Aspheric mirror withconstant angular magnification II.

Zeljko Andrei6 and Nikola Radi6

Differential equations for a constant-angular-magnification aspheric-mirror surface shape are derived fora general dependence of the camera image height on the camera field angle. The explicit equations of theconstant-angular-magnification mirror surface are given for some particular values of the angularmagnification. For a series of odd integer values of the angular magnification, 10th-order polynomialapproximations of the mirror surface are presented. The imaging performance of such a mirror with anangular magnification of 7 is analyzed and compared with a spherical mirror. The main cause of imageblur in all-sky cameras at the edge of the field of view was found to be a strong image curvature. We showthat increasing the camera-to-mirror distance and/or stopping down the camera lens reduces the imageblur.

Key words: Mirrors, aspheric mirrors, all-sky camera.

IntroductionIn our previous papers' 2 we described asphericalmirrors with a constant angular magnification over a1800 or larger field of view. Such a mirror wasmeant to be part of an inexpensive all-sky camerawhose other part would be an ordinary camera thatphotographs the virtual image of the sky formed bythe mirror (see Fig. 1). Similar all-sky cameras withconvex spherical mirrors are well known in the fieldof meteor photography. The advantage of anaspheric mirror over the spherical mirror in thisapplication is that the angular magnification of thecamera is constant over the field of view, whichsimplifies image analysis. In our previous papers,equations for the surface of a constant-angular-magnification aspheric mirror were found with theassumption that the approximation F tan(a) F isvalid for the camera objective (see Fig. 2 for notation).This approximation was shown to be true within atolerance of 1% for normal and telephoto lenses thatwould normally be used in the construction of such acamera. The tolerance of 1% was derived from thefact that many mass-produced photographic lensesshow residual distortion of this order of magnitude.

The authors are with the Department of Materials Research andElectronics, Ruder Boskovbi Institute, Bijenicka 54, 41000 Zagreb,Croatia.

Received 14 June 1993; revised manuscript received 12 October1993.

0003-6935/94/194179-05$06.00/0.© 1994 Optical Society of America.

In this paper, equations for an aspherical mirrorsurface are derived for any case in which image heightcan be described by an analytical function of the form

h = g(a), (1)

where h is the image height measured from theoptical axis in the radial direction in the image planeand a is the object angle, again measured from theoptical axis. This general equation is then used tofind the differential equation of the mirror surface forthe case of a distortion-free camera lens that strictlysatisfies

h = F tan(a), (2)

where F is the focal length of the lens. Also, it isshown that implicit equations for the mirror-surfaceshape 2 can be converted into explicit equations for afew discrete values of angular magnification M. Theadvantage of an explicit surface equation is that it canbe used to ray trace the mirror with a ray-tracingprogram and to evaluate its imaging performance.Last, the spot diagrams for a mirror with M = 7 aregenerated in this way, and the performance of aspherical, a parabolic (an approximation of a constant-angular-magnification mirror 2), and a true constant-angular-magnification mirror are discussed and com-pared with one another.

Mirror Shape in a General CaseWe start the derivation of the differential equation ofthe mirror surface with the assumption that the

1 July 1994 / Vol. 33, No. 19 / APPLIED OPTICS 4179

E over the whole field of view. When the procedureused in the derivation of the differential equation ofthe constant-angular-magnification mirror is fol-lowed,' it is not difficult to prove that the differential

| s equation of the mirror surface when Eqs. (1) and (3)hold is

dy -Mg(a) - ar= tan 2

WFig. 1. All-sky camera imaging the complete celestial hemisphereabove the horizon. During the imaging process the observer'sspherical coordinate system is transformed into a plane polarcoordinate system. In the imaging process the azimuth AZ is leftintact and is equal to the polar angle t, but the zenith distance ZDis transformed into radius vector p. The idea behind this researchis to produce a camera that would satisfy ZD = constant x p, whichrequires use of an aspheric mirror whose properties are discussedbelow.

dependence of the image height on the object anglecan be described by an analytical function of the formgiven by Eq. (1). The geometry of the problem isdefined in Fig. 2. We also must recall that thecamera reimages the virtual image of the sky pro-duced by the aspheric mirror onto the film plane andthat we want to achieve a proportionality of the imageheight h and the object angle 13 in the object space ofthe mirror:

1h=T

mirror . -.virtualimage

(3)

y

camera

with

a = -arctan(y _ d)

Putting g(a) = a produces the equationconstant-angular-magnification mirror:

dy M-1 x \1d =-tan 2 arctan d y

as expected. The distortion-free lens satisfies Eq. (2)strictly. By putting g(a) = tan(a) and by using Eq.(5), we transform Eq. (4) into

dy Mxdx= _-tan 2d-y

For most functions g(a) Eqs. (4) and (5) must besolved numerically.

Explicit Form of the Mirror-Surface Equation for SomeDiscrete MIt has been shown2 that Eq. (6) has an analyticalsolution given by

d sin(a)x(a) = cos[(N + 1)at]}1(N+

d cos(a)y(a) = d - {cos[(N + )ot]ll(N+

for which the absolute value of the variable a exists inthe interval [0, r/(M + 1)].

However, to be able to ray trace the asphericsurface shape, the existing optical ray-tracing pro-grams require an explicit polynomial equation of theaspheric surface shape. Such a solution cannot beprovided for the general case of Eqs. (8). For onlysome particular values of the angular magnificationM can Eqs. (8) be rewritten in a closed form or, moregenerally, as a very good power-series approximation.To this end we first note that, for the odd integervalues of M, Eqs. (8) can be rewritten as an implicitrelation in Cartesian coordinates:

TN+ 1[X2 + (1 - Y)2]1/2l =

Fig. 2. Geometry of an all-sky camera. An ordinary photo-graphic camera looks down onto a convex mirror and photographsthe virtual image of the sky produced by the mirror. All signs arepositive as shown.

1

[X2 + (1 - y)2](N+1)/2

Here X = x/d and Y = y/d are the reduced Cartesiancoordinates, parameter N = (M - 1)/2 and Tn(x) isthe Chebyshev polynomial of the first kind. When

4180 APPLIED OPTICS / Vol. 33, No. 19 / 1 July 1994

ZENITH

NSV

OBSERVER (4)

(5)

of the

(6)

- arctantd _Y)] (7)

d

(8)

(9)

T0(x) is written explicitly, Eq. (9) yields an algebraicequation of the (N + 1)/2nd order for the (1 - Y)2function. For only M = 3 and M = 7 are thesolutions of this implicit equation given by simpleequations:

M= 3, Y= 1 - (1 + X2)1/2, (10)

M= 7, Y = 1 - [(1 + 8X4 )1 /2 + 3X 2]1 /2 . (11)

For use in optical design programs, the above andother odd M solutions are conveniently approximatedby the 10th-order polynomials of X:

/1 2 X4 5X6 3s\M=3, 1 + =5xX)M = 3 Y= - 2- 8 +16 128 1286)

+ 0(X12), (12)

M=5, Y =X2 1- + - + owl2b (13)

3 7X 2 19X4 203X6 3381X 8\M= 7, Y= -X\t 2 + 8 16 128 + 256 )

+ 0(X12 ), (14)

M = 9, Y= -X2(2 + 3X2 21X6 - 22X)

+ 0(X12), (15)

1, _X2 5 55X2 583X4 158977X6 \2+ 8 48 384

+ 0(X1 0), (16)

where O(Xn) is a higher-order polynomial residual.

Imaging Performance of an All-Sky Camera

We have used the explicit equation of the mirrorsurface for M = 7 [Eqs. (14)] to ray trace the mirrorand to analyze its performance. We assume animaging lens with a 50-mm focal length and a speed ofF/2 placed 1 m above the mirror surface. We haveray traced the mirror in these conditions and pro-duced spot diagrams of the mirror's virtual image.The linear scale on the graphs corresponds to themirror focal plane. The dimensions of the corre-sponding spots produced by the camera in the filmplane are reduced by the ratio of the distance of thefilm plane from the camera lens to the distance of thecamera lens from the virtual image produced by themirror. The former distance is slightly greater thanthe focal length of the camera lens since the virtualimage of the sky is fairly close to the camera. In ourcase spots on the film are 22.4 times smaller thanthose in the virtual image.

The spot diagrams are calculated for the best-fitcurved (spherical) focal plane, as found automaticallyby the ray-tracing program. They are given for aspherical mirror (Fig. 3), a constant-angular-magnifi-cation aspheric mirror (Fig. 4), and a parabolic mirror(Fig. 5). Unfortunately, the ray-tracing programwas not able to ray trace the mirrors when the imageangle approached 90°, so the spot diagrams are givenfor the largest object angle possible.

Finally, we traced the shape of the surface of thecircles of least confusion by observing spot diagramsfor various positions of the image plane (defocusing).The results of this analysis are in Fig. 6.

The most interesting fact is that the best imageplane is more strongly curved for an aspheric mirrorthan for the corresponding spherical mirror. Sincethe required depth of the field of the camera is

FILE ALLSKY7S ALL-SKY CAMERA M=7 04-21-19930.50008mm De=5.080 Inc=5.8000mm

-5.0000 0.0008 5.8000 10.0000 15.0000

~E At I AL 6

., .........

Fig. 3. Spot diagrams of a spherical mirror with paraxial regionM =7. They-axis parameter is an object angle (in degrees), and thex-axisparameter is defocused (in millimeters) from the spherical image surface (R = 330.3 mm).

1 July 1994 / Vol. 33, No. 19 / APPLIED OPTICS 4181

FILE ALLSXY7 ALL-SKY CAMERA M=7 04-21-19938.5868mm Def=x10.000 Inc=5.0008mn

0.0000 5.8888 18.0000 15.0000 28.0000

9 * AD Hi; .. .. ..!i' .!.' t. so~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~......

s~~~~~~~~~~~~i & , W*

X (§"~\R~lll}R*;$ @8||1]2;9; .11 'P t -.11 15

Fig. 4. Spot diagrams of a constant-angular-magnification aspheric mirror with M = 7. The y-axis parameter is an object angle (indegrees), and the x-axis parameter is defocused (in millimeters) from the spherical image surface (R = 330.3 mm).

considerable in any case, it is obvious that manycompact designs of all-sky cameras failed because ofthis fact alone. The mirror aberrations (almost pureastigmatism at large field angles) are quite unimpor-tant because of the large reduction in spot size whenthe virtual image onto the camera film plane isreimaged. Thus in our case the worst spot sizeobserved at the edge of the field of a spherical mirroris 0.55 mm in diameter. The reimaging to the filmplane reduces this diameter to 25 [um. Note that

spot sizes of fast 50-mm lenses are usually larger than50 pm.

The spot shape on the surface of the best image isroughly spherical. The camera lens will add its ownaberration to this shape. Since there are literallythousands of various 50-mm lens designs, it is impos-sible to find a typical design and to use it in theray-trace analysis. Thus it is left to the interestedreaders to make such an analysis. However, we donot expect a large increase in the spot sizes. Recent

FILE ALLSXY7A ALL-SKY CAMERA M=7 04-21-1993| s 0.50680mm Def=x18.808 Inc=5.0800mm

0.0000 5.0000 18.0000 15.0000 20.0000

~~~~~~~ :-lr. ' Al N

~~~~~~ I~~~~~~~~"

Fig. 5. Spot diagrams of a parabolic mirror with paraxial region M = 7. The y-axis parameter is an object angle (in degrees), and thex-axis parameter is defocused (in millimeters) from the spherical image surface (R = 330.3 mm).

4182 APPLIED OPTICS / Vol. 33, No. 19 / 1 July 1994

1000-

S

P

A

20 40 60Object angle (deg )

Fig. 6. Shape of the surface of circles of least confusion deter-mined from spot diagrams. It can be seen that the required depthof field of the camera is larger for aspheric mirrors. The camera-to-mirror distance was 1000 mm in this case, and the angularmagnification was M = 7. S, spherical mirror; P, parabolicmirror; A, aspheric mirror.

tests3 performed by photographing actual star fieldsthrough some common 50-mm lenses available on theAmerican market show that one can remove anyresidual lens aberrations by closing the lens to F/2.8or F/4 at most. Also, the all-sky camera does notrequire a high image resolution to be used successfully.Terminal flares of bright meteors, i.e., those thatsuch a camera can record, produce overexposed traceson film that can be more than a millimeter thick.

Figure 7 permits us to compare the depth of thefield of the camera lens (a 50-mm focal-length lens issupposed) with the depth of the field required by thevirtual image curvature. The smallest camera-to-mirror distance can easily be determined from thisgraph. For an aspheric mirror, note that the dis-tance drops from more than 2000 mm for the lens

_ 800-EE

600-

0._!

400*0

° 200a)

5.6

4

2.8

2

200 600 1000 1400 1800Camera to mirror distance (mm )

Fig. 7. Depth of field of a 50-mm focal-length camera lens as afunction of the camera-to-mirror distance and the lens aperture(solid curves). A blur circle of 50 plm was used in the calculations.The depth of field required for an all-sky camera is indicated by thedashed curves for a spherical (), parabolic (P), and aspheric (A)mirror (M = 7).

aperture of F/2 to 900 mm for F/5.6. A sphericalmirror permits a more compact camera since thecorresponding distance is in the range from 1600 to600 mm as the lens is closed down. However, theangular distortions introduced by the spherical mir-ror are then present.

References1. Z. Andreid, "Simple 1800 field of view F-theta all-sky camera,"

in Innovative Optics and Phase Conjugate Optics, R. Ahlers andT. T. Tschudi, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1500,293-304 (1991).

2. N. Radid and Z. Andreid, "Aspheric mirror with constantangular magnification," Appl. Opt. 31, 5915-5917 (1992).

3. A. Dyer, "Seeking the best 35 mm camera," Astronomy 21,74-79 (1993).

1 July 1994 / Vol. 33, No. 19 / APPLIED OPTICS 4183

EE

0

0

0~

U)

a)

c