Abstract

We examine the consequences for an adaptive-optics system of the fact that the turbulence-induced wave-front distortion for two propagation paths with only slightly different propagation directions can be significantly different. We consider the implications of this fact for a compensated imaging system and for an adaptive-optics laser transmitter. Theory and numerical results are presented. The basic results are presented in terms of the average optical transfer function of a compensated imaging system and in terms of the average antenna gain of an adaptive-optics laser transmitter, each expressed as a function of the angular separation ϑ between the propagation path along which the reference signal arrives and the propagation path along which the adaptive-optics system is to provide performance. It is shown that for high spatial frequencies (for the compensated imaging system) and for large-aperture diameters (for the adaptive-laser optics transmitter), i.e., large compared with r0/λ and with r0, respectively, the magnitude of the anisoplanatism effect can be characterized by an isoplanatic patch angular size, which we denote by ϑ0. If the angular separation between the two propagation paths is ϑ, it is shown that the optical transfer function and the antenna gain are each reduced by a factor of exp[−(ϑ/ϑ0)5/3]. This simply expressed performance-reduction factor represents an asymptotic limit for high spatial frequencies and for large transmitter diameters. For lower spatial frequencies and smaller transmitter diameters the reduction factor is not so severe. Numerical results are presented to illustrate this.

© 1982 Optical Society of America

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Figures (7)

Fig. 1
Fig. 1

S(Q) for constant CN2. Results are shown for c = 0.00, 0.25, 0.50, 0.75, 1.00. The lowest of the curves corresponds to c =0.00 (with the c = 0.25 curve so nearly coincident with it as to be indistinguishable), whereas the highest of the curves corresponds to c = 1.00.

Fig. 2
Fig. 2

Asymptotic approximations to S(Q) for constant CN2. The asymptotic approximations to S(Q) given by Eqs. (41) and (42) are shown here as the dashed curves for the two cases of c = 0.0, 1.00. The corresponding exact results, taken from Fig. 1, are shown as the solid curves. The upper curve corresponds to c = 1.00. As can be seen for c = 1.00, we may consider the asymptotic approximations to be quite useful for Q < 0.1 and Q > 1.0, whereas for c = 0.00 the corresponding limits are Q < 0.02 and Q > 5.0.

Fig. 3
Fig. 3

Effect of anisoplanatism on the normalized antenna gain of an adaptive-optics laser transmitter for constant CN2. The normalized antenna gain 〈G〉/GDL (where GDL is diffraction-limited antenna gain) is shown as a function of the normalized transmitter-aperture diameter D/r0 for various values of the normalized angular separation ϑ/ϑ0. Each curve corresponds to a value of ϑ/ϑ0 running over the set of values 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, and 6.0.

Fig. 4
Fig. 4

Vertical distribution of the refractive-index structure constant CN2. Above 1-km altitude the value of CN2 is independent of whether it is daytime or nighttime. Below 1 km distinct values of CN2 are shown for daytime and nighttime.

Fig. 5
Fig. 5

S(Q) for vertical propagation. Results are calculated with Eq. (52) using the values of CN2 shown in Fig. 4, with (a) based on the daytime values of CN2, and with (b) based on the nighttime values. Results are shown for c = 0.00, 0.25, 0.50, 0.75, 1.00. The lowest of the curves in each figure corresponds to c = 0.00, whereas the highest is for c = 1.00. The curve for c = 0.25 is so close to that for c = 0.00 as to be barely distinguishable near the peak.

Fig. 6
Fig. 6

Asymptotic approximation to S(Q) for vertical propagation. The asymptotic approximations to S(Q) given by Eqs. (41) and (42), with the values of α and β taken from Eqs. (60) and (61), are shown here as the dashed curves for the two cases of c = 0.00, 1.00. The corresponding exact results, taken from Figs. 5(a) and 5(b), are shown as the solid curves. The upper curve in each figure is for c = 1.00.

Fig. 7
Fig. 7

Effect of anisoplanatism on the normalized antenna gain of an adaptive-optics laser transmitter for vertical propagation. The normalized antenna gain 〈G〉/GDL is shown as a function of the normalized aperture diameter D/r0 for various values of the normalized angular separation ϑ/ϑ0. Each curve corresponds to a value of ϑ/ϑ0 running over the set of values 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, 6.0. The results shown here are for daytime turbulence conditions and a zero-degree zenith angle, but they are identical (to within linewidth-type accuracy) with results obtained for other zenith angles and for nighttime conditions. Moreover, the results shown here appear to be identical with those obtained for propagation over a path along which CN2 is constant, which results are shown in Fig. 3.

Tables (1)

Tables Icon

Table 1 Effect of Anisoplanatism on the Normalizesd Antenna Gain of an Adaptive-Optics Laser Transmittera

Equations (61)

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T DL ( f ) = K ( λ f / D ) ,
K ( x ) = { 2 / π [ cos - 1 ( x ) - x ( 1 - x 2 ) 1 / 2 ] if x 1 0 if x > 1 .
G DL = ¼ π ( D / λ ) 2 .
C l ( r - r ) = [ l ( r ) - l ¯ ] [ l ( r ) - l ¯ ] ,
l ¯ = l ( r ) .
T ( f ) = T DL ( f ) exp [ - C l ( 0 ) + C l ( λ f ) ] .
T ( f ) T DL ( f ) exp [ - C l ( 0 ) ] .
T ( f ) = T DL ( f ) .
G = G DL exp [ - C l ( 0 ) ] { d r K ( r / D ) exp [ C l ( r ) ] d r K ( r / D ) } ,
G G DL exp [ - C l ( 0 ) ] .
M = exp [ 2 μ ( μ + 1 ) C l ( 0 ) ] .
G = G DL exp [ - ( 1 + μ ) 2 C l ( 0 ) ] × { d r K ( r / D ) exp [ ( 1 - μ ) 2 C l ( r ) ] d r K ( r / D ) } .
G = G DL = d r K ( r / D ) exp [ 4 C l ( r ) ] d r K ( r / D ) .
G = G DL exp [ - 4 μ C l ( 0 ) ] .
ϑ = θ 1 - θ 2 .
C l ( r - r , ϑ ) = [ l ( r , θ 1 ) - l ¯ ] [ l ( r , θ 2 ) - l ¯ ] .
C l ( r , 0 ) = C l ( r ) .
C l ( r , ϑ ) = 8.16 4 π k 2 PATH d v C N 2 0 d σ σ - 8 / 3 × [ 1 - cos ( σ 2 v / k ) ] J 0 { σ r [ 1 - ( v / L ) ] - ϑ v } ,
S ( r , ϑ ) = 2.905 k 2 PATH d v C N 2 ( { r [ 1 - ( v / L ) ] } 5 / 3 + { ϑ v } 5 / 3 - ½ { r [ 1 - ( v / L ) ] } 2 + 2 r ϑ [ 1 - ( v / L ) ] v c + { ϑ v } 2 5 / 6 - ½ { r [ 1 - ( v / L ) ] } 2 - 2 r ϑ [ 1 - ( v / L ) ] v c + { ϑ v } 2 5 / 6 ) ,
c = r · ϑ / ( r ϑ ) .
T ( f ) = T DL ( f ) exp [ - S ( λ f , ϑ ) + C l ( 0 , 0 ) - C l ( λ f , 0 ) - 2 C l ( 0 , ϑ ) + C l ( λ f , ϑ ) + C l ( λ f , - ϑ ) ] .
T ( f ) T DL ( f ) exp [ - S ( λ f , ϑ ) + C l ( 0 , 0 ) - 2 C l ( 0 , ϑ ) ] .
T ( f ) = T DL ( f ) exp { - S ( λ f , ϑ ) + 4 [ C l ( 0 , 0 ) - C l ( 0 , ϑ ) ] } .
T ( f ) T DL ( f ) exp [ - S ( λ f , ϑ ) ] .
G = G DL { d r K ( r / D ) exp [ - S ( r , ϑ ) + C l ( 0 , 0 ) - C l ( r , 0 ) - 2 C l ( 0 , ϑ ) + C l ( r , ϑ ) + C l ( r , - ϑ ) ] d r K ( r / d ) } .
G G DL exp [ C l ( 0 , 0 ) - 2 C l ( 0 , ϑ ) ] × { d r K ( r / D ) exp [ - S ( r , ϑ ) ] d r K ( r / D ) } .
G = G DL × d r K ( r / D ) exp { - S ( r , ϑ ) + 2 [ C l ( r , ϑ ) + C l ( r , - ϑ ) ] } d r K ( r / D ) .
G G DL { d r K ( r / D ) exp [ - S ( r , ϑ ) ] d r K ( r / D ) } .
lim r / ϑ S ( r , ϑ ) = 2.905 k 2 PATH d v C N 2 ( ϑ v ) 5 / 3 .
lim r / ϑ 0 S ( r , ϑ ) = 2.905 k 2 PATH d v C N 2 { r [ 1 - ( v / L ) ] } 5 / 3 .
r 0 = { ( 2.905 / 6.88 ) k 2 PATH d v C N 2 [ 1 - ( v / R ) ] 5 / 3 } - 3 / 5
ϑ 0 = { 2.905 k 2 PATH d v C N 2 v 5 / 3 } - 3 / 5 ,
lim r / ϑ S ( r , ϑ ) = ( ϑ / ϑ 0 ) 5 / 3
lim r / ϑ 0 S ( r , ϑ ) = 6.88 ( r / r 0 ) 5 / 3 .
S ( r , ϑ ) = [ ( r / r 0 ) ( ϑ / ϑ 0 ) ] 5 / 6 S [ ( r / r 0 ) / ( ϑ / ϑ 0 ) ] ,
S ( Q ) = A PATH d v C N 2 ( Q 5 / 6 [ 1 - ( v / L ) ] 5 / 3 + Q - 5 / 6 v 5 / 3 L 0 - 5 / 3 - ½ { Q [ 1 - ( v / L ) ] 2 + 2 [ 1 - ( v / L ) ] v L 0 - 1 c + Q - 1 v 2 L 0 - 2 } 5 / 6 - ½ { Q [ 1 - ( v / L ) ] 2 - 2 [ 1 - ( v / L ) ] v L 0 - 1 c + Q - 1 v 2 L 0 - 2 } 5 / 6 ) ,
L 0 = r 0 / ϑ 0
A = 2.905 k 2 ( r 0 ϑ 0 L 0 ) 5 / 6
A = 6.88 { PATH d v C N 2 [ 1 - ( v / L ) ] 5 / 3 } - 1 ,
S ( Q ) = 6.88 { PATH d v C N 2 [ 1 - ( v / L ) ] 5 / 3 } - 1 × PATH d v C N 2 × ( Q 5 / 6 [ 1 - ( v / L ) ] 5 / 3 + Q - 5 / 6 v 5 / 3 L 0 - 5 / 3 - ½ [ Q [ 1 - ( v / L ) ] 2 + 2 [ 1 - ( v / L ) ] v L 0 - 1 c + Q - 1 v 2 L 0 - 2 } 5 / 6 - ½ { Q [ 1 - ( v / L ) ] 2 - 2 [ 1 - ( v / L ) ] v L 0 - 1 c + Q - 1 v 2 L 0 - 2 } 5 / 6 ) .
S ( Q ) Q - 5 / 6 [ 1 - α ( 1 - c 2 ) Q - 1 / 3 ] ,             Q 1 ,
S ( Q ) 6.88 Q 5 / 6 [ 1 - β ( 1 - c 2 ) Q 1 / 3 ] ,             Q 1 ,
α = L 0 - 1 / 3 { PATH d v C N 2 [ 1 - ( v / L ) ] 1 / 3 v 2 } × { PATH d v C N 2 v 5 / 3 } - 1
β = L 0 1 / 3 { PATH d v C N 2 [ 1 - ( v / L ) ] 2 v - 1 / 3 } × { PATH d v C N 2 [ 1 - ( v / L ) ] 5 / 3 } - 1 .
r 0 = 3.02 k - 6 / 5 L - 3 / 5 ( C N 2 ) - 3 / 5 ,
ϑ 0 = 0.950 k - 6 / 5 L - 8 / 5 ( C N 2 ) - 3 / 5 ,
L 0 = 3.18 L ,
α = 1.020 ,
β = 2.206.
r 0 = { ( 2.905 / 6.88 ) k 2 sec ( ψ ) 0 d h C N 2 } - 3 / 5 ,
ϑ 0 = { 2.05 k 2 [ sec ( ψ ) ] 8 / 3 0 d h C N 2 h 5 / 3 } - 3 / 5 ,
S ( Q ) = 6.88 { 0 d h C N 2 } - 1 × 0 d h C N 2 { Q 5 / 6 + Q - 5 / 6 ( h / H 0 ) 5 / 3 - ½ [ Q + 2 ( h / H 0 ) c + Q - 1 ( h / H 0 ) 2 ] 5 / 6 - ½ [ Q - 2 ( h / H 0 ) c + Q - 1 ( h / H 0 ) 2 ] 5 / 6 } ,
α = H 0 - 1 / 3 { 0 d h C N 2 h 2 } { 0 d h C N 2 h 5 / 3 } - 1 ,
β = H 0 1 / 3 { 0 d h C N 2 h - 1 / 3 } { 0 d h C N 2 } - 1 ,
H 0 = ( 6.88 ) 3 / 5 { 0 d h C N 2 h 5 / 3 } 3 / 5 { 0 d h C N 2 } - 3 / 5 .
r 0 = { 1.666 × 10 7 3.369 × 10 7 } k - 6 / 5 [ sec ( ψ ) ] - 3 / 5 ,
ϑ 0 = { 4.188 × 10 3 4.604 × 10 3 } k - 6 / 5 [ sec ( ψ ) ] - 3 / 5 ,
L 0 = H 0 sec ( ψ ) ,
H 0 = { 3979 7319 }
α = { 0.955 0.846 } ,
β = { 2.651 2.930 } .