Vol. 1, No. 7/July 1984/J. Opt. Soc. Am. A 785

Comparison of diffractive and refractive effects intwo-wavelength adaptive transmission

Edward P. Wallner

Itek Optical Systems, 10 Maguire Road, Lexington, Massachusetts 02173

Received November 7, 1983; accepted March 7, 1984

Holmes and Rao Gudimetla [J. Opt. Soc. Am. 73,1119-1122 (1983)] analyze the performance of an adaptive optical

system in which the optical path differences caused by atmospheric turbulence are measured at one wavelengthand used to compensate an outgoing laser beam of a different wavelength. The wavelength dependence of diffrac-

tion is shown to lead to imperfect correction. Earlier work [E. P. Wallner, Proc. Soc. Photo-Opt. Instrum. Eng. 75,119-125 (1976); J. Opt. Soc. Am. 67, 407-409 (1977)] shows that the wavelength dependence of the index of refrac-

tion of air also leads to imperfect correction for beams not directed vertically. In this Communication I comparethe refractive and diffractive effects and show that either may be more important, depending on the geometry ofthe case and the wavelengths used.

INTRODUCTION

Holmes and Rao Gudimetla1 analyze the performance of anadaptive optical system in which the optical path differences(OPD's) caused by atmospheric turbulence are measured atone wavelength and used to compensate an outgoing laserbeam of a different wavelength. The ratio of the intensity ofthe return beam at the beacon to that for a single-wavelengthsystem is taken as the measure of performance. Any loss may

be attributed to dispersion in the diffraction of the beam bythe turbulence; i.e., because diffraction is wavelength de-pendent, after being diffracted by turbulence near thetransmitter, rays of different wavelengths will subsequentlyfollow different paths and therefore encounter different tur-bulence. For horizontal paths of less than about 10 km andfor vertical paths, this is the dominant chromatic effect. Forother upward paths and for long horizontal paths, atmosphericrefraction will cause the beam to curve, and, since the atmo-sphere is dispersive, rays of different wavelength will follow

different paths and encounter different turbulence purely asa result of the effects of geometric optics.

REFRACTIVE DISPERSION

Refractive dispersion was treated 2' 3 for the case correspondingto a beacon and an outgoing beam with equal but finite in-tensity distributions. The case for a large aperture and abeacon at an altitude much greater than the depth of the at-mosphere was covered, but the extension to the case of finiteaperture, finite range, and discrete intensity distributions isstraightforward. This extension is carried out in AppendixA, with the mean-square residual error being given by Eq.(A9).

In Eq. (A9) it is assumed that the turbulence follows aKolmogorov spectrum, i.e., the inner scale is zero and the outer

scale is infinite. In the cases evaluated here, the separationof rays of different wavelengths generally lies between theinner and outer scales assumed in Ref. 1, so that this as-sumption does not appreciably affect the comparison of re-sults. (The outer scale of 1 m used in Ref. 1 appears smallwhen compared with the values quoted in Ref. 4. A larger

value of outer scale would lead to slightly more performancedegradation.)

In Eq. (A9) it is also assumed that the elevation of the lineof sight is not a small angle. [In the horizontal case, the ver-tical gradient of refractive index causes a fixed ray curvaturethat is proportional to refractivity; this leads to an expressionfor the mean-square error that is similar to Eq. (A9) with theray separation being a quadratic function of range.]

For rms wave-front errors of the order of 1 rad, the Strehlratio may be approximated by

S - exp(-kl2 uopD2 ),

where S is the Strehl ratio, k, = 27r/Xl, and UOPD is the rmsoptical phase difference.

NUMERICAL EVALUATION

The integrals in Eq. (A9) have been evaluated for the casesillustrated in Fig. 4 of Ref. 1, using the same modified Huf-nagel model of turbulence, the International Civil AviationOrganization (ICAO) standard atmosphere,5 and the refrac-tivity of dry air as given by Owens.6 The results of this eval-uation are shown by the solid curves in Fig. 1; the curves of Fig.4 of Ref. 1 have been replotted as dotted curves for compar-ison.

Except for the vertical case for which the total effect issmall, the effects of refraction are greater than those of dif-fraction for wavelength ratios less than about 1.5, and they areespecially so for ratios less than unity. This reflects thevariation in the index of air, which falls relatively rapidly inthe near ultraviolet but slowly in the near infrared.

The results in Fig. 1 are for a fixed range of only 20 km.Another case of interest is that in which the beacon is far be-yond the atmosphere so that all transmitted rays are nomi-nally parallel. The performance, considering refraction only,in this case is shown in Fig. 2. (Note the change in verticalscale.) Here, the effect is much more severe, and the ratio ofwavelengths is restricted to near unity at zenith angles largerthan 45 deg.

Obviously, both of these effects must be considered forcorrect results, and a complete analysis would include both

0740-3232/84/070785-03$02.00 © 1984 Optical Society of America

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786 J. Opt. Soc. Am. A/Vol. 1, No. 7/July 1984

L = 20 km = path lengthZ =zenith angleA, = 0.532 jum = transmitter wavelength - Refraction onlyX, = beacon wavelength Refraction onlyao=Ro = 0.5 m = beam size … Diffraction only

0 1 2 3Beacon Wavelength/Transmitter Wavelength, X2/X1

Fig. 1. Comparison of refractive and diffractive dispersion.

0:

co

1.0 , \0.8 \0.6 - \

0.4 I \.4Z = 45° = 60° | H . 107 m

0.2 { I R = 0.5 m0 =/ 60° \, ~ X = 0.532 .r

0I I0 1 2 3

Beacon Wavelength/Transmitter Wavelength, A2/X1

Fig. 2. Refractive dispersion for distant beacon.

from the start. Considering the overall Strehl ratio as theproduct of those given by the separate analyses would prob-able not be wide of the mark.

APPENDIX A. DERIVATION OF MEAN-SQUARE WAVE-FRONT ERROR

If the wave-front error caused by atmospheric turbulence ismeasured at wavelength X,, scaled by the ratio of the atmo-.

c(x) = W(x) - | dxiW(x1)6(xi)

= _dxjW(xO)[6(x) -(xi)], (A2)

where e(x) is the error at x with the aperture average removed,She dx indicates integration over the aperture plane, W(x)is the aperture weighting function [W(x) = lh7rR2 for IxI <R and W(x) = 0 elsewhere], and R is the radius of aperture.

By squaring the error, taking its expected value, and aver-aging over the aperture, we get the desired mean-squarewave-front error:

CoPD2= l dx 3 W(x 3 )(E(x 3 )2)

- 5 dxl 5 dx 2 5 dx3W(xW(x 2)W(x 3 )

X ([(X1) - (6(X2)] [6(X1) -6(X3), (A3)

where (f ) is the expected value of f.In a homogeneous atmosphere, the random OPD caused by

turbulence between two rays will depend on their wavelengthsand on their separation but not on their absolute positions.The mean-square OPD between them can then be expressedby a structure function:

D(xi - X2 , X1 , X2) = ([8(X1, X1) - O(x2, X2)]2), (A4)

where D(x, X,, X2) is the structure function for OPD.Equation (A3) can be expanded by using Eq. (Al) and can

be expressed in terms of the structure function [Eq. (A4)].The symmetry of the structure function and permutation ofthe variables of integration allow the 16 terms so generatedto be reduced to f6ur and one integration to be eliminated; thisresults in

cTOPD2

= D(0, Xl, N2)- 5 dxl dx 2 W(xlW(x 2)

X [1/2 D(x 1 - X2 , X1, N2) + 1/2 D(xl - X2 , N2 , XI)

- D(xi - X2 , Xi, NI)]. (A5)

Equation (A5) may be simplified further by defining theoverlap area of two apertures displaced by y as

= l dxW(x + y/2)W(x - y/2)

2R2 {cos-'(y/2R) - (y/2R)[1 - (y/2R)2 11/2 ), 0 < 2 < y10 , y >2 (M

spheric refractivities at Ni and N2, and used to correct a beamtransmitted at wavelength N2, a residual error will remain ifthe rays follow different paths through the atmosphere. Theresidual OPD at point x is

6(X) = O(X, XI) - 8(X, N2 ), (Al)

where x is the position in the aperture plane, 6(x) is the OPDat x, and 0(x, N) is the optical path length from x to the targetpoint at wavelength N.

Any part of this error that is constant over the aperture willnot affect performance and may be removed. This leaves

'where y = I yl.Changing variables in Eq. (A5) and substituting Eq. (A6)

into it yields

0O0PD2 = D(O, Ni, N2) - 5 dyo(y)['/ 2D(y, N1 , N2)

+ %/2D(y, N2, XI) - D(y, XI, XA)]. (A7)

The structure function of OPD between two rays traversingan atmosphere with Kolmogorov turbulence may be computedas

D(y, N,, X2) = 2.914 secz 5 dhC. 2 (h)Is(y, Xi, N2, h)15/3 ,

(A8)

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Vol. 1, No. 7/July 1984/J. Opt. Soc. Am. A 787

where D (y, N1, N2) is the structure function of OPD betweenrays of wavelengths N1 and N2, y is the separation of rays at the

aperture, z is the zenith angle of the rays, h is the altitude, His the altitude of the target, C 2(h) is the structure parameterfor the index of refraction, and s(y, XI, N2, h) is the separationof rays at altitude h.

Substituting Eq. (A8) into Eq. (A7) gives the integral forthe mean-square error in terms of the separation of therays:

aOPD2 = 2.914 sec z 4 dhCn2 (h) {ls(O x1, X2, hI 5/3

- S dyO(y)[ 1/ 21 s(y, x1, N2, h) 15

/3

+ 1/2 1 s(y, X2, Xi, h) 15/3- s(y, Ni, Xi, h) 15/3]

(A9)

The separation of the rays has two components. First, in the

absence of any atmosphere, the rays would converge uniformlyon the target point. The separation would then be

s1(y, h) = (1 - h/H)y. (A10)

Second, a separation arises because of dispersion in the indexof refraction of the atmosphere. In Ref. 2, the lateral devia-tion of a ray in the vertical plane from its vacuum path wasshown to be

b(h, X) = N (X)sec(z_)tan(z_)[P(h)/gpJ]= bo(X)P(h)/Po, (All)

where b (h, N) is the deviation at altitude h for wavelength X,b (N) = N8 (X)sec(z -)tan(z ) (Po/gs), Ns(X) is the refractivity

of air at sea level for wavelength N, z - is the zenith angle of theray in a vacuum, P(h) is the atmospheric pressure at altitudeh, Po is the atmospheric pressure at sea level (Po = 1.013 X 105

N/M2), g is the acceleration of gravity (g = 9.807 m/sec2 ), and

Ps is the density of air at sea level (Ps = 1.2250 kg/M3 ).Two rays, which, in the absence of the atmosphere, would

be separated by a small distance y in the vertical plane at theaperture and which have a small difference in zenith angle 5z,

will actually have a separation in the atmosphere of

S2(Xl, N2, h) = ly + h'z sec(z-)

+ [bo(N1) - bo(X2)]P(h)/Po~uo, (A12)

where u, is the unit vector in the vertical plane normal to theline of sight. Adjusting y and 5z so that the rays converge atboth the aperture and the target yields

S2 (X1 , N2, h) = [bo(N1) - bo(X2 )]

X {1 - P(h)/Po - [1 - P(H)/Po]h/H~u,. (A13)

The total separation is the sum of the two components:

s(y, XI, N2, h) = s1 (y, h) + s 2 (Nl, N2, h)

= (1 - h/H)yi + [1 - P(h)/P(H)]Y2 , (A14)

where

Yi = [bo(Ni) - bo(N2)][1 -P(H)/Po]u + Y,

Y2 = [bo(Xi) - bo(N2)][P(h)/Po]uv

This expression for separation is used in the integrand of Eq.(A9), allowing the mean-square error to be computed for anyspecified atmospheric model.

REFERENCES

1. J. F. Holmes and V. S. Rao Gudimetla, "Strehl's ratio for a two-wavelength continuously deformable optical adaptive transmitter,"J. Opt. Soc. Am. 73, 1119-1122 (1983).

2. E. P. Wallner, "The effects of atmospheric refraction on com-pensated imaging," Proc. Soc. Photo-Opt. Instrum. Eng. 75,119-125 (1976).

3. E. P. Wallner, "Minimizing atmospheric dispersion effects incompensated imaging," J. Opt. Soc. Am. 67, 407-409 (1977).

4. R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air prop-agation effects relevant to optical communications," Proc. IEEE56, 1523-1545 (1970).

5. National Advisory Committee for Aeronautics, "Manual of theICAO standard atmosphere," NACA Tech. Note 3182, Washing-ton, D.C., 1954.

6. J. C. Owens, "Optical refractive index of air: dependence onpressure, temperature, and composition," Appl. Opt. 6, 51-59(1967).

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