Minimizing atmospheric dispersion effects in compensated imagingEdward P. Wallner

Itek Corporation, Lexington, Massachusetts 02173(Received 13 August 1976) -

A compensated imaging system that corrects optical path length distortions due to atmospheric turbulence bymeans of an achromatic corrector will have residual errors caused by the dispersion of the atmosphere. Theseerrors become significant for astronomical objects at large zenith angles, but they may be minimized by specialdispersion correctors.

The variation of the index of refraction of the atmo-sphere with wavelength affects the ability to compen-sate for the effects of atmospheric turbulence in twoways. First, rays of different wavelength traversingthe same atmospheric path will suffer optical pathlength distortions proportional to the refractivity of airat the respective wavelengths. If the active opticsused to compensate the wave front are achromatic,e. g., a deformable mirror, the compensation can beperfect only at a single wavelength leaving a residualchromatic error. Second, rays of light at differentwavelengths will, in general, traverse different atmo-spheric paths and, therefore, suffer randomly differ-ent optical path length distortions. Again, perfectcompensation is impossible with a simple corrector.The magnitude and minimization of these errors istreated here (Fig. 1).

CHROMATIC ERROR

The composition of air in the tropopause is fixed andtherefore its refractivity is proportional to density(Ref. 1, p. 88):

N(X) = n(X) - 1 =Ns(X)p/p8 , (1)

where N(X) is the refractivity at wavelength X, n(X) theindex of refraction, and p is the density, and subscripts refers to standard temperature and pressure.

The form of (1) makes the analysis of dispersionrelatively simple and leads to readily evaluated coef-ficients for the residual errors involved.

In order to minimize the residual chromatic errordue to dispersion, wave-front compensation should becorrect at the mean wavelength X0 defined by

N(xo) = f dXI(X)N(X) , (2)

where X0 is the weighted mean wavelength, and I(X) isthe effective intensity distribution of the source, normal-ized to f dXI()k) = 1.

Analysis of the residual chromatic error leads to themean-square error:

a ch = a 2 (N2(oX) f dXI(X)N 2 (X) - 1) , (3)0 J O A V 19

407 J. Opt. Soc. Am., Vol. 67, No. 3, March 1977

where a2h is the mean-square chromatic wave-fronterror due to atmospheric dispersion, and a2 is themean-square wave-front error for uncorrected atmo-sphere.

For Kolmogorov turbulence characterized by the co-herence length ro and a circular aperture, the root-mean-square uncorrected error is

auc=0 0162 (D/ro)51 6 (waves), (4)

where D is the aperture diameter and ro is the atmo-spheric turbulence coherence length.

The coefficient of Eq. (3) was evaluated numericallyfor the intensity distribution from a 5500 K black-bodyradiation transmitted through 1 atm and detected by anS-20 photosurface (see Ref. 2). The resulting coeffi-cient for the error variance is 1. Ox 10-4. The rmschromatic error for a 1 m telescope is, therefore,only 510 wave at the rather strong turbulence of ro = 0. 05M. This error source can, therefore, usually be ne-glected.

REFRACTION-INDUCED ERROR

The second error term arises because rays enteringthe atmospheric other than vertically are bent by atmo-spheric refraction. Dispersion then causes rays which

UNREFRACTED RAYS ATTOP OF ATMOSPHERE

X2 > X1

REFRACTED RAYS

\ ?k2 TELESCOPE APERTURE

LATERAL SHIFTAbo

CORRECTED SHIFTCAbo

IMAGE POINT

FOCAL PLANE

FIG. 1. Ray paths through atmosphere and image compensa-tion system.

Copyright © 1977 by the Optical Society of America 407

5.0

- -̂ iWITHTROPOPAUSE

- ICAO STrANDARD \ /ATMOSPHERE L/

U.

X -SEA LEV EL SITE *ro = 10 CENTIMETE RS \

0.5 l I I0 0.2 0.4 0.6 0.8 1.0

FRACTIONAL DISPERSIVE SHIFT REMAINING (C)

FIG. 2. Turbulence factor in dispersion error.

are coincident at the top of the atmosphere to enter theaperture at different points. (They also enter at dif-ferent angles so that an angular dispersion correctionmust also be made to remove chromatic error fromthe image. This is a standard correction, separatefrom the phase correction treated here.)

The lateral displacement of a ray at the aperture asa function of wavelength is derived in Ref. 2, Eq. (10)as

Ab, = [N,(X) - N5 (X')] sect,. tang, (Po/gp5 ) , (5)

where Abo is the lateral displacement between rays ofwavelength X and X' at aperture, .,, is the true zenithangle of object, P0 the atmospheric pressure at alti-tude of telescope, g the acceleration of gravity, and psis the density of air at standard conditions.

For a horizontally stratified atmosphere, this dis-placement will lie in a vertical plane.

If the adaptive optical system simply makes the bestpossible achromatic phase correction to rays of allcolors, there will remain a residual mean-squarephase error of

(E2 ) =' sec"8 3 e.. tan,5/ 3 t.(P 0/gp5 )5 /3

x f d x d AI (X) I (X ) I N(X) -N(X')|

x 2.dh 921 C2(h)I 1 -P(h)/PoI 51 3 (6)

where h is the altitude above telescope, P(h) the pres-sure at altitude h, X0 the reference wavelength, andCn(h) is the refractive index structure parameter (m213 )[See Ref. 2, Eq. (23). J

For a sea level site and a standard atmosphere, theP0 term in (6) is 3.49 x 106 m5/. For an S-20

408 J. Opt. Soc. Am., Vol. 67, No. 3, March 1977

photo surface detecting a 55000 black body through1 atm and a filter cutting off sharply at 0. 4 and0.75 plm, the spectral integral in (6) has the value4. 19 x 10-10.

The altitude integration in (6) has been evaluated fortwo model turbulence distributions, each correspond-ing to an r0 of 10 cm. For an exponential turbulencedistribution with a scale height of 3. 2 km, the integralis 1.30 waves2 m- 1/3 at a reference wavelength of 0.55gim. This leads to an rms error of 0. 049 waves at 450zenith angle and 0.12 waves at 600 zenith angle.

If the low-altitude turbulence is reduced by half anda tropopause layer at 15 km with equal total strength isadded, the integral has a value of 3.95 waves2 m5/3 andthe corresponding errors become 0.085 and 0. 21 waves.This error can therefore limit system performance atlarge zenith angles.

MINIMIZING REFRACTION-INDUCED ERROR

A way to reduce this error can be derived by con-sidering two extreme cases. If all of the turbulencewere concentrated in a thin layer at the aperture, raysof different wavelength entering the aperture at thesame point would suffer the same optical path lengthdistortion (except for the trivial difference discussedpreviously under chromatic error) and the correctionwould be perfect. Substitution of C2(h) =C 6(h,) yields0 for the last integral in Eq. (6) in agreement with thisresult.

On the other hand. consider the case in which all ofthe turbulence is concentrated in a thin layer at theupper limit of the atmosphere before atmospheric re-fraction has caused rays of different color to diverge.Here rays of different color entering the aperture at aseparation Abo given by Eq. (4) will have the same op-tical path length distortion. If these rays were shiftedback into coincidence before compensation, by meansof a dispersion corrector which shifted them laterallyaccording to Eq. (5), the compensation would again beperfect. In this case, C2(h) =C,6(oo) and in order forthe last integral in (6) to vanish, its last argumentmust be changed to I P(h)/P01 .

In general the turbulence is distributed in altitudebetween these limits and one expects the optimum per-formance to occur with a lateral dispersion correctorwhich reduces the displacement of originally coincidentrays to some intermediate fraction C of the distanceAbo. When this is done the argument in (6) becomesI C - P(h)/P1l , where C is a free parameter whichmay be used to minimize the error.

Figure 2 shows how the value of this integral varieswith the parameter C for the atmospheric models de-scribed above. By selecting the dispersive shift of thewave front properly, the mean-square error can be re-duced by a factor of 0. 44 for the case of low-altitudeturbulence only, and by a factor of 0. 32 for the casewith a tropopause layer. This reduces the rms errorsat 600 zenith angle to 0.08 and 0.12 wave rms,

Edward P. Wallner 408

respectively, which is consistent with high-qualityimaging.

CONCLUSION

The error due to atmospheric dispersion on verticalpaths is generally negligible. That at large zenithangles may limit system performance, but can be mini-mized by making the proper dispersive lateral shift of

the incident wave fronts before optical path length com-pensation.

'M. Born and E. Wolf, Principles of Optics, 3rd ed. (McGraw-Hill, New York, 1972).

2E. Wallner, "The Effects of Atmospheric Dispersion on Com-pensated Imaging, " Proceedings of the SPIE/SPSE TechnicalSymposium East, Vol. 75, No. 19, 1976.

409 J. Opt. Soc. Am., Vol. 67, No. 3, March 1977 Edward P. Wallner 409