Abstract

Tracking apertures which are on axis, off axis, and annular with respect to the pointer optics are considered in terms of their effectiveness in canceling atmospheric turbulence-induced wave-front tilt errors. The off-axis tracker is found to be the least effective, whereras the annular configuration is least sensitive to the wind profile and slewing conditions. The key to minimizing the centroid wander in the focal plane is the proper setting of the low-pass cutoff frequency of the tracking servo. That setting is based on wind velocity, slew rate, and aperture diameters. A too-high setting of the cutoff frequency can actually degrade tracker performance when the tracking aperture is small.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front-compensative systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
    [Crossref]
  2. C. B. Hogge and R. R. Butts, “Frequency Spectra for the Geometric Representation of Wavefront Distortions Due to Atmospheric Turbulence,” IEEE Trans. Ant. Prop. AP-24, 144–154 (1976).
    [Crossref]
  3. R. W. Lee and J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375–406 (1969).
    [Crossref]
  4. S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. 61, 1285–1292 (1971).
    [Crossref]

1976 (2)

D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front-compensative systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
[Crossref]

C. B. Hogge and R. R. Butts, “Frequency Spectra for the Geometric Representation of Wavefront Distortions Due to Atmospheric Turbulence,” IEEE Trans. Ant. Prop. AP-24, 144–154 (1976).
[Crossref]

1971 (1)

1969 (1)

R. W. Lee and J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

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. Ant. Prop. AP-24, 144–154 (1976).
[Crossref]

Clifford, S. F.

Fried, D. L.

Greenwood, D. P.

Harp, J. C.

R. W. Lee and J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

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. Ant. Prop. AP-24, 144–154 (1976).
[Crossref]

Lee, R. W.

R. W. Lee and J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

IEEE Trans. Ant. Prop. (1)

C. B. Hogge and R. R. Butts, “Frequency Spectra for the Geometric Representation of Wavefront Distortions Due to Atmospheric Turbulence,” IEEE Trans. Ant. Prop. AP-24, 144–154 (1976).
[Crossref]

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

R. W. Lee and J. C. Harp, “Weak Scattering in Random Media, with Applications to Remote Probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

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 (18)

FIG. 1
FIG. 1

Geometry: filled configuration.

FIG. 2
FIG. 2

Geometry: annular configuration.

FIG. 3
FIG. 3

Geometry: interior configuration.

FIG. 4
FIG. 4

Error variance vs cutoff frequency: filled on-axis configuration, wind case 1.

FIG. 5
FIG. 5

Error variance vs cutoff frequency: filled contiguous off-axis configuration, tilt x, wind case 1.

FIG. 6
FIG. 6

Error variance vs cutoff frequency: filled contiguous off-axis configuration, tilt y, wind case 1.

FIG. 7
FIG. 7

Error variance vs cutoff frequency: filled on-axis configuration, wind case 2.

FIG. 8
FIG. 8

Error variance vs cutoff frequency: filled contiguous off-axis configuration, tilt x, wind case 2.

FIG. 9
FIG. 9

Error variance vs cutoff frequency: filled contiguous off-axis configuration, tilt y, wind case 2.

FIG. 10
FIG. 10

Error variance vs cutoff frequency: filled on-axis configuration, wind case 3.

FIG. 11
FIG. 11

Error variance vs cutoff frequency: filled contiguous off-axis configuration, tilt x, wind case 3.

FIG. 12
FIG. 12

Error variance vs cutoff frequency: filled contiguous off-axis configuration, tilt y, wind case 3.

FIG. 13
FIG. 13

Error variance vs cutoff frequency: annular configuration, wind case 1.

FIG. 14
FIG. 14

Error variance vs cutoff frequency: annular configuration, wind case 2.

FIG. 15
FIG. 15

Error variance vs cutoff frequency: annular configuration, wind case 3.

FIG. 16
FIG. 16

Error variance vs cutoff frequency: interior configuration, wind case 1.

FIG. 17
FIG. 17

Error variance vs cutoff frequency: interior configuration, wind case 2.

FIG. 18
FIG. 18

Error variance vs cutoff frequency: interior configuration, wind case 3.

Equations (36)

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

( t ) = α p ( t ) - α t ( t ) * h ( t ) .
σ 2 ( f co ) = 0 W p ( f ) d f + 0 H ( f , f co ) W t ( f ) d f - 2 0 H 1 / 2 ( f , f co ) W t p ( f ) d f ,
α D ( t ) = 2 λ A D - 2 d x ˜ W D ( x ˜ ) ϕ ( x ˜ , t ) x ,
W D ( x ˜ ) = { 1 , x ˜ 1 2 D 0 , x ˜ > 1 2 D
α d ( t ; r ˜ 1 ) = 2 λ A d - 2 d x ˜ W d ( x ˜ , r ˜ 1 ) ϕ ( x ˜ , t ) ( x - r 1 x ) ,
W d ( x ˜ , r ˜ 1 ) = { 1 , x ˜ - r ˜ 1 1 2 d 0 , x ˜ - r ˜ 1 > 1 2 d
C d ( τ ) = α d ( t ) α d ( t + τ ) , C D ( τ ) = α D ( t ) α D ( t + τ ) , C d D ( τ ) = α d ( t ) α D ( t + τ ) ,
W i ( f ) = 4 0 d τ cos ( 2 π f τ ) C i ( τ ) .
σ i 2 = 0 d f W i ( f ) .
W i ( f ) = - 1 2 d r ˜ T i ( r ˜ ) W δ ϕ ( r ˜ ; f ) ,
T d ( r , θ ) = 2 ( λ A d π ) 2 { cos - 1 ( r d ) - ( r d ) [ 1 - ( r d ) 2 ] 1 / 2 - 2 3 r d [ 1 - ( r d ) 2 ] 3 / 2 ( 1 + 4 cos 2 θ ) ,             0 r d 0 ,             d r
T D ( r , θ ) = T d ( r , θ ) with d D ,
T d D ( r , θ ) = ( λ A D π ) 2 { π ,             0 r 2 D - d 2 ( D d ) 4 cos - 1 γ 1 + cos - 1 γ 2 - [ γ 2 + γ 1 ( D d ) 3 ] ( 1 - γ 2 2 ) 1 / 2 - 4 γ 2 3 d ( 1 - γ 2 2 ) 3 / 2 [ 1 + 4 r 2 2 ( r cos θ + r 1 cos δ ) 2 ] ,             D - d 2 r 2 D + d 2 0 ,             D + d 2 r 2
r 2 = [ r 2 + r 1 2 + 2 r r 1 cos ( δ - θ ) ] 1 / 2 , γ 1 = [ r 2 2 + ( 1 2 D ) 2 - ( 1 2 d ) 2 ] / r 2 D , γ 2 = [ r 2 2 + ( 1 2 d ) 2 - ( 1 2 D ) 2 ] / r 2 d .
W p = W D , W t = W d , W t p = W d D .
α A ( t ) = 2 λ A D 2 - A d 2 ( d x ˜ W D ( x ˜ ) ϕ ( x ˜ , t ) x - d x ˜ W d ( x ˜ , 0 ) ϕ ( x ˜ , t ) x ) ,
α A ( t ) = 1 A D 2 - A d 2 [ A D 2 α D ( t ) - A d 2 α d ( t ; 0 ) ] .
W p = W d , W t = b 2 2 W D + b 1 2 W d - 2 b 1 b 2 W d D , W t p = b 2 W d D - b 1 W d ,
b 1 = [ ( D / d ) 4 - 1 ] - 1 , b 2 = [ 1 - ( d / D ) 4 ] - 1 .
σ D - d 2 = σ d 2 + σ D 2 - 2 σ d D 2 .
σ 2 ( 0 ) = σ D 2 , σ 2 ( ) = σ D - d 2 .
σ 2 ( 0 ) = σ d 2 , σ 2 ( ) = b 2 2 σ D - d 2 .
σ 2 ( 0 ) = b 2 2 σ D 2 + b 1 2 σ d 2 - 2 b 1 b 2 σ d D 2 , σ 2 ( ) = b 2 2 σ D - d 2 .
W δ ϕ ( r ; f ) = 0.1305 k 2 L f - 8 / 3 × 0 1 d s v 5 / 3 ( s ) sin 2 [ π r f s / v ( s ) ] C n 2 ( s ) ,
v ( s ) = v a .
v ( s ) = ω L ( 1 - s ) ,
v ( s ) = ω L s .
W δ ϕ ( r ; f ) = 0.330 k 2 L C n 2 D 5 / 3 ( f / f 0 ) - 8 / 3 f 0 - 1 K ( f r / f 0 D ) ,
f 0 = v a / π D K ( u ) = 4 3 ( 1 - sin ( 2 u ) 2 u ) .
f 0 = ω L / π D K ( u ) = 8 3 0 1 d s ( 1 - s ) 5 / 3 sin 2 ( s u 1 - s ) .
f 0 = ω L / π D K ( u ) = sin 2 u
p 2 = σ 2 / ( 13.0 D - 1 / 3 L C n 2 ) ,             and x co = f co / f 0 .
H ( f , f co ) = [ 1 + ( f / f co ) 2 ] - 1 .
R = min [ p 2 ( x co ) ] / p 2 ( 0 ) ,
p 2 ( 0 ) = 0.0885.
p 2 ( 0 ) = 0.0885 ( d / D ) - 1 / 3 .