STREHL RATIO WITH ANISOPLANATISM

For a perfect correction the paths of the beacon signal and the imaging or projected laser should be the same. In general, this is not possible to achieve, and there is a degradation in performance caused by time delays, displacement of the two paths by translation and angle, and differences in wavelength of the beacon and the measurement or projecting systems. The effects of displacement, angular mispointing, time delay, and atmospheric dispersion can each be treated as an anisoplanatic effect. In fact, if all the effects are present simultaneously, they can be combined to get a total offset of the measurement from the imaging paths. In this section the effect of a general displacement on the Strehl ratio is determined.

The Strehl ratio (SR) for a circular aperture [#!7!#] from the Huygens–Fresnel approximation is

SR = $\displaystyle {1 \over {2\pi }}$$\displaystyle \int$d$\displaystyle \bbox$α K(α) exp $\displaystyle \left[\vphantom{ {-{{{\cal D}\left( {\bbox \alpha } \right)} \over 2}} }\right.$$\displaystyle {-{{{\cal D}\left( {\bbox \alpha } \right)} \over 2}}$$\displaystyle \left.\vphantom{ {-{{{\cal D}\left( {\bbox \alpha } \right)} \over 2}} }\right]$.     (1)

The integral is over a circular aperture of unit radius, $\cal {D}$($\bbox$α) is the structure function, and K(α) is a factor times the optical transfer function given by
K(α) = $\displaystyle {{16} \over \pi}$$\displaystyle \left[\vphantom{ {\cos ^{- 1}(\alpha )-\alpha \left( {1-\alpha ^2} \right)^{1/ 2}} }\right.$cos-1(α)-α$\displaystyle \left(\vphantom{ {1-\alpha ^2} }\right.$1-α2$\displaystyle \left.\vphantom{ {1-\alpha ^2} }\right)^{{1/ 2}}_{}$$\displaystyle \left.\vphantom{ {\cos ^{- 1}(\alpha )-\alpha \left( {1-\alpha ^2} \right)^{1/ 2}} }\right]$ U(1 - α),     (2)

where U$\left(\vphantom{ x }\right.$x$\left.\vphantom{ x }\right)$ is the unit step function defined as
U(x) = 1        for    x≥0 ,  
U(x) = 0        for    x < 0  . (3)

To find the Strehl ratio, one must first determine the structure function. It was found by Fried[#!4!#] for angular anisoplanatism. If the source is collimated and a general displacement is introduced, his expression for a wave propagating from ground to space becomes

$\displaystyle \cal {D}$(αD) = 2(2.91) k02$\displaystyle \int\limits_{{\,\,\; 0}}^{{\,\,\,\,\,\; \infty}}$dz Cn2(z)$\displaystyle \left[\vphantom{ {( {\alpha \kern 1ptD} )^{5/ 3}+d^{5/ 3}(z)}}\right.$(αD)5/3+d5/3(z)  
    $\displaystyle \left.\vphantom{ {-{\slantfrac{1}{2}}\,\left\vert {{\bbox \alpha}... ...vert {\,{\bbox \alpha} \kern 1ptD-{\bbox d}(z)\,} \right\vert^{5 / 3}} }\right.$-$\displaystyle \slantfrac$12 $\displaystyle \left\vert\vphantom{ {{\bbox \alpha} \kern 1ptD+{\bbox d}(z)\,} }\right.$$\displaystyle \bbox$αD+$\displaystyle \bbox$d(z) $\displaystyle \left.\vphantom{ {{\bbox \alpha} \kern 1ptD+{\bbox d}(z)\,} }\right\vert^{{5/ 3}}_{}$-$\slantfrac$12$\displaystyle \left\vert\vphantom{ {\,{\bbox \alpha} \kern 1ptD-{\bbox d}(z)\,} }\right.$ $\displaystyle \bbox$αD-$\displaystyle \bbox$d(z) $\displaystyle \left.\vphantom{ {\,{\bbox \alpha} \kern 1ptD-{\bbox d}(z)\,} }\right\vert^{{5 / 3}}_{}$$\displaystyle \left.\vphantom{ {-{\slantfrac{1}{2}}\,\left\vert {{\bbox \alpha}... ...vert {\,{\bbox \alpha} \kern 1ptD-{\bbox d}(z)\,} \right\vert^{5 / 3}} }\right]$, (4)

where Cn2(z) is the turbulence strength as a function of altitude; k0 = 2π/λ, where λ is the wavelength of operation; D is the aperture diameter; and $\bbox$d(z) is the vector displacement of the two paths.

The sums of the terms in brackets almost cancel, thus causing difficulties if one tries to evaluate this integral numerically. The terms in the absolute-value sign are equal to

$\displaystyle \left\vert\vphantom{ {\,{\bbox \alpha} \kern 1ptD\pm {\bbox d}(z)\,} }\right.$ $\displaystyle \bbox$αD±$\displaystyle \bbox$d(z) $\displaystyle \left.\vphantom{ {\,{\bbox \alpha} \kern 1ptD\pm {\bbox d}(z)\,} }\right\vert^{{5/ 3}}_{}$ = $\displaystyle \left[\vphantom{ {\left( {\alpha \kern 1ptD} \right)^2\pm 2\alpha \kern 1ptD\,d(z)\cos \left( \varphi \right)+d^2(z)} }\right.$$\displaystyle \left(\vphantom{ {\alpha \kern 1ptD} }\right.$αD$\displaystyle \left.\vphantom{ {\alpha \kern 1ptD} }\right)^{2}_{}$±2αD d (z)cos$\displaystyle \left(\vphantom{ \varphi }\right.$$\displaystyle \varphi$$\displaystyle \left.\vphantom{ \varphi }\right)$+d2(z)$\displaystyle \left.\vphantom{ {\left( {\alpha \kern 1ptD} \right)^2\pm 2\alpha \kern 1ptD\,d(z)\cos \left( \varphi \right)+d^2(z)} }\right]^{{5/ 6}}_{}$,     (5)

where is the angle between $\bbox$α and $\bbox$d(z) . This expression can be simplified and the numerical difficulties can be eliminated by using Gegenbauer polynomials.[#!8!#] Their generating function is
$\displaystyle \left(\vphantom{ {1-2ax+a^2} }\right.$1-2ax+a2$\displaystyle \left.\vphantom{ {1-2ax+a^2} }\right)^{{-\lambda }}_{}$ = $\displaystyle \sum\limits_{{p=0}}^{\infty}$Cpλ(x) ap.     (6)

These functions are sometimes referred to as ultraspherical functions because they are a generalization of the Legendre polynomials Pn(t) , whose generating function is
$\displaystyle \left(\vphantom{ {1-2ax+a^2} }\right.$1-2ax+a2$\displaystyle \left.\vphantom{ {1- 2ax+a^2} }\right)^{{-1/ 2}}_{}$ = $\displaystyle \sum\limits_{{p=0}}^{\infty}$Pp(x) ap.     (7)

The Gegenbauer polynomials with the cosine of a variable as the argument are given in Eq. (8.934 #2) of Ref. 8 and can be rewritten as
Cpλ$\displaystyle \left[\vphantom{ {\cos \left( \varphi \right)} }\right.$cos$\displaystyle \left(\vphantom{ \varphi }\right.$$\displaystyle \varphi$$\displaystyle \left.\vphantom{ \varphi }\right)$$\displaystyle \left.\vphantom{ {\cos \left( \varphi \right)} }\right]$ = $\displaystyle \sum\limits_{{m=0}}^{p}$$\displaystyle {{\Gamma\,\left[ {\lambda +m} \right]\,\Gamma\,\left[ {\lambda +p... ...\right]} \over {m!\,(p-m)!\,\left( {\Gamma\,\left[ \lambda \right]} \right)^2}}$,     (8)

where - Γ$\left[\vphantom{ x }\right.$x$\left.\vphantom{ x }\right]$ is the gamma function. A particular Gegenbauer polynomial that is required is
C2-5/6$\displaystyle \left[\vphantom{ {\cos (\varphi )} }\right.$cos($\displaystyle \varphi$)$\displaystyle \left.\vphantom{ {\cos (\varphi )} }\right]$ = $\slantfrac$56$\displaystyle \left[\vphantom{ {1- {\textstyle{ \slantfrac{1}{3}}}\cos ^2\left( \varphi \right)} }\right.$1-$\slantfrac$13cos2$\displaystyle \left(\vphantom{ \varphi }\right.$$\displaystyle \varphi$$\displaystyle \left.\vphantom{ \varphi }\right)$$\displaystyle \left.\vphantom{ {1- {\textstyle{ \slantfrac{1}{3}}}\cos ^2\left( \varphi \right)} }\right]$.     (9)

For αD > d (z) , the terms in the structure function can be expanded in Gegenbauer polynomials. The zeroth- and all odd-order terms cancel. When the summation index is changed by the substitution p→2p the result is
$\displaystyle \cal {D}$(αD) = 2(2.91) k02$\displaystyle \int\limits_{{\,\,\, 0}}^{{\,\,\,\,\,\,\infty}}$dz Cn2(z)$\displaystyle \left\{\vphantom{ {d^{5/ 3}(z)- (\alpha \kern 1ptD)^{5/ 3}\sum\li... ...)} \right]}\,\left[ {{{d(z)} \over {\alpha \kern 1ptD}}} \right]^{2p}} }\right.$$\displaystyle {d^{5/ 3}(z)- (\alpha \kern 1ptD)^{5/ 3}\sum\limits_{p=1}^\infty ... ...hi \right)} \right]}\,\left[ {{{d(z)} \over {\alpha \kern 1ptD}}} \right]^{2p}}$$\displaystyle \left.\vphantom{ {d^{5/ 3}(z)- (\alpha \kern 1ptD)^{5/ 3}\sum\lim... ...} \right]}\,\left[ {{{d(z)} \over {\alpha \kern 1ptD}}} \right]^{2p}} }\right\}$.     (10)

It is this canceling of the first two terms of the power series that would cause numerical difficulties. Define a distance moment as
dm≡2.91 k02$\displaystyle \int\limits_{{\,\,\, 0}}^{{\,\,\,\,\,\,\infty}}$dz Cn2(z) dm(z)     (11)

and a phase variance as
σ$\scriptstyle \varphi$2 = d5/3.     (12)

Unlike the calculation for Strehl ratio for uncorrected turbulence and for corrected turbulence with tilt jitter, an exact analytical solution cannot be found for anisoplanatism. Fortunately, for adaptive-optics systems, the Strehl ratio should be fairly high by design, which requires the structure function to be small. This assumption allows one to retain only the first term of the Gegenbauer expansion to give
$\displaystyle \cal {D}$(αD) = 2σ$\scriptstyle \varphi$2 - 2x,     (13)

where
x = d2$\displaystyle \left[\vphantom{ {1- {\textstyle{ \slantfrac{1}{3}}}\cos ^2\left( \varphi \right)} }\right.$1-$\slantfrac$13cos2$\displaystyle \left(\vphantom{ \varphi }\right.$$\displaystyle \varphi$$\displaystyle \left.\vphantom{ \varphi }\right)$$\displaystyle \left.\vphantom{ {1- {\textstyle{ \slantfrac{1}{3}}}\cos ^2\left( \varphi \right)} }\right]$$\displaystyle \slantfrac$56(αD)-1/3.     (14)

We justify this single-term approximation below by showing that it produces a result close to the exact result.
...
The Strehl ratio with the six term approximation is
SR $\displaystyle \approx$ $\displaystyle {{\exp \left( {-\sigma _\varphi} ^2 \right)} \over {2\pi }}$$\displaystyle \int$d$\displaystyle \bbox$α K(α) $\displaystyle \left(\vphantom{ {1+x+{{x^2} \over 2}+{{x^3} \over 6}+{{x^4} \over {24}}+{{x^5} \over {120}}} }\right.$$\displaystyle {1+x+{{x^2} \over 2}+{{x^3} \over 6}+{{x^4} \over {24}}+{{x^5} \over {120}}}$$\displaystyle \left.\vphantom{ {1+x+{{x^2} \over 2}+{{x^3} \over 6}+{{x^4} \over {24}}+{{x^5} \over {120}}} }\right)$.     (15)

If just the first term in the last parenthetical expression is retained, the result is equivalent to the extended Maréchal approximation. It is shown below that the six-term approximation is best for aperture sizes normally encountered. The angle integral for the nth term, after use of the binomial theorem, is proportional to
Φ(n) = $\displaystyle {1 \over {2\pi }}$$\displaystyle \int\limits_{{\,\,\, 0}}^{{\,\,\,\,\,\,\,\, 2\pi }}$d$\displaystyle \varphi$ $\displaystyle \left[\vphantom{ {1-\slantfrac{1}{3}} \cos ^2\left( \varphi \right) }\right.$1-$\displaystyle \slantfrac$13cos2$\displaystyle \left(\vphantom{ \varphi }\right.$$\displaystyle \varphi$$\displaystyle \left.\vphantom{ \varphi }\right)$$\displaystyle \left.\vphantom{ {1-\slantfrac{1}{3}} \cos ^2\left( \varphi \right) }\right]^{n}_{}$ = $\displaystyle {1 \over {2\pi }}$$\displaystyle \sum\limits_{{m=0}}^{n}$$\displaystyle \left(\vphantom{ \begin{array}{c} n \\ n-m\end{array} }\right.$$\displaystyle \begin{array}{c} n \\ n-m\end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} n \\ n-m\end{array} }\right)$ 3-m$\displaystyle \int\limits_{{\,\,\, 0}}^{{\,\,\,\,\,\, 2\pi }}$d$\displaystyle \varphi$ cos2m$\displaystyle \left(\vphantom{ \varphi }\right.$$\displaystyle \varphi$$\displaystyle \left.\vphantom{ \varphi }\right)$,     (16)

where
$\displaystyle \left(\vphantom{ \begin{array}{c} n \\ n-m\end{array} }\right.$$\displaystyle \begin{array}{c} n \\ n-m\end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} n \\ n-m\end{array} }\right)$ = $\displaystyle {{n!} \over {\left( {n-m} \right)!\,\,m!}}$.     (17)

Equation (4.641 # 4) in Gradshteyn and Ryzhik[#!8!#] is
$\displaystyle \int\limits_{{\,\,\, 0}}^{{\,\,\,\,\,\, \pi / 2}}$d$\displaystyle \varphi$ cos2m$\displaystyle \left(\vphantom{ \varphi }\right.$$\displaystyle \varphi$$\displaystyle \left.\vphantom{ \varphi }\right)$ = $\displaystyle {{\pi (2m-1)!!} \over {2(2m)!!}}$,     (18)

where
(2m - 1)!! = (2m - 1)(2m - 3)…(3)(1), (19)
(2m)!! = (2m)(2m - 2)…(4)(2). (20)

With these relations, the angle integral is equal to
Φ(n) = 1 - $\displaystyle \sum\limits_{{m=1}}^{n}$$\displaystyle \left(\vphantom{ \begin{array}{c} n \\ n-m\end{array} }\right.$$\displaystyle \begin{array}{c} n \\ n-m\end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} n \\ n-m\end{array} }\right)$ 3-m$\displaystyle {{(2m-1)!!} \over {(2m)!!}}$.     (21)

The values of interest to us are Φ(0) = 1, Φ(1) = 0.8333, Φ(2) = 0.7083, Φ(3) = 0.6134, Φ(4) = 0.5404, and Φ(5) = 0.4836. The aperture integration for the nth term is proportional to
Y(n) = $\displaystyle \int\limits_{{\,\,\, 0}}^{{\,\,\,\,\,\, 1}}$dα α1-n/3K(α).     (22)

This is a generalization of the integral evaluated by Tatarski in Sec. 55, Eq. (22) of Ref. 9. Its value is
Y$\displaystyle \left(\vphantom{ n }\right.$n$\displaystyle \left.\vphantom{ n }\right)$ = $\displaystyle {8 \over {(2-n/ 3)\,\sqrt \pi }}$ Γ $\displaystyle \left[\vphantom{ \begin{array}{c} -n/ 6+{3 \over 2} \\ { -n/ 6+3} \end{array} }\right.$$\displaystyle \begin{array}{c} -n/ 6+{3 \over 2} \\ { -n/ 6+3} \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} -n/ 6+{3 \over 2} \\ { -n/ 6+3} \end{array} }\right]$              for   n < 6.     (23)

The values of interest to us are Y$\left(\vphantom{ 0 }\right.$ 0$\left.\vphantom{ 0 }\right)$ = 1, Y$\left(\vphantom{ 1 }\right.$1$\left.\vphantom{ 1 }\right)$ = 1.402, Y$\left(\vphantom{ 2 }\right.$2$\left.\vphantom{ 2 }\right)$ = 2.087, Y$\left(\vphantom{ 3 }\right.$3$\left.\vphantom{ 3 }\right)$ = 3.396, Y$\left(\vphantom{ 4 }\right.$4$\left.\vphantom{ 4 }\right)$ = 6.419, and Y$\left(\vphantom{ 5 }\right.$5$\left.\vphantom{ 5 }\right)$ = 16.94. With these values for the integral, the Strehl ratio approximation is
SR $\displaystyle \approx$ (1+0.9736 E+0.5133 E2+0.2009 E3+0.0697 E4+0.02744 E5)exp(-σ$\scriptstyle \varphi$2),     (24)

where
E = $\displaystyle {{d_{\,2}} \over {D^{1/ 3}}}$.     (25)

...
There is an error made in using this approximation for the central part of the aperture that increases with each term in the approximation. One has to determine whether this error is less than or greater than the increased accuracy achieved over the remainder of the aperture by using additional series terms. To resolve these uncertainties, I compared the Strehl ratio, using various numbers of terms, with exact calculations.

I calculated the Strehl ratio numerically for the case in which the displacement does not vary with propagation distance. In Fig. [*] are plotted the exact Strehl ratio versus displacement for the Hufnagel–Valley 21 (HV-21) model of turbulence[#!10!#,#!11!#,#!12!#] and the Strehl ratio from relation (24) for D/ro = 1, with only the unity term in parenthesis (extended Marechal approximation) and with different numbers of terms in the parenthesis.
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