ) G * ( r 1 2 ρ ) = exp { 1 2 D ( ρ ) 1 2 [ Φ ( r + 1 2 ρ ) Φ ( r 1 2 ρ ) ] 2 + [ φ ( r + 1 2 ρ ) φ ( r 1 2 ρ ) ] [ Φ ( r + 1 2 ρ ) Φ ( r 1 2 ρ ) ] } = exp { 1 2 D ( ρ ) + 1 2 i = 2 N a i 2 [ F i ( r + 1 2 ρ ) F i ( r 1 2 ρ ) ] 2 + i = 2 N j = N + 1 a i a j [ F i ( r + 1 2 ρ ) F i ( r 1 2 ρ ) ] [ F j ( r + 1 2 ρ ) F j ( r 1 2 ρ ) ] } .
F ( r ) even i = [ 8 ( n + 1 ) ] 1 / 2 R n m ( r ) cos m θ F ( r ) odd i = [ 8 ( n + 1 ) ] 1 / 2 R n m ( r ) sin m θ } if m 0 , F ( r ) i = [ 4 ( n + 1 ) ] 1 / 2 R n o ( r ) if m = 0 ,
R n m ( r ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! ( 2 r D ) n 2 s .
a i [ φ ( r ) i = 1 N a i F i ( r ) ] 2 W ( r ) d 2 r = 0.
W ( r ) F i ( r ) F j ( r ) d 2 r = K δ i j ,
a i = W ( r ) F i ( r ) φ ( r ) d 2 r W ( r ) F i 2 ( r ) d 2 r .
W ( r ) = { 1 for | r | D / 2 , 0 for | r | > D / 2 .
τ ( ρ ) = 4 D 2 exp [ 1 2 D ( ρ ) ] × φ π / 2 + φ d θ 0 L ( θ ) r d r U ( r + 1 2 ρ ) U * ( r 1 2 ρ ) × exp ( i k z ρ · r ) exp { 1 2 i = 2 N a i 2 [ F i ( r + 1 2 ρ ) F i ( r 1 2 ρ ) ] 2 + i = 2 N j = N + 1 M a i a j [ F i ( r + 1 2 ρ ) F i ( r 1 2 ρ ) ] × [ F j ( r + 1 2 ρ ) F j ( r 1 2 ρ ) ] } ,
L ( θ ) = 1 2 ( ρ D ) cos ( θ ϕ ) + 1 2 ( ρ D ) [ ( ρ D ) 2 sin 2 ( θ ϕ ) ] 1 / 2 ,
I ( p ) = ( k 2 π z ) 2 τ ( ρ ) exp ( i k z ρ · p ) d 2 ρ .
I ( α ) = N F 2 2 π 0 1 J 0 ( 2 α x ) τ ( x ) x d x ,
I ( α ) N = 0 1 J 0 ( 2 α x ) τ ( x ) x d x 0 1 τ 0 ( x ) x d x ,
τ 0 ( ρ ) = 4 D 2 ϕ π / 2 + ϕ d θ × 0 L ( θ ) r d r U ( r + 1 2 ρ ) U * ( r 1 2 ρ ) exp ( i k z ρ · r ) .
SR = 0 1 τ ( x ) x d x 0 1 τ 0 ( x ) x d x .
I ( α r ) / I ( 0 ) = e 1 .
( α r ) 2 = 0 I ( α ) α 3 d α 0 I ( α ) α d α .
E = 0 α e I ( α ) α d α 0 I ( α ) α d α .
r 0 = ( 6.88 / a ) 3 / 5 ,
a = 2.91 k 2 C n 2 ( s ) Q ( s ) d s .
U ( r ) = { exp [ 1 2 r 2 ( 4 D 2 + i k f ) ] for | r | D / 2 , 0 for | r | < D / 2 ,
τ ( ρ ) = 4 D 2 exp [ 1 2 D ( ρ ) ] ϕ π / 2 + ϕ d θ 0 L ( θ ) r d r G ( r , θ ) × exp [ 4 ( r D ) 2 ( ρ D ) 2 ] .
τ ( ρ ) = 4 D 2 exp [ 1 2 D ( ρ ) ] ϕ π / 2 + ϕ d θ 0 L ( θ ) r d r G ( r , θ ) × cos [ N F ( r D ) ( ρ D ) cos ( θ ϕ ) ] .
SR = e σ 2 .
U ( r ) = { exp [ 1 2 r 2 ( 4 D 2 + i k f ) ] for d / 2 | r | D / 2 0 for | r | > D / 2 ,
τ ( ρ ) = G ( r + 1 2 ρ ) G * ( r 1 2 ρ ) U ( r + 1 2 ρ ) U * ( r 1 2 ρ ) × [ cos ( k z ρ · r ) + i sin ( k z ρ · r ) ] d 2 r .
G ( r + 1 2 ρ ) G * ( r 1 2 ρ ) = exp { 1 2 D ( ρ ) + 1 2 i = 2 N a i 2 [ F i ( r + 1 2 ρ ) F i ( r 1 2 ρ ) ] 2 + i = 2 N j = N + 1 a i a j [ F i ( r + 1 2 ρ ) F i ( r 1 2 ρ ) ] × [ F j ( r + 1 2 ρ ) F j ( r 1 2 ρ ) ] } .
U ( r ) = R ( r ) + i I ( r ) ,
U ( r + 1 2 ρ ) U * ( r 1 2 ρ ) = [ R ( r + 1 2 ρ ) R ( r 1 2 ρ ) + I ( r + 1 2 ρ ) I ( r 1 2 ρ ) ] + i [ R ( r 1 2 ρ ) I ( r + 1 2 ρ ) R ( r + 1 2 ρ ) I ( r 1 2 ρ ) ] .
R ( r + 1 2 ρ ) R ( r 1 2 ρ ) + I ( r + 1 2 ρ ) I ( r 1 2 ρ ) = R ( r + 1 2 ρ ) R ( r 1 2 ρ ) + I ( r + 1 2 ρ ) I ( r 1 2 ρ ) ,
R ( r 1 2 ρ ) I ( r + 1 2 ρ ) R ( r + 1 2 ρ ) I ( r 1 2 ρ ) = R ( r 1 2 ρ ) I ( r + 1 2 ρ ) + R ( r + 1 2 ρ ) I ( r 1 2 ρ ) .
U ( r ) = A ( r , θ ) exp [ i ψ ( r , θ ) ] ,
A ( r , θ ) cos [ ψ ( r , θ ) ] = ( SIGN ) × A ( r , θ ) cos [ ψ ( r , π + θ ) ] ,
A ( r , θ ) sin [ ψ ( r , θ ) ] = ( SIGN ) × A ( r , θ ) sin [ ψ ( r , π + θ ) ] ,
ψ ( r , θ ) = ψ ( r , π + θ ) ,
A ( r , θ ) = ± A ( r , π + θ ) ,
I ( p ) = ( k 2 π z ) 2 × G ( r + 1 2 ρ ) G * ( r 1 2 ρ ) U ( r + 1 2 ρ ) U * ( r 1 2 ρ ) × { cos [ k z ρ · ( r p ) ] + i sin [ k z ρ · ( r p ) ] } d 2 r d 2 ρ .
I m = cos [ k z ρ · ( r p ) ] [ R ( r 1 2 ρ ) I ( r + 1 2 ρ ) R ( r + 1 2 ρ ) I ( r 1 2 ρ ) + sin [ k z ρ · ( r p ) ] × [ R ( r + 1 2 ρ ) R ( r 1 2 ρ ) + I ( r + 1 2 ρ ) I ( r 1 2 ρ ) ] .

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