Solar disk sextant optical configuration

Hong-Yee Chiu, Eugene Maier, Kenneth H. Schatten, and Sabatino Sofia

In this paper we evaluate the performance of a plausible configuration for the solar disk sextant, an instru-ment to be used to monitor the solar diameter. Overall system requirements are evaluated, and tolerableuncertainties are obtained. We conclude that by using a beam splitting wedge, a folded optics design canbe used to measure the solar diameter to an accuracy of 10-6, despite the greater aberrations present in suchoptical systems.

1. Introduction

In accompanying articles",2 the underlying principleof an instrument for monitoring the diameter of the sunis proposed. In the proposed experiment, the solar disksextant (SDS), the diameter of the solar disk will bemeasured in the optical continuum where the effects ofsolar surface activity are minimal. The specificationson detector and optics are discussed separately.2 Thepresent work deals with an evaluation of the opticalconfiguration used in the system.

II. Optimum Detector and Optical SystemConfiguration

The SDS consists of three parts as depicted in Fig. 1.A beam splitting wedge (BSW) divides the incomingsolar ray into two components separated by an angle 0which is somewhat greater than the apparent solar an-gular diameter 00. The two component rays are thenfocused by an optical system of equivalent focal lengthF onto a focal plane, forming two solar images of di-ameter D, separated by a distance S = OF betweencenters of the two solar images, as shown in Fig. 2.

To determine the radius of curvature and center ofcurvature of the solar image, at least three points on thecircumference must be measured. A plausible detectorarray arrangement that can achieve this purpose isshown in Fig. 3. In principle, a knowledge of the edgeinformation in either set of detector arrays, (RA,dA,dB,dC,RC) or (RB,dA,dB,dC,RD), is adequate to deter-mine the radius of curvature R. Henceforth the radiusof curvature and the solar radius will be used inter-changeably. In addition, the measurements will be

The authors are with NASA Goddard Space Flight Center, Labo-ratory for Planetary Atmospheres, Solar Radiation Office, Greenbelt,Maryland 20771.

Received 26 September 1983.

used to obtain a value for the solar diameter, defined inEq. (2), as a measure of chord length through the sun'scenter in a particular direction. However, as a differ-ential measurement can often give greater accuracy thanan absolute one, the difference in the location of the twoadjacent solar edges measured by the central arraysdA,dB,dC can determine the minimum distance ofseparation d, between the two solar images, and fromthis the solar diameter. d is related to 0, 00, and F viathe simple relation

d = (0-Oo)F,

yielding 00 in terms of 0 and dIF:

Oo = 0- dIF.

(1)

(2)

Assuming that 0 is strictly a constant, the angular di-ameter of the sun, 0o, is now experimentally measuredin terms of dIF. Thus, the relative experimental errorin the measurement of 00, (00)/Oo, is given by

0° [(D) (FD)J [(d) (F ] D

where D = OoF is the diameter of the sun. This is to becompared with the error in O0 associated with a directmeasurement of D via the relationship D = OoF:

6% = [56Df2 + )2]1/2 .(4)

We find that to achieve the same accuracy in 00, therelative errors (d)/d and (6F)/F may be relaxed by afactor of Did. To be more exact, the major contributionto uncertainties in the above equation will come froman uncertainty in the distance between the nodal pointof the objective to the detector plane, rather than theactual focal length F. Further, because d < D, re-quirements on dimensional stabilities of the focal lengthF and detector plane are reduced by a factor of Did.However, due to eccentricity of the earth's orbit aroundthe sun, the earth-sun distance changes within a period(1 yr) by 3.34% and if we require that the two solar im-

1230 APPLIED OPTICS / Vol. 23, No. 8 / 15 April 1984

I ~~~~~~~~~~D- - - -L...FOCUSING -N 0 _ * TOPTICAL-- - BSW SYSTEM T

SUN BS'.. FOCAL LENGTH F

I U 1 2 SOLAR IMAGES

Fig. 1. SDS optical configuration. The solar rays enter into a beamsplitting wedge (BSW) and are split into two rays which are then fo-cused by a focusing optical system (FOS) into two solar images withminimal separation between the two images. The beam splittingangle 0 is chosen to be greater than the apparent size of the sun 00

so that the two solar images do not overlap.

A B

E, F,

E0 F0

E 2F

Image 1 Image 2

Fig. 2. Geometry of determination of solar diameters. Separationbetween two adjacent edges of the two nonoverlapping solar imagesis measured from EiFi (i = 0,1,2). At least three points are neededto determine the center of curvature and the radius of curvature.

Additional measurement points improve the accuracy.

ages never overlap, a maximum value of Did is 33. Avalue of 20 may not be unrealistic. Thus, an error limitof ADID = 10-6 will imply an error limit of 6d/d = 2 X10-5.

In Fig. 3, the important parameter d characterizingthe distance of separation between two adjacent edgesof the sun is detected by the same detector arrays, thussimplifying the problem of registration and other di-mensional stability problems.

A study of the relationship between entrance apertureof the optical system, the detector element spacing, andthe nodal point to the detector plane distance has beenmade. 2 Using the value of 10 ,m for the detector ele-ment spacing (from commercially available detectorarrays), an optimum configuration is obtained as fol-lows: entrance aperture, 10-20 cm; focal length F,10-20 m. Future refinements are not expected tochange these parameters drastically.

With the above-mentioned system parameters andthe detector configuration depicted in Fig. 3, we will nowstudy the requirements of the optical componentsneeded to build the SDS.

I1. Beam Splitting Wedge (BSW)

After some preliminary studies it was decided to usea wedge-shaped etalon to split the entrance solar raysinto two components. The configuration is shown inFig. 4. The ray configurations are self-explanatory.These emerging rays (Fig. 4 shows three major compo-nents ray 1, ray 2, and ray 3) are then focused by theoptical system to form a series of solar images. Centers

:RB

RD

Fig. 3. Plausible detector array arrangement. Three detector arraysdA, dB, and dC detect adjacent edges of two nonoverlapping solarimages. The outer arrays RA,RB,RC,RD in conjunction withdA,dB,dC determine the approximate solar radius and the centers

of the two solar images.

-RAY 2

- RAY 3

Fig. 4. Multireflection ray paths in the BSW. An incoming ray iseventually split into multicomponents ray 1, ray 2, ray 3, etc. by twoparallel plates E1 and E2 which are at an angle 41 with respect to eachother. Only ray 1 and ray 2 are used to form the solar images in Figs.

2 and 3.

of adjacent images are separated by the angle 0 2c11,where P1 is the wedge angle. If 0 > 00, all images donot overlap.

Let the reflectivities and transmission coefficients ofE1B and E2B be R1,T1 and R2,T2, respectively. A littlealgebra yields the intensities I2,15,18, ... in terms of Ioas follows:

I2 = TlT 210,

15 = RlR 2 TlT 2 IO, (5)

18 = RR 2 TlT 2 IO

Thus, I5/I2 = R1R2 . To reduce exposure time (in orderto avoid image motion during exposure) it might benecessary to adjust R1 nd R2 to maximize overall lighttransmission.

15 April 1984 / Vol. 23, No. 8 / APPLIED OPTICS 1231

E2

Fig. 5. Possible design for the BSW to minimize changes in beamsplitting angle 0. El and E2 are plane-parallel plates, while C andD are cut out from a wedge. If El, E2, C, and D are made from thesame material and if the temperatures of El, E2, C, and D are the

same, the wedge angle Dl should be independent of thetemperature.

N2

13

INI30

N1

A f_

4

x

Fig. 6. Geometry of multireflection within the BSW. N and N2are normals to the surfaces El and E2 (Fig. 5), respectively. Afteremerging from the first etalon plate El, the entrance ray To becomesI, with unaltered direction. 11 is reflected by E2 into 13, such that theangles (I1,N2) and (N2 ,I3 ) are equal. I3 is reflected byE 1 into l4 suchthat the angles ( 3,N1 ) and (N1 ,14 ) are equal. All angles are measuredwith respect to unit vectors in the direction away from reflectingsurfaces in conformity with the usual definition of incidence and exit

angles.

IV. Stability Requirements of the BSW

In the proposed experiment the apparent solar di-ameter is obtained in terms of the wedge angle '1l, plusa small correction term

studied in detail; in the past stabilities >10-7 have beenreported.3 Although there is no reason to believe thatthis problem will upset the SDS project, a study ofthermal structure of the BSW is an essential next stepin this project.

V. Pointing Accuracy Requirement of the BSW

The pointing position of the BSW affects its beamsplitting characteristics. Rotation of the BSW relativeto the image plane causes the two beams R1 and R2 torotate about each other. We assume that rotation ofthe BSW with respect to the detector arrays may beprevented by a rigid mounting. Flexure of the opticalstructure, however, can still take place, changing thepointing direction.

Consider the geometry shown in Fig. 6. N,N 2 arenormals to the reflection surfaces E1 ,E2 shown in Fig.4. Let the entrance ray Io (which after exiting from E1becomes I, which remains parallel to I) coincide withthe Z axis and let N2 be in the X-Z plane. The surfaceE2 reflects I into I3, which is then reflected by thesurface E1 into I4 (the designation of rays are the sameas in Fig. 4). Let the angles be defined as follows:

Il,N2 = 02,

13 ,N1 = , (7)

NlN2 = 4i3 (wedge angle).

The polar coordinates (r,0,0) of the unit vectorsN1,N2,J3,J4 are as follows:

N2 = (1,0,02),

13 = (1,0,202),

N1 = (1,01,4,), (8)

14 = (1,04,04)-

We want to determine the direction of I4 in terms ofother angles. By evaluating scalar products betweenunit vectors we obtain the required relations, and usingsmall angle approximations we find the relationshipsfor the beam splitting angle 04 [which is the same as 0in Eq. (1)]:

04 = Z11,14 2Pl + O(x3 ) (9)

whereOo = O - d/F = 2 - d/F. (6)

It is essential to maintain a constancy in i or to be ableto monitor any variations to or beyond the target ac-curacy of the solar diameter measurement. Thus, witha target relative accuracy of 1/106, it is necessary tomaintain the wedge angle to an accuracy of at least 1/106. It is possible to achieve this accuracy by using awedge made of a single material so that, in isothermalconditions, the wedge will expand or contract homolo-gously resulting in no change of shape, thus preservingthe wedge angle. A number of low expansion opticalmaterials, such as quartz, have the desired propertieswith relatively relaxed requirements on thermal equal-ization (to 1C). A possible design is shown in Fig. 5.Dimensional stability of optical materials has been

O(x3 ) = a1O1t14, (10)

and a is a numerical factor -1. That is, variations inthe pointing direction will generate second-order er-rors.

In order that the two solar images will remain fixedin the focal plane when the telescope rotates with re-spect to its optical axis, I should be in the N1,N2 planeand at equal angles to N1 and N2. Departure from thisgeometry generates errors of the third order. Let 0be the error in 0 due to pointing errors, then

5O 0204

= 0al e '(11)

since01 0, the average misalignment of angles mea-sured in terms of 02 and 4 is of the order of

1232 APPLIED OPTICS / Vol. 23, No. 8 / 15 April 1984

0202¢4. (12)0

If (0/0 10-6, both 02 and 04 must be <10-3 or -3 minof arc. Thus the tolerable error in the pointing anglesof the BSW is several minutes of arc, an easily attainablegoal.

VI. Optical System Requirements

While reduction in stability may be compensated forby the use of a wedge design as described earlier, limi-tations of field of view reduce the quality of data.Generally speaking, one must consider off-axis imageaberration. In our case, off-axis aberration is most se-rious in the outer detector arrays whose chief functionis to determine the location of the center of the solarimages. The central arrays, which are used to deter-mine d, suffer rKelatively small off-axis aberration.Effects of image aberration on data reduction will bestudied when a final optical design is available. Wemust know, however, the tolerance limit on image dis-tortion in order to be able to reach the target accu-racy.

A. Tolerable Error in Outer Detectors

According to the detector configuration depicted inFig. 3, the central region of the detector plane containingthree detector arrays, dA,dB,dC, is chiefly responsiblefor the accurate determination of d; in this region theoptical distortion is likely to be low. The four outerarrays, RA, RB, RC, and RD, are used to determine thelocation of two solar images. In the detector geometryshown, the main error in the determination of the lo-cations of the centers of images occurs in the directionperpendicular to the central arrays. Let AD be such anerror; then the error 5dp (due to positional uncertainty)in the measurement of D is

6d, = D 15fl2 (13)

The diameter D is essentially measured from thequantity Ay (distance of the chord to the solar edge inFig. 3) via the relation

1/2D(1 - cosn) = Ay. (14)

Thus bAy/Ay = AD/D and Eq. (13) becomes

bdp=Dt {A/y)2 (15)d d NY Ay)

and the tolerance limit on ((3Ay/Ay)i is

16Ay) =d dP~1/2. (16)

As an example, let d/D = 1/20, bdp/d = 10-6, then(bIAy)/lAy = 2.2 X 10-4. If n = 300, then Ay = 0.07D.If Did = 20, then Ay d. The limit for bd/d wasplaced to be 2 X 10-5 for Did = 20. Thus the tolerableerror in the measurement of Ay is ten times that for(d.

B. Tolerance Limit Due to Geometrical Aberrations ofImages

Any optical system will generate aberrations to acertain extent. Let us define the geometrical image asthe projection of the original object through a pinholeof infinitesimal size. The optical image formed at thefocal plane will differ from the geometrical image.Assuming the absence of astigmatism, the geometricalimage can be linked to the optical image through a 1-Dmapping function. In the absence of astigmatism, themapping function is symmetrical around the optical axisin the focal plane. Let R be the radial coordinate (fromthe optical axis) of a geometrical point and let r be theradial coordinate in the focal plane measured from theoptical axis. The mapping function can be written as

R = r[l + a(r)]. (17)

We will disregard the trivial case a (r) = constant,hence the first significant term in a (r) is kr where k isa constant.

An uncertainty in a(r), ba(d), will then produce anuncertainty in d and D. This gives

bd = dba(d). (18)

If (did < 2 X 10-5, then (a(d) < 2 X 10-5 in the vicinityof d to be measured. Because the tolerable error of Ayis approximately ten times that for d, the tolerable errorin Oa(d) at RA, RB, RC, and RD is ba(d) < 2 X 10-4.

In addition to this uncertainty, a lack of knowledgeof the exact location of the optical axis with respect tothe images will introduce an error in application of Eq.(17) even if a(r) is exactly and precisely known. Ingeneral if the two solar images are aligned symmetricallywith respect to the optical center, this type of error isof the second or higher order because the lowest-orderterm in a (r) is kr. Two cases will be considered: (1)On-axis case: the resulting uncertainty when the linejoining two points passes through the optical axis. (2)Off-axis case: the uncertainty when the line joining twopoints does not pass through the optical axis. We as-sume that the two solar images are symmetrically placedwith respect to the optical axis so that only the uncer-tainty in the location of the optical axis will be consid-ered.

(1) On-axis case: Consider two optical image pointsrl,r 2 corresponding to two geometrical image pointsR1,R 2 (Fig. 7) along a line passing through the opticalaxis 0. From Eq. (17) we find

R1 + R2 = rl(l + kri) + r2(1 + kr2). (19)

An uncertainty br1 in the position of the optical axisalong the line r1r2 will introduce an error b(r1 + r2) givenby

6(r1 + r2) = -kb(r2 + r2)= -2h[(ri - r2)br + br21.

We assume that the measurement is made symmetri-cally around the optical axis, hence r = r 2. Then

6d = (r, + r2) -2k~r 2 . (21)

(20)

15 April 1984 / Vol. 23, No. 8 / APPLIED OPTICS 1233

R3

8x I I

Fig. 7. Schematics showing optical distortion. R, R 2, R 3, and R 4are undistorted (geometrical) image points in the focal plane, which

through distortion become r, r2, r3, and r4.

As a numerical example let us consider the followingcase:

k = 0.01/ro (1% aberration at ro),

r = 10 cm (1/2 field of view),

D = 10 cm, d = 0.5 cm, Ad/d = 2 X 10-5.

Then the tolerance limit on br is

br - = 007 cm. (22)V 2k

(2) Off-axis case: Let R3 and R 4 be the two geomet-rical image points at equal distance to the optical axis,let r3 and r4 be the corresponding optical image points.Let X 3,X 4 and X3,X4 be the distances of the lines joiningR3,R4 and r3,r4 to the optical axis, and let y3 + y4 = r3r4and Y3 + Y4 = R3R4. Let the optical axis be subjectedto an uncertainty x in the direction along X or x.Using Eq. (17) and trigonometric relations we find

-6(W + Y4) - -k X6X . (23)Y3 + Y4 r3

This will generate an uncertainty in the measurementof d via the relationship Eq. (15), so that

5d D 'X XX2

d d lr3 J (24)Using the same numerical example as in the on-axis

case and with r3 = 10 cm, y + y = 1.5 cm, we find thetolerance limit as

6x = 1 cm, (25)

which is a rather lax condition.To summarize: if k = 0.01 at the edge of the field, the

most stringent limit in the optical design calls for aknowledge of the distortion function to an accuracy of2 X 10-5 in the vicinity of d and to 2 X 10-4 at the outerdetectors. To achieve the desired accuracy, the imagemust be centered to 0.07 cm with respect to the centraldetector arrays. On account of the close spacing of thedetector arrays, this may be easily achieved. The ver-tical alignment requirement is quite relaxed, being 1cm.

On the other hand, the wedge angle should be stableto within 10-6. Studies of the stability of the shape ofthe wedge should be the next step in the developmentof this experiment.

In the above examples it was assumed that the maincontribution comes from each individual term underconsideration. If contribution from all terms is thesame, all the above limits for first-order terms must bereduced by an additional factor of 6 _ 2.5, and sec-ond-order terms by an additional factor of i6 _ 1.6.

VII. Discussion

We conclude that the use of the beam splitting wedge(BSW) is a viable concept in measuring the solar di-ameter. Effectively, the use of a BSW reduces thesystem requirement by a factor of dD, where d is theshortest distance between the two solar images and Dis the diameter of the solar image. Allowing a 3.3%annual change in the apparent solar image, dD may bemade as small as, say, 1/20. Thus, to reach a targetaccuracy of D/D 10-6, FIF must be measured onlyto a fraction 2 X 10-5 instead of 10-6. If several opticalsystems with different BSW angles are used, systemrequirements can be even less stringent. However, inall cases, dimensional stability of the BSW is of utmostimportance.

Optical distortions will generally introduce second-order effects if the distortion function is known. Errorsthat arise in the beam splitting angle 0 due to pointinguncertainties are much smaller than other sources.

The effect of spatial dependence of the point spreadfunction is more involved; it has the direct effect ofmodifying the FFTD (finite Fourier transform defini-tion) integral.4 This effect is most easily evaluatedexperimentally when instruments are available.

References1. S. Sofia, H.-Y. Chiu, E. Maier, K. H. Schatten, P. Minott, and A.

S. Endal, Appl. Opt. 23, 1235 (1984).2. H. Y. Chiu, Appl. Opt. 23,1226 (1984).3. J. W. Berthold III and S. F. Jacobs, Appl. Opt. 15, 2344 (1976).4. H. A. Hill, R. T. Stebbins, and J. R. Oleson, Astrophys. J. 200, 484

(1975).

1234 APPLIED OPTICS / Vol. 23, No. 8 / 15 April 1984