Strehl ratios with various types of anisoplanatism

Richard J. Sasiela

Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02173-9108

Abstract

There are many ways in which the paths of two waves through turbulence can become separated, thereby leading to anisoplanatic effects. Among these are a parallel path separation, an angular separation, one caused by a time delay, and one that is due to differential refraction at two wavelengths. All these effects can be treated in the same manner. Gegenbauer polynomials are used to obtain an approximation for the Strehl ratio for these anisoplanatic effects, yielding a greater range of applicability than the Maréchal approximation.

1. INTRODUCTION

Adaptive-optics systems are used to correct images of objects. These systems work by measuring the phase distortion on a downpropagating wave called a beacon and applying the negative of that phase to a deformable mirror. If this is done well, then the image of the beacon is close to diffraction limited; and if a laser beam is projected along the corrected path, it will have propagation characteristics approaching those of a wave propagating in vacuum. It is not possible to make a perfect correction; one of the major error sources is due to the fact that the rays of the object to be imaged or the laser beam to be propagated are along a path displaced from that of the beacon. A measurement of this degradation is the Strehl ratio, which is the ratio of the intensity of the actual beam on axis to that of a diffraction-limited beam.

\begin{matrix}c \\ ◯ \end{matrix} Optical Society of America, 1992.

This displacement can have several causes. The receiving and the transmitting apertures may be displaced from each other owing to misalignment or vignetting of the beams. The paths can be separated in angle, for instance, when the object to be imaged is different from the beacon. The correction is applied with a time delay after the measurements. In this time the turbulence is displaced by winds and slewing of the telescope. The paths may be separated because the beacon and the imaging wavelengths differ, in which case refraction operates differently on the two waves. All the effects are typically present simultaneously.

These anisoplanatisms have been treated separately in the past^{1–7}; however, they are all manifestations of the same effect. … A better analytic approximation that applies in the range of operation of a typical adaptive-optics system is developed here. This is applied to obtain expressions for the various types of anisoplanatism discussed above.

In Section 2 the general formula for the Strehl ratio with any type of anisoplanatism is derived. Gegenbauer polynomials provide a convenient way to keep track of the series terms and to cancel terms that lead to numerical difficulties if the integral is evaluated numerically. In Sections 3–6 the general formula is applied to obtain the Strehl ratio for various types of anisoplanatism. The cases considered are parallel path displacements, angular offsets, timedelay-induced offsets, and offsets that are due to refractive effects that vary with wavelength. The Strehl ratio in the presence of several effects is examined in Section 7. It is shown that, depending on the direction of the relative displacements, one can get a cancellation of the displacements so that the Strehl ratio is high or an enhancement so that the Strehl ratio is less than the product of the Strehl ratios of the individual terms.

2. 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 = \frac{1}{2π} ∫ dα K(α) exp [−\frac{D (α)}{2}] .  (1)

The integral is over a circular aperture of unit radius, D(α) is the structure function, and K(α) is a factor times the optical transfer function given by

K(α) = \frac{16}{π} [cos^{−1}(α) − α (1 − α^{2})^{1/2}] U(1 − α),  (2)

where U (x) 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

D(αD) = 2(2.91) k_{0}^{2} ∫_{0}^{∞}dz C_{n}^{2}(z) [(αD)^{5/3} + d^{5/3}(z)
− ^{1}/_{2} |αD + d(z) |^{5/3} − ^{1}/_{2} | αD − d(z) |^{5/3}] ,  (4)

where C_{n}^{2}(z) is the turbulence strength as a function of altitude; k_{0} = 2π/λ, where λ is the wavelength of operation; D is the aperture diameter; and 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

| αD ± d(z) |^{5/3} = [(αD)^{2} ± 2αD d(z) cos (φ) + d^{2}(z)]^{5/6} ,  (5)

where is the angle between α and d(z) . This expression can be simplified and the numerical difficulties can be eliminated by using Gegenbauer polynomials.^{8} Their generating function is

(1 − 2ax + a^{2})^{−λ} = ∑_{p=0}^{∞} C_{p}^{λ}(x) a^{p}.  (6)

These functions are sometimes referred to as ultraspherical functions because they are a generalization of the Legendre polynomials P_{n}(t) , whose generating function is

(1 − 2ax + a^{2})^{−1/2} = ∑_{p=0}^{∞} P_{p}(x) a^{p}.  (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

C_{p}^{λ} [cos (φ)] = ∑_{m=0}^{p} \frac{Γ [λ + m] Γ [λ + p − m] cos [(p − 2m)φ]}{m! (p − m)! (Γ [λ])^{2}},  (8)

where - Γ [x] is the gamma function. A particular Gegenbauer polynomial that is required is

C_{2}^{−5/6} [cos(φ)] = ^{5}/_{6} [1 − ^{1}/_{3} cos^{2} (φ)] .  (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

D(αD) = 2(2.91) k_{0}^{2} ∫_{0}^{∞}dz C_{n}^{2}(z) \{d^{5/3}(z) − (αD)^{5/3} ∑_{p=1}^{∞} C_{2p}^{−5/6} [cos (φ)] [\frac{d(z)}{αD}]^{2p}\} . (10)

It is this canceling of the first two terms of the power series that would cause numerical difficulties. Define a distance moment as

d_{m} ≡ 2.91 k_{0}^{2} ∫_{0}^{∞}dz C_{n}^{2}(z) d^{m}(z)  (11)

and a phase variance as

σ_{φ}^{2} = d_{5/3}.  (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

D(αD) = 2σ_{φ}^{2} − 2x,  (13)

where

x = d_{2} [1 − ^{1}/_{3} cos^{2} (φ)] ^{5}/_{6}(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 ≈ \frac{exp (−σ_{φ}^{2})}{2π} ∫ dα K(α)(1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + \frac{x^{5}}{120}) .  (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) = \frac{1}{2π} ∫_{0}^{2π}dφ [1 − ^{1}/_{3} cos^{2} (φ)]^{n} = \frac{1}{2π} ∑_{m=0}^{n} ( \begin{matrix}n \\ n − m \end{matrix} ) 3^{−m} ∫_{0}^{2π}dφ cos^{2m} (φ) ,  (16)

where

( \begin{matrix}n \\ n − m \end{matrix} ) = \frac{n!}{(n − m)! m!}.  (17)

Equation (4.641 # 4) in Gradshteyn and Ryzhik^{8} is

∫_{0}^{π/2}dφ cos^{2m} (φ) = \frac{π(2m − 1)!!}{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 − ∑_{m=1}^{n} ( \begin{matrix}n \\ n − m \end{matrix} ) 3^{−m} \frac{(2m − 1)!!}{(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) = ∫_{0}^{1}dα α^{1−n/3}K(α).  (22)

This is a generalization of the integral evaluated by Tatarski in Sec. 55, Eq. (22) of Ref. 9. Its value is

Y (n) = \frac{8}{(2 − n/3) \sqrt{π}} Γ [ \begin{matrix}−n/6 + \frac{3}{2} \\ −n/6 + 3 \end{matrix} ]  for n < 6.  (23)

The values of interest to us are Y (0) = 1, Y (1) = 1.402, Y (2) = 2.087, Y (3) = 3.396, Y (4) = 6.419, and Y (5) = 16.94. With these values for the integral, the Strehl ratio approximation is

SR ≈ (1 + 0.9736 E + 0.5133 E^{2} + 0.2009 E^{3} + 0.0697 E^{4} + 0.02744 E^{5}) exp(−σ_{φ}^{2}),  (24)

where

E = \frac{d_{2}}{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. 1 are plotted the exact Strehl ratio versus displacement for the Hufnagel–Valley 21 (HV-21) model of turbulence^{10–12} and the Strehl ratio from relation (24) for D/r_{o} = 1, with only the unity term in parenthesis (extended Marechal approximation) and with different numbers of terms in the parenthesis. …

3. DISPLACEMENT ANISOPLANATISM

In the simplest case of displacement anisoplanatism, which was treated in Section 2, the displacement is constant along the propagation direction. The terms to use to find the Strehl ratio are

d(z) = d,  (26)
d_{2} = 2.91 k_{0}^{2} μ_{0} d^{2},  (27)
E = 6.88 (\frac{d}{D} )^{2} (\frac{D}{r_{o}} )^{5/3} ,  (28)
σ_{φ}^{2} = 2.91 k_{0}^{2} μ_{0} d^{5/3} = 6.88 (\frac{d}{r_{o}} )^{5/3} .  (29)

The Strehl ratios are plotted in Figs. 2 and 3.

4. ANGULAR ANISOPLANATISM

When the propagation beam is offset by a constant angle from the direction along which turbulence is measured, the effect is called angular anisoplanatism.^{4} It arises naturally when one is tracking a satellite target and directing a laser beam at it. Because of the finite speed of light, the laser beam has to lead the tracking direction, resulting in an angular difference between the direction along which the target is tracked and the one along which the laser beam is directed. This error can be eliminated if the target has a reflector for the beacon that extends a suitable distance in the point-ahead direction. For the case of an angular error

d(z) = θ z,  (30)
d_{2} = 2.91 k_{0}^{2} μ_{2} θ^{2},  (31)
E = 6.88 \frac{μ_{2}}{μ_{0}} (\frac{θ}{D} )^{2} (\frac{D}{r_{o}} )^{5/3} ,  (32)
σ_{φ}^{2} = 2.91 k_{0}^{2} θ^{5/3} ∫_{0}^{L}dz C_{n}^{2}(z)z^{5/3} = (θ/θ_{o})^{5/3} ,  (33)

where the isoplanatic angle is defined by

θ_{o} = (2.91 k_{0}^{2} μ_{5/3})^{−3/5} .  (34)

...

5. TIME DELAY

If there is a time delay between when turbulence is measured and when a correction is applied to the deformable mirror, there is a degradation in performance.^{7} This effect is not often thought of as an anisoplanatic effect; however, it can be treated as such. ...

d(z) = v(z)τ,  (35)
d_{2} = 2.91 k_{0}^{2} ∫_{0}^{L}dz C_{n}^{2}(z) v^{2}(z) τ^{2} = (τ/τ_{2})^{2} ,  (36)
E = \frac{τ^{2}}{τ_{2}^{2}D^{1/3}} ,  (37)
σ_{φ}^{2} = 2.91 k_{0}^{2} ∫_{0}^{L}dz C_{n}^{2}(z) v^{5/3}(z) τ^{5/3} = (τ/τ_{5/3})^{5/3} ,  (38)

where the temporal moment is defined as

1/τ_{m}^{5/3} = 2.91 k_{0}^{2} ∫_{0}^{L}dz C_{n}^{2}(z) v^{m}(z).  (39)

…

6. CHROMATIC ANISOPLANATISM

If the beacon beam that senses the turbulence has a wavelength different from that of the laser beam that is sent out, then the two beams will follow different paths through the atmosphere because of the dispersive properties of the atmosphere. The analysis given here parallels that given by Belsher and Fried.^{1}

… The change of refractive index with wavelength has been given by Allen^{16} as

Δn_{0} = (λ_{1}^{2} − λ_{2}^{2}) [\frac{29 498.1}{(146λ_{2}^{2} − 1) (146λ_{1}^{2} − 1)} + \frac{255.4}{(41λ_{2}^{2} − 1) (41λ_{1}^{2} − 1)}] 10^{−6}.  (40)

The atmospheric density versus altitude is given by Cole.^{17} The ratio of the ... . Thus the beam displacement along the path is

d d d_{c}(z) = −\frac{ξ sin (ξ) Δn_{0}}{ξ cos^{2} (ξ)} [ ∫_{0}^{z}dz′α (z′) − \frac{z}{L} ∫_{0}^{L}dz′α (z′)] .  (41)

Define the integral of the air density as

I (z) = ∫_{0}^{z}dz′α (z′) .  (42)

Evaluating the integral and … The moments of this displacement are

d_{m} = [\frac{sin (ξ) Δn_{0}}{cos^{2} (ξ)} ]^{m} T_{m},  (43)

where

T_{m} = 2.91 k_{0}^{2} sec (ξ) ∫_{0}^{H}dh C_{n}^{2}(h) [I(h) − \frac{h sec (ξ)}{L}I(L)]^{m} .  (44)

H is the altitude of the target. The last term in brackets goes to zero as the range becomes infinite. … For the infinite range, this reduces to

T_{m} = 2.91 k_{0}^{2} sec (ξ) ∫_{0}^{H}dh C_{n}^{2}(h) I^{m}(h).  (45)

…

7. COMBINED DISPLACEMENT

If there are several anisoplanatic effects present, with each not decreasing the Strehl ratio much, it is a common practice to multiply the Strehl ratios for the individual effects to get a combined Strehl ratio. The validity of this assumption is now examined. The total displacement that is due to a translation, an angular offset, a time delay, and a chromatic offset is

d d d_{t}(z) = d d d + θz + vv v(z)τ + d d d_{c}(z),  (46)

where chromatic displacement is given in Eq. (50). The two terms necessary for calculating the Strehl ratio are

E = \frac{d_{2}}{D^{1/3}} ,  (47)
σ_{φ}^{2} = d_{5/3},  (48)

where

d_{m} = 2.91 k_{0}^{2} ∫_{0}^{∞}dz C_{n}^{2}(z) |d_{t}(z)|^{m} .  (49)

…

… Tyler et al.^{18} took advantage of the vector nature of the displacement almost to eliminate the effect of chromatic anisoplanatism on an adaptive-optics system by choosing an optimal offset angle of a beacon from the propagation direction.

8. SUMMARY

An approximate expression for the Strehl ratio that is easily evaluated for any turbulence distribution was derived. It applies for various anisoplanatic effects. This expression was shown to give much better agreement with the exact answer than the extended Marechal approximation. The zenith dependence is included in the formula. This approximation was applied to parallel path displacements, angular offsets, time-delay induced offsets, and offsets owing to refractive effects that vary with wavelength. Examples for each type of anisoplanatism at various zenith angles were evaluated.

The Strehl ratio in the presence of several effects was examined. It was shown that, depending on the direction of the relative displacements, one can get a cancellation or an enhancement of the effect of the displacements. Therefore it is possible for there to be little reduction in the Strehl ratio if there is little net path displacement. If the displacements are in the same direction, the Strehl ratio is less than the product of the Strehl ratios of the individual terms.

ACKNOWLEDGMENTS

This research was sponsored by the Strategic Defense Initiative Organization through the U.S. Department of the Air Force.

REFERENCES

1. J. Belsher and D. Fried, “Chromatic refraction induced pseudo anisoplanatism,” tOSC Rep. TR-433 (Optical Sciences Co., Placentia, Calif., 1981).

2. B. L. Ellerbroek and P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” tOSC Rep. TR-613 (Optical Sciences Co., Placentia, Calif., 1984).

3. D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71-76 (1975).

4. D. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52-61 (1982).

5. D. Korff, G. Druden, and R. P. Leavitt, “Isoplanicity: the translation invariance of the atmospheric Green’s function,” J. Opt. Soc. Am. 65, 1321-1330 (1975).

6. J. H. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am. 65, 65-68 (1975).

7. G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251-262 (1984).

8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

9. V. I. Tatarski, The Effects Of The Turbulent Atmosphere On Wave Propagation (U. S. Department of Commerce, Washington, D.C., 1971).

10. R. E. Hufnagel, Optical Propagation through Turbulence (Optical Society of America, Washington, D. C., 1974).

11. J. L. Bufton, P. O. Minott, M. W. Fitzmaurice, and P. J. Titterton, “Measurements of turbulence profiles in the troposphere,” J. Opt. Soc. Am. 62, 1068-1070 (1972).

12. G. C. Valley, “Isoplanatic degradation of tilt correction and short-term imaging system,” Appl. Opt. 19, 574-577 (1980).

13. M. G. Miller and P. L. Zieske, “Turbulence environmental characterization,” RADC-TR-79-131 (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1979).

14. D. P. Greenwood, “Bandwidth specifications for adaptive optics systems,” J. Opt. Soc. Am. 67, 390-393 (1977).

15. D. L. Fried, “Time-delay-induced mean-square error in adaptive optics,” J. Opt. Soc. Am. A 7, 1224-1225 (1990).

16. C. W. Allen, Astrophysical Quantities (Athlone, London, 1963).

17. A. E. Cole, A. Court, and A. J. Kantor, Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965).

18. G. Tyler, J. Belsher and D. Fried, “Amelioration of chromatic refraction induced pseudoanisoplanatism,” tOSC Rep. TR-465 (Optical Sciences Co., Placentia, Calif., 1982).

FIGURES

Fig. 1. Comparison of the Maréchal and the two- to six-term approximations with the exact value of the Strell ratio, for an anisoplanatic displacement, for D/r_{0} equal to 1.

Fig. 2. Comparison of the Maréchal and the two- to six-term approximations with the exact value of the Strell ratio, for an anisoplanatic displacement, for D/r_{0} equal to 5.

Fig. 3. Comparison of the Maréchal and the two- to six-term approximations with the exact value of the Strell ratio, for an anisoplanatic displacement, for D/r_{0} equal to 10.

Fig. 4. Strehl ratio for angular anisoplanatic error at zenith, for various turbulence models, versus separation angle for a 0.6-m system. Upper-altitude turbulence has a strong effect on the Strehl ratio.

Fig. 5. Strehl ratio for angular anisoplanatism at 30^{∘} for a 0.6-m system.

Fig. 6. Strehl ratio versus time delay at zenith for a 0.6-m system.

Fig. 7. Strehl ratio versus time delay for a 0.6-m system at 30^{∘} zenith angle. Strehl ratio at 30^{∘} for a 0.6-m system.

Fig. 8. Difference (×10^{6}) in refractive index between 0.5 μm and other wavelengths.

TABLES

Table 1. Values of T_{2} and T_{5/3} to Solve for the Chromatic Displacement for Various Turbulence Models for a Wavelength of 0.5 μm

Model  T_{2}^{a}  T_{5/3}^{b}
SLC-Day  2.71 × 10^{−6}  2.00 × 10^{−7}
HV-21  6.16 × 10^{−6}  3.60 × 10^{−7}
HV-54  3.40 × 10^{−5}  1.87 × 10^{−6}
HV-72  5.95 × 10^{−5}  3.25 × 10^{−6}

aThe units of T_{2} are m^{1/3}.

bT_{5/3} is dimensionless.