Abstract

In this paper we study quantitatively the effects of atmospheric turbulence on the measurement sensitivity of a quadrant detecting array, along with the measurement variance induced by atmospheric turbulence. Our results show that the stronger the turbulence strength, the lower the measurement sensitivity, and the larger the measurement variance. This agrees with the experimental phenomena we have observed up to now. In addition, the measurement sensitivity in the X direction could be affected by the deviation of light spots in the Y direction due to the existence of gaps between the quadrant elements. Such effects still exist for the measurement variance even when there are no gaps at all. This effect is analyzed quantitatively.

© 1989 Optical Society of America

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References

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  1. J. F. Rands, D. R. Zankowsky, “Laser Applications in Construction, Agriculture and Surveying,” Proc. IEEE 70, 635 (1982).
    [CrossRef]
  2. P. W. Harrison, “Measurement of Dam Deflection by Laser,” International Water Power & Dam Construction 30, 52 (Apr.1978).
  3. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980).
  4. L. G. Kazovsky, “Tracking Accuracy of Laser Systems: Theory,” Opt. Eng. 22, 339 (1983).
    [CrossRef]
  5. T. J. Gilmartin, J. Z. Holtz, “Focused Beam and Atmospheric Coherence Measurements at 10.6 μm and 0.63 μm,” Appl. Opt. 13, 1906 (1974).
    [CrossRef] [PubMed]
  6. J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).
    [CrossRef]
  7. V. I. Tatarskii, Wave Propagation in the Atmosphere (McGraw-Hill, New York, 1961).
  8. Special volume of Laser Propagation in the Atmosphere (Anhui Institute of Optics & Fine Mechanics,AcademiaSinica, 1976), p. 50.

1983 (1)

L. G. Kazovsky, “Tracking Accuracy of Laser Systems: Theory,” Opt. Eng. 22, 339 (1983).
[CrossRef]

1982 (1)

J. F. Rands, D. R. Zankowsky, “Laser Applications in Construction, Agriculture and Surveying,” Proc. IEEE 70, 635 (1982).
[CrossRef]

1978 (1)

P. W. Harrison, “Measurement of Dam Deflection by Laser,” International Water Power & Dam Construction 30, 52 (Apr.1978).

1974 (1)

Gilmartin, T. J.

Harrison, P. W.

P. W. Harrison, “Measurement of Dam Deflection by Laser,” International Water Power & Dam Construction 30, 52 (Apr.1978).

Holtz, J. Z.

Kazovsky, L. G.

L. G. Kazovsky, “Tracking Accuracy of Laser Systems: Theory,” Opt. Eng. 22, 339 (1983).
[CrossRef]

Rands, J. F.

J. F. Rands, D. R. Zankowsky, “Laser Applications in Construction, Agriculture and Surveying,” Proc. IEEE 70, 635 (1982).
[CrossRef]

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980).

Strohbehn, J. W.

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in the Atmosphere (McGraw-Hill, New York, 1961).

Zankowsky, D. R.

J. F. Rands, D. R. Zankowsky, “Laser Applications in Construction, Agriculture and Surveying,” Proc. IEEE 70, 635 (1982).
[CrossRef]

Appl. Opt. (1)

International Water Power & Dam Construction (1)

P. W. Harrison, “Measurement of Dam Deflection by Laser,” International Water Power & Dam Construction 30, 52 (Apr.1978).

Opt. Eng. (1)

L. G. Kazovsky, “Tracking Accuracy of Laser Systems: Theory,” Opt. Eng. 22, 339 (1983).
[CrossRef]

Proc. IEEE (1)

J. F. Rands, D. R. Zankowsky, “Laser Applications in Construction, Agriculture and Surveying,” Proc. IEEE 70, 635 (1982).
[CrossRef]

Other (4)

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1980).

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).
[CrossRef]

V. I. Tatarskii, Wave Propagation in the Atmosphere (McGraw-Hill, New York, 1961).

Special volume of Laser Propagation in the Atmosphere (Anhui Institute of Optics & Fine Mechanics,AcademiaSinica, 1976), p. 50.

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Figures (5)

Fig. 1
Fig. 1

Quadrant detector system.

Fig. 2
Fig. 2

Normalized Sx for various Y (values of light spot deviation from the QD center in the y direction) vs X/W0.

Fig. 3
Fig. 3

Normalized Sx for various C n 2 values vs X/W0: ——, theoretical curves; •, ○,experimental points.

Fig. 4
Fig. 4

Various Y values; variance var(Sx) vs X/W0.

Fig. 5
Fig. 5

Various C n 2 values; variance var(Sx) vs X/W0.

Equations (21)

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I ( ρ , L ) = ( W 0 2 / W 2 ) I 0 exp ( 2 ρ 2 / W 2 ) ,
I ( L , ρ ) = ( W 2 / W b 2 ) I 0 exp ( 2 ρ 2 / W 0 ) ,
W b = W 2 / ( 1 f ) ; W 2 = W 0 2 [ ( 1 α 2 L ) 2 + ( α 1 L ) 2 ] , f = 1 . 36 C n 2 k 7 / 6 L 11 / 6 { α 1 L / [ ( α 1 L ) 2 + ( 1 α 2 L ) 2 ] } 5 / 6 , }
W 2 = W 0 2 [ 1 + ( α 1 L ) 2 ] , f = 1 . 36 C n 2 k 7 / 6 L 11 / 6 { α 1 L / [ 1 + ( α 1 L ) 2 ] } 5 / 6 . }
W 2 = W 0 2 ( α 1 L ) 2 , f = 1 . 36 C n 2 k 7 / 6 L α 1 . }
var ( Δ α ) = lim ρ 0 D s ( ρ ) / ( K 2 ρ 2 ) ,
D s ( ρ ) = 1 . 72 C n 2 k 2 L l 0 ( 1 / 3 ) ρ 2 ,
var ( Δ α ) = 1 . 72 C n 2 L l 0 ( 1 / 3 ) .
var ( Δ X ) = ( d Δ X / d Δ α ) 2 var ( Δ α ) = 1 . 72 C n 2 L 3 l 0 ( 1 / 3 ) .
I ( x , y ) = ( W 0 2 / W b 2 ) I 0 exp [ 2 ( x 2 + y 2 ) / W b 2 ] .
i k = ( η q / h ν ) S k I ( x , y ) d x d y , ( k = 1 , 2 , 3 , 4 ) ,
i 1 = [ ( η q W 0 2 ) / ( h ν W b 2 ) ] I 0 x + Δ l x / 2 d x y + Δ l y / 2 d y × exp [ 2 ( x 2 + y 2 ) / W b 2 ] , i 2 = [ ( η q W 0 2 ) / ( h ν W b 2 ) ] I 0 x Δ l x / 2 d x y + Δ l y / 2 d y × exp [ 2 ( x 2 + y 2 ) / W b 2 ] , i 3 = [ ( η q W 0 2 ) / ( h ν W b 2 ) ] I 0 x Δ l x / 2 d x y Δ l y / 2 d y × exp [ 2 ( x 2 + y 2 ) / W b 2 ] , i 4 = [ ( η q W 0 2 ) / ( h ν W b 2 ) ] I 0 x + Δ l x / 2 d x y Δ l y / 2 d y × exp [ 2 ( x 2 + y 2 ) / W b 2 ] . }
S x = i 1 + i 4 i 2 i 3 = [ ( η q π W 0 2 ) / ( 8 h ν ) ] × I 0 { 2 + P [ ( 2 Y Δ l y ) / W b ] P [ ( 2 Y + Δ l y ) / W b ] } { P [ ( 2 X Δ l x ) / W b ] + P [ ( 2 X Δ l x ) / W b ] } .
S y = i 1 + i 2 i 3 i 4 = [ ( η q π W 0 2 ) / ( 8 h ν ) ] × I 0 { 2 + P [ ( 2 X Δ l x ) / W b ] P [ ( 2 X + Δ l x ) / W b ] } { P [ ( 2 Y Δ l y ) / W b ] + P [ ( 2 Y Δ l y ) / W b ] } ,
P ( x ) = ( 1 / 2 π ) x x exp ( t / 2 ) d t .
i 1 = [ ( η q π W 0 2 ) / ( 8 h ν ) ] I 0 [ 1 P ( 2 X / W b ) ] [ 1 P ( 2 Y / W b ) ] , i 2 = [ ( η q π W 0 2 ) / ( 8 h ν ) ] I 0 [ 1 + P ( 2 X / W b ) ] [ 1 P ( 2 Y / W b ) ] , i 3 = [ ( η q π W 0 2 ) / ( 8 h ν ) ] I 0 [ 1 + P ( 2 X / W b ) ] [ 1 + P ( 2 Y / W b ) ] , i 4 = [ ( η q π W 0 2 ) / ( 8 h ν ) ] I 0 [ 1 P ( 2 X / W b ) ] [ 1 + P ( 2 Y / W b ) ] , }
var ( i k ) = ( d i k / d X ) 2 var ( Δ X ) .
var ( i 1 ) = var ( i 2 ) = [ ( η q π W 0 2 I 0 ) / ( 8 h ν ) ] 2 [ 1 P ( 2 Y / W b ) ] 2 × exp ( 4 X 2 / W b 2 ) var ( Δ X ) ,
var ( i 3 ) = var ( i 4 ) = [ ( η q π W 0 2 I 0 ) / ( 8 h ν ) ] 2 [ 1 + P ( 2 Y / W b ) ] 2 × exp ( 4 X 2 / W b 2 ) var ( Δ X ) .
var ( S x ) = k var ( i k ) = 2 [ ( η q π W 0 2 I 0 ) / ( 8 h ν ) ] 2 exp ( 4 X 2 / W b 2 ) × [ 1 P ( 2 Y / W b ) ] 2 + [ 1 + P ( 2 Y / W b ) ] 2 var ( Δ X ) .
var ( S x ) = 1 . 72 [ ( η q π W 0 2 I 0 ) / ( 4 h ν ) ] 2 exp ( 4 X 2 / W b 2 ) C n 2 L 3 l 0 1 / 3 .

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