## 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 *r*_{0}/λ and with *r*_{0}, respectively, the magnitude of the anisoplanatism effect can be characterized by an isoplanatic patch angular size, which we denote by *ϑ*_{0}. 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*/*ϑ*_{0})^{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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### Equations (61)

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(1)
$${T}_{\text{DL}(\mathbf{f})}=K(\mathrm{\lambda}f/D),$$
(2)
$$K(x)=\{\begin{array}{ll}2/\pi [{\text{cos}}^{-1}(x)-x{(1-{x}^{2})}^{1/2}]\hfill & \text{if}\hspace{0.17em}x\le 1\hfill \\ 0\hfill & \text{if}\hspace{0.17em}x>1\hfill \end{array}.$$
(3)
$${G}_{\text{DL}}=\xbc\pi {(D/\mathrm{\lambda})}^{2}.$$
(4)
$${C}_{l}(\mid \mathbf{r}-{\mathbf{r}}^{\prime}\mid )=\u3008[l(\mathbf{r})-\overline{l}][l({\mathbf{r}}^{\prime})-\overline{l}]\u3009,$$
(5)
$$\overline{l}=\u3008l(\mathbf{r})\u3009.$$
(6)
$$\u3008T(\mathbf{f})\u3009={T}_{\text{DL}}(\mathbf{f})\text{exp}[-{C}_{l}(0)+{C}_{l}(\mathrm{\lambda}f)].$$
(7)
$$\u3008T(\mathbf{f})\u3009\approx {T}_{\text{DL}}(\mathbf{f})\text{exp}[-{C}_{l}(0)].$$
(8)
$$\u3008T(\mathbf{f})\u3009={T}_{\text{DL}}(\mathbf{f}).$$
(9)
$$\u3008G\u3009={G}_{\text{DL}}\hspace{0.17em}\text{exp}[-{C}_{l}(0)]\hspace{0.17em}\left\{\frac{\int \text{d}\mathbf{r}K(r/D)\text{exp}[{C}_{l}(\mathbf{r})]}{\int \text{d}\mathbf{r}K(r/D)}\right\},$$
(10)
$$\u3008G\u3009\approx {G}_{\text{DL}}\hspace{0.17em}\text{exp}[-{C}_{l}(0)].$$
(11)
$$M=\text{exp}[2\mu (\mu +1){C}_{l}(0)].$$
(12)
$$\u3008G\u3009={G}_{\text{DL}}\hspace{0.17em}\text{exp}[-{(1+\mu )}^{2}{C}_{l}(0)]\times \left\{\frac{\int \text{d}\mathbf{r}K(\mathbf{r}/D)\text{exp}[{(1-\mu )}^{2}{C}_{l}(\mathbf{r})]}{\int \text{d}\mathbf{r}K(\mathbf{r}/D)}\right\}.$$
(13)
$$\u3008G\u3009={G}_{\text{DL}}=\frac{\int \text{d}\mathbf{r}K(r/D)\text{exp}[4{C}_{l}(\mathbf{r})]}{\int \text{d}\mathbf{r}K(\mathbf{r}/D)}.$$
(14)
$$\u3008G\u3009={G}_{\text{DL}}\hspace{0.17em}\text{exp}[-4\mu {C}_{l}(0)].$$
(15)
$$\mathit{\vartheta}={\mathit{\theta}}_{1}-{\mathit{\theta}}_{2}.$$
(16)
$${C}_{l}(\mathbf{r}-{\mathbf{r}}^{\prime},\mathit{\vartheta})=\u3008[l(\mathbf{r},{\mathit{\theta}}_{1})-\overline{l}][l(\mathbf{r},{\mathit{\theta}}_{2})-\overline{l}]\u3009.$$
(17)
$${C}_{l}(\mathbf{r},0)={C}_{l}(r).$$
(18)
$${C}_{l}(\mathbf{r},\mathit{\vartheta})=\frac{8.16}{4\pi}{k}^{2}{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{\int}_{0}^{\infty}\text{d}\sigma {\sigma}^{-8/3}\times [1-\text{cos}({\sigma}^{2}v/k)]\hspace{0.17em}{J}_{0}\{\sigma \mid \mathbf{r}[1-(v/L)]-\mathit{\vartheta}v\mid \},$$
(19)
$$\begin{array}{l}S(\mathbf{r},\mathit{\vartheta})=2.905{k}^{2}{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}({\{r[1-(v/L)]\}}^{5/3}+{\{\vartheta v\}}^{5/3}\\ -\hspace{0.17em}\xbd\mid {\{r[1-(v/L)]\}}^{2}+2r\vartheta [1-(v/L)]vc+{\{\vartheta v\}}^{2}{\mid}^{5/6}\\ -\hspace{0.17em}\xbd\mid {\{r[1-(v/L)]\}}^{2}-2r\vartheta [1-(v/L)]vc+{\{\vartheta v\}}^{2\mid 5/6}),\end{array}$$
(20)
$$c=\mathbf{r}\xb7\mathit{\vartheta}/(r\vartheta ).$$
(21)
$$\u3008T(\mathbf{f})\u3009={T}_{\text{DL}}(\mathbf{f})\text{exp}[-S(\mathrm{\lambda}\mathbf{f},\mathit{\vartheta})+{C}_{l}(0,0)-{C}_{l}(\mathrm{\lambda}\mathbf{f},0)-2{C}_{l}(0,\mathit{\vartheta})+{C}_{l}(\mathrm{\lambda}\mathbf{f},\mathit{\vartheta})+{C}_{l}(\mathrm{\lambda}\mathbf{f},-\mathit{\vartheta})].$$
(22)
$$\u3008T(\mathbf{f})\u3009\approx {T}_{\text{DL}}(\mathbf{f})\hspace{0.17em}\text{exp}[-S(\mathrm{\lambda}\mathbf{f},\mathit{\vartheta})+{C}_{l}(0,0)-2{C}_{l}(0,\mathit{\vartheta})].$$
(23)
$$\u3008T(\mathbf{f})\u3009={T}_{\text{DL}}(\mathbf{f})\hspace{0.17em}\text{exp}\{-S(\mathrm{\lambda}\mathbf{f},\vartheta )+4[{C}_{l}(0,0)-{C}_{l}(0,\mathit{\vartheta})]\}.$$
(24)
$$\u3008T(\mathbf{f})\u3009\approx {T}_{\text{DL}}(\mathbf{f})\text{exp}[-S(\mathrm{\lambda}\mathbf{f},\mathit{\vartheta})].$$
(25)
$$\u3008G\u3009={G}_{\text{DL}}\hspace{0.17em}\left\{\frac{\int \text{d}\mathbf{r}K(r/D)\text{exp}[-S(\mathbf{r},\mathit{\vartheta})+{C}_{l}(0,0)-{C}_{l}(\mathbf{r},0)-2{C}_{l}(0,\mathit{\vartheta})+{C}_{l}(\mathbf{r},\mathit{\vartheta})+{C}_{l}(\mathbf{r},-\mathit{\vartheta})]}{\int \text{d}\mathbf{r}K(r/d)}\right\}.$$
(26)
$$\u3008G\u3009\approx {G}_{\text{DL}}\hspace{0.17em}\text{exp}[{C}_{l}(0,0)-2{C}_{l}(0,\mathit{\vartheta})]\times \left\{\frac{\int \text{d}\mathbf{r}K(r/D)\text{exp}[-S(\mathbf{r},\mathit{\vartheta})]}{\int \text{d}\mathbf{r}K(r/D)}\right\}.$$
(27)
$$\u3008G\u3009={G}_{\text{DL}}\times \frac{\int \text{d}\mathbf{r}K(r/D)\text{exp}\{-S(\mathbf{r},\mathit{\vartheta})+2[{C}_{l}(\mathbf{r},\mathit{\vartheta})+{C}_{l}(\mathbf{r},-\mathit{\vartheta})]\}}{\int \text{d}\mathbf{r}K(r/D)}.$$
(28)
$$\u3008G\u3009\approx {G}_{\text{DL}}\left\{\frac{\int \text{d}\mathbf{r}K(r/D)\text{exp}[-S(\mathbf{r},\mathit{\vartheta})]}{\int \text{d}\mathbf{r}K(r/D)}\right\}.$$
(29)
$$\underset{r/\vartheta \to \infty}{\text{lim}}\hspace{0.17em}S(\mathbf{r},\mathit{\vartheta})=2.905{k}^{2}{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{(\vartheta v)}^{5/3}.$$
(30)
$$\underset{r/\vartheta \to 0}{\text{lim}}\hspace{0.17em}S(\mathbf{r},\mathit{\vartheta})=2.905{k}^{2}{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{\{r[1-(v/L)]\}}^{5/3}.$$
(31)
$${r}_{0}={\left\{(2.905/6.88){k}^{2}{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{[1-(v/R)]}^{5/3}\right\}}^{-3/5}$$
(32)
$${\vartheta}_{0}={\left\{2.905\hspace{0.17em}{k}^{2}{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{v}^{5/3}\right\}}^{-3/5},$$
(33)
$$\underset{r/\vartheta \to \infty}{\text{lim}}\hspace{0.17em}S(\mathbf{r},\mathit{\vartheta})={(\vartheta /{\vartheta}_{0})}^{5/3}$$
(34)
$$\underset{r/\vartheta \to 0}{\text{lim}}\hspace{0.17em}S(\mathbf{r},\mathit{\vartheta})=6.88{(r/{r}_{0})}^{5/3}.$$
(35)
$$S(\mathbf{r},\vartheta )={[(r/{r}_{0})(\vartheta /{\vartheta}_{0})]}^{5/6}\mathcal{S}[(r/{r}_{0})/(\vartheta /{\vartheta}_{0})],$$
(36)
$$\begin{array}{l}\mathcal{S}(Q)=A{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}({Q}^{5/6}{[1-(v/L)]}^{5/3}\\ +\hspace{0.17em}{Q}^{-5/6}{v}^{5/3}{{\mathcal{L}}_{0}}^{-5/3}-\xbd\{Q{[1-(v/L)]}^{2}\\ +\hspace{0.17em}2[1-(v/L)]v{{\mathcal{L}}_{0}}^{-1}c+{Q}^{-1}{v}^{2}{{\mathcal{L}}_{0}}^{-2}{\}}^{5/6}\\ -\xbd\{Q{[1-(v/L)]}^{2}-2[1-(v/L)]v{{\mathcal{L}}_{0}}^{-1}c\\ +\hspace{0.17em}{Q}^{-1}{v}^{2}{{\mathcal{L}}_{0}}^{-2}{\}}^{5/6}),\end{array}$$
(37)
$${\mathcal{L}}_{0}={\mathbf{r}}_{0}/{\vartheta}_{0}$$
(38)
$$A=2.905{k}^{2}{({r}_{0}{\vartheta}_{0}{\mathcal{L}}_{0})}^{5/6}$$
(39)
$$A=6.88\hspace{0.17em}{\left\{{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{[1-(v/L)]}^{5/3}\right\}}^{-1},$$
(40)
$$\begin{array}{l}\mathcal{S}(Q)=6.88\hspace{0.17em}{\left\{{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{[1-(v/L)]}^{5/3}\right\}}^{-1}\\ \times \hspace{0.17em}{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}\times ({Q}^{5/6}{[1-(v/L)]}^{5/3}\\ +\hspace{0.17em}{Q}^{-5/6}{v}^{5/3}{{\mathcal{L}}_{0}}^{-5/3}-\xbd[Q{[1-(v/L)]}^{2}\\ +\hspace{0.17em}2[1-(v/L)]v{{\mathcal{L}}_{0}}^{-1}c+{Q}^{-1}{v}^{2}{{\mathcal{L}}_{0}}^{-2}{\}}^{5/6}\\ -\hspace{0.17em}\xbd\{Q{[1-(v/L)]}^{2}-2[1-(v/L)]v{{\mathcal{L}}_{0}}^{-1}c\\ +\hspace{0.17em}{Q}^{-1}{v}^{2}{{\mathcal{L}}_{0}}^{-2}{\}}^{5/6}).\end{array}$$
(41)
$$\mathcal{S}(Q)\approx {Q}^{-5/6}[1-\alpha (1-\u2153{c}^{2}){Q}^{-1/3}],\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}Q\gg 1,$$
(42)
$$\mathcal{S}(Q)\approx 6.88{Q}^{5/6}[1-\beta (1-\u2153{c}^{2}){Q}^{1/3}],\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}Q\ll 1,$$
(43)
$$\alpha =\u215a{{\mathcal{L}}_{0}}^{-1/3}\left\{{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{[1-(v/L)]}^{1/3}{v}^{2}\right\}\times {\left\{{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{v}^{5/3}\right\}}^{-1}$$
(44)
$$\beta =\u215a{{\mathcal{L}}_{0}}^{1/3}\left\{{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{[1-(v/L)]}^{2}{v}^{-1/3}\right\}\times {\left\{{\int}_{\text{PATH}}\text{d}v{{C}_{N}}^{2}{[1-(v/L)]}^{5/3}\right\}}^{-1}.$$
(45)
$${r}_{0}=3.02{k}^{-6/5}{L}^{-3/5}{({{C}_{N}}^{2})}^{-3/5},$$
(46)
$${\vartheta}_{0}=0.950{k}^{-6/5}{L}^{-8/5}{({{C}_{N}}^{2})}^{-3/5},$$
(47)
$${\mathcal{L}}_{0}=3.18L,$$
(49)
$$\beta =\mathrm{2.206.}$$
(50)
$${r}_{0}={\left\{(2.905/6.88){k}^{2}\hspace{0.17em}\text{sec}(\psi ){\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}\right\}}^{-3/5},$$
(51)
$${\vartheta}_{0}={\left\{2.05{k}^{2}{[\text{sec}(\psi )]}^{8/3}{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}{h}^{5/3}\right\}}^{-3/5},$$
(52)
$$\begin{array}{l}\mathcal{S}(Q)=6.88\hspace{0.17em}{\left\{{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}\right\}}^{-1}\\ \times \hspace{0.17em}{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}\{{Q}^{5/6}+{Q}^{-5/6}{(h/{H}_{0})}^{5/3}\\ -\xbd{[Q+2(h/{H}_{0})c+{Q}^{-1}{(h/{H}_{0})}^{2}]}^{5/6}\\ -\xbd{[Q-2(h/{H}_{0})c+{Q}^{-1}{(h/{H}_{0})}^{2}]}^{5/6}\},\end{array}$$
(53)
$$\alpha =\u215a{{H}_{0}}^{-1/3}\left\{{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}{h}^{2}\right\}\hspace{0.17em}{\left\{{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}{h}^{5/3}\right\}}^{-1},$$
(54)
$$\beta =\u215a{{H}_{0}}^{1/3}\hspace{0.17em}\left\{{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}{h}^{-1/3}\right\}\hspace{0.17em}{\left\{{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}\right\}}^{-1},$$
(55)
$${H}_{0}={(6.88)}^{3/5}{\left\{{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}{h}^{5/3}\right\}}^{3/5}\hspace{0.17em}{\left\{{\int}_{0}^{\infty}\text{d}h{{C}_{N}}^{2}\right\}}^{-3/5}.$$
(56)
$${r}_{0}=\{1.666\times {10}^{7}\mid 3.369\times {10}^{7}\}{k}^{-6/5}{[\text{sec}(\psi )]}^{-3/5},$$
(57)
$${\vartheta}_{0}=\{4.188\times {10}^{3}\mid 4.604\times {10}^{3}\}{k}^{-6/5}{[\text{sec}(\psi )]}^{-3/5},$$
(58)
$${\mathcal{L}}_{0}={H}_{0}\hspace{0.17em}\text{sec}(\psi ),$$
(59)
$${H}_{0}=\{3979\mid 7319\}$$
(60)
$$\alpha =\{0.955\mid 0.846\},$$
(61)
$$\beta =\{2.651\mid 2.930\}.$$