Analysis of He-Ne Laser Surface Reflectionsfrom an Off-Axis Parabolic MirrorJames D. Evans

Surface reflections from an off-axis parabolic mirror are analyzed in three dimensions applying the tech-niques of vector analysis and ray tracing. A He-Ne laser is directed onto the mirror surface and heldfixed while the parabolic mirror is rotated through an angle of 3600 on an axis that is parallel to that ofthe original paraboloid. The reflection from the mirror traces a closed figure in a plane perpendicular tothe paraboloidal axis when the mirror undergoes one complete revolution about its center axis. Theshape of this figure depends on the value of the off-axis distance and the focal length; consequently, thisfigure yields information about the parameters specified in the construction of off-axis parabolic mirrors.

1. Introduction

The off-axis parabolic reflector' is used in ir opti-cal systems that require no obstruction of the inci-dent collimated signal. In an optical informationsystem, for example, placing a detector at the focalpoint of an on-axis mirror will alter the signal in anunknown fashion before it can be detected. Toeliminate this obscuration, the off-axis parabolicmirror is used that has the property that its focalpoint is completely removed from the path of the in-coming radiation. However, the use of the off-axisparabolic mirror requires special alignment proce-dures. Generally, a visible light source and an opti-cal collimator are used so that collimated light willbe shone onto the entirety of the mirror surface.The mirror is then oriented until an acceptablefocus is obtained with the reflected rays; this indi-cates the radiation is parallel to the parabolic axis,and the mirror is ready for use in an optical system.The focal length and off-axis distance may be direct-ly measured when these alignment conditions aremet.

In this paper, however, the alignment2 of the off-axis parabolic mirror is achieved by using only aHe-Ne laser rather than an optical collimator. Anexperimental technique is given whereby the experi-menter can go through successive translations of themirror until alignment is achieved. This alignedcondition yields experimental estimates of the off-axis distance and focal length.

The author is with Laser Window Group, Air Force MaterialsLaboratory, Air Force Systems Command, Wright-PattersonAFB, Ohio 45433.

Received 24 January 1972.

After alignment is attained, a second method fordetermining the off-axis distance and focal lengthcan be applied that requires that the mirror be suc-cessively translated and rotated. Applying the tech-niques of vector analysis and ray tracing to thissetup yields some useful mathematical relationships.Rotating the mirror 360° on an axis through its cen-ter which is parallel to the paraboloidal axis and bynoting characteristics of the closed figure that isformed by the reflected laser beam, one can obtainestimates of the off-axis distance and focal length si-multaneously.

Hence, this technique allows the mirror user tocheck the optical manufacturer's specifications forthe off-axis distance and focal length should thesevalues become questionable. The equations derivedalso guide the person aligning the mirror towardproper alignment since certain conditions must holdwhen proper alignment is reached.

II. Theory

A. Alignment by Mirror Translation (Two-Dimensional Analysis)

The experimental setup for obtaining mirror align-ment is shown in Fig. 1. The He-Ne laser is placeda distance K from the mirror surface and is directedonto this surface, thus producing a reflection oflength S on the screen. The value of S dependsupon the angle of misalignment . As the angle ofmisalignment approaches zero, one can set up the re-lationships,

S/[K - (d2/4F)] = (S - d)/(K - F), (1)

which if rewritten is

(F/4)(d/F) - (S4)(d/F) - K(d/F) + S = 0. (2)

By conducting the experiment so that d < F and

212 APPLIED OPTICS / Vol. 12, No. 2 / February 1973

K

Fig. 1. Experimental setup for obtaining mirror alignment (two-dimensional analysis).

K > S >> F, we can safely drop the cubed term.Solving the remaining equation for d/F gives

which I is reflected upon mirror rotation 0 and aredefined as

a = B sinO,

b = A + D/2 + B cosO.

For this analysis, the laser beam is confined to strikethe surface constrained by the equations

a + b = r

Z = (a + b2 )/4F.

The vector I, formulated by the points P and Ps,lies in the Y-Z plane and is

I = i(O) + (O) + c[K - (a2 + b2)/4F],

where i, j, and k are the unit vectors in rectangularCartesian coordinates.

The radius of curvature vector R originates at Psand is dependent only upon the value of r (since theparaboloid is a surface of revolution) given by Eq.(7). Hence, 4

IRI = 2F(1 + r2/4F2)21 2,

d/F = 2(K/S) 2 + 11'/ - K/St. (3)

From Fig. 1, the off-axis distance is denoted by A.Hence, one can write

d = A + a, (4)

where is the distance from the edge that the laserbeam strikes the mirror. Substituting this value ofd yields

q(w) = A/F + wF = 2[(K/S)2 + 111/2 - K/Sj. (5)

Hence, if corresponding values of w and S are mea-sured as the mirror is translated, a straight line plotwill occur when the mirror is aligned.

B. Alternate Determination of F and A (Three-

Dimensional Analysis)

The vector I in Fig. 2 represents the incidentHe-Ne laser beam that is shone onto the mirror sur-face. It is assumed that this beam is parallel to theparabolic axis (in this case the Z axis). The experi-mental setup yields the capability of mirror rotationabout the X and Y axes; hence, by proper mirrorrotation, the incoming beam may be made parallelto the parabolic axis. The focal length, off-axis dis-tance, and mirror diameter are represented by thequantities F, A, and D, respectively. The point P isthe originating point of the visible radiation and hasthe coordinates P1 = P[O,(A + D/2 + B),K] and liesin the plane Z = K. Since the mirror surface is aparaboloid of revolution, 3 which is a special case el-liptic paraboloid, it is governed by the equations

x2 + Y2 = r2

Z = r2/4F.

(13)

which is the scalar magnitude of the vector R. Thequantity r is defined as the perpendicular distancefrom the paraboloidal axis to the point P5 and is afunction of 0 since r2 = a2 + b2. From Fig. 3, how-ever, one can think of r as being a vector originatingfrom PF as

r = - i(A + D/2) sinO + i[(A + D/2) cosO + B] + k(O). (14)

Note: To obtain this value of r, one must think ofthe point PF as a translation defined by

X, = X + (A + D/2) sinO,

YPF = Y + (A + D/2)(1 - cosO),

(15)

(16)

z

(6)

(7)

Hence the incident beam strikes the mirror surfaceat the point Ps = P[O,(A + D/2 + B), (a2 + b2)/4F],where a and b are points on the mirror surface from

Fig. 2. Experimental setup for determining the off-axis distanceand focal length.

February 1973 / Vol. 12, No. 2 / APPLIED OPTICS 213

l' I S

LASER(8)

(9)

A -H

(10)

(I)

(12)

- - - - - 7 -

rather than a rotation of axes as shown in Fig. 3.Combining Eqs. (6) and (7) gives

Z = (X2 + Y2)/4F. (17)

The vector r touches this surface at PR where

PI = P - (A + D/2) sinO, [(A +

D/2) cosO + B [(A + D/2)2 + B + 2B(A + D/2) cos0]/4F.

(18)

If we define the position vector5 to any point onthe surface of Eq. (17) to be

rp = iu + iv + k(u2 + v)/4F,

the normal vector N at this point is

N = Orp/8u X rp/av

(19)

(20)

XZ-AXIS INTO PAGE

Fig. 3. New focal point position after a rotation of 0.= i(A + D/2) sinO/2F - [(A + D/2) cosO + B]/2F + k(1). (21)

This normal vector N is the same normal vector thatoriginates at the point P and is collinear with theradius vector R. The vector point function of N liesat the point PN where

PN= P((A + D/2) sin0/2F, - [(A + D/2 cos + B]/2Fj

+ (A + D/2 + B), [1 + (a2 + b2)/4F]). (22)

Since the radius vector R and the normal vector Nare collinear, the relationship

Nk = R = iR, + jR + kR, (23)

can be written where k is the appropriate multiplica-tive factor. Hence,

(k2N,2 + k2N, + k2N,2)" 2 = 2F(1 + r 2/4F2)31 2, (24)

which gives as its solution

k = 2F(1 + r2/4F2). (25)

Hence, the radius vector R is given as

R = i(A + D/2) sinO(l + r2/4F2 )

- [(A + D/2) cos + Bi + r2/4F2) (26)

+ k2F(1 + r/4F).

The vectors I and R have now been formulated interms of a and b, and the reflected vector V remainsto be found. By using similar vector analysis tech-niques, one can show that the V vector has V andVy components that are found to be

V =

14KF - [(A + D/2) + B + 2B(A + D/2) cosO]l (A + D/2) sinO14F - [(A + D/2)2 + B + 2B(A + D/2) cosOll

(27)

and

Fig. 4. Experimental setup.

These components are the ones we have been lookingfor and describe the closed figure obtained in termsof the unknown quantities, F and A, and the mea-surable quantities, B, 0, D, and K. Hence by mea-suring V and V as a function of these other quan-tities, each pair of points on the closed curve willyield simultaneous estimates of F and A.

111. Experiment

The experimental apparatus consists of an off-axisparabolic mirror, translating mount, Lansing mirrormount, 90° rigid angle plate, Hardinge Bros. rotatingcollet chuck, Gaertner optical bench, and TEMoomode He-Ne laser. The parabolic mirror has a12.7-cm diam, 50.8-cm focal length, and a 6.35-cmoff-axis distance and was purchased from FersonOptics.

The apparatus is set up by placing the mirror intoa cylindrical sleeve (see Fig. 4). The mirror is heldwith set screws, and this combination is bolted into

(28)

214 APPLIED OPTICS / Vol. 12, No. 2 / February 1973

4KF - [(A + D/2)2 + B2 + 2B(A + D/2) cos0]I[(A + D/2) cosO + B]v, =- 14F2 _ [(A + D/2)2 + B2 + 2B(A + D/2) cosO]j

3

0.00 4.00 8.00 12.00SMALL W(CM)

16.00 20.00

Fig. 5. Plots of q(w) vswfor values of 40, 00, and -2.

the Lansing mount so that it can be rotated by ad-justing the micrometer dials. The Lansing mount isthen connected to the rotating collet chuck by meansof a rigid 900 angle plate. The rotating chuck hasone-degree graduations so that it can be rotated toany whole angle 0. The chuck is mounted on a trans-lating mount that is clamped to one end of a Gaert-ner Optical bench. The He-Ne laser is mounted onthe opposite end of the optical bench to the left (notshown) and is collinear with the axis of rotation ofthe rotating chuck. The translating mount has a16-cm travel that is perpendicular to the incomingbeam and moves with a precision of 0.025 cm. Byturning the crank on the translator, the desiredvalue of B can be obtained. From Fig. 4, one cansee that the value of B ranges from -D/2 to +D/2,and when the beam strikes the center of the mirror,B = 0.

A. Alignment by Mirror Translation (Two-Dimensional Analysis)

The experimental alignment of the mirror is at-tained by setting the rotating chuck on 0 andtranslating the mirror so that the laser beam passesacross the mirror diameter and is reflected onto thescreen. As long as the reflected ray and the incom-ing laser beam fall in the same plane for differenttranslations of the mirror, the experimental align-ment reduces to a two-dimensional problem.

From Fig. 1, we notice the relationship of Eqs.(1)-(5). Three different sets of data for q(w) and Swere taken and are shown plotted in Fig. 5. Thedata for curves A, B, and C correspond to 0 values of+4°, 00, and -2°, respectively. Curve B was linearlyleast squares fitted and yielded the equation

q(w) = 2006 X 10 'w + 1239 X 10'.

Hence A/F = 1.239 X 10-l and 1/F = 2.006 X 10-2cm-1, which yields A = 6.17 cm, F = 49.9 cm, whichis within 2-3% of the known values of A = 6.35 cmand F = 50.8 cm.

Hence, the techniques of finding the straight lineplot gives, in addition to proper mirror alignment,estimates of the focal length F and off-axis distance

B. Alternate Determination of F and A (Three-Dimensional Analysis)

Initially, it was not known that correct mirroralignment, without using rotation, would also givesimultaneous estimates of F and A. For this reason,the techniques of vector analysis and ray tracingwere applied to the rotating mirror in order to deter-mine mathematical relationships between the closedfigure formed on the plane Z = K, the focal length,and the off-axis distance. As a result of this analy-sis, the equations for V and Vy were derived. Dueto experimental limitations, however, data weretaken from 0 = 50 to 0 = 350 rather than for 0-360°.The data obtained are listed in Table I. If desired,one could solve Eqs. (27) and (28) for A and F andobtain values that would satisfy these equations si-multaneously for that particular data set. Repeat-ing this for the remaining data sets would give tenestimates each of A and F. An alternate approach,however, is to find the values of A and F that mini-mize the cumulative error for all ten sets of equa-tions simultaneously. This is, in fact, what wasdone; the values of A and F that yield minimumerror are A = 6.02 cm and F = 48.3 cm.

IV. Results and Conclusions

In this paper as in others,6 ,7 the He-Ne laser isused as an investigating tool for determining curva-ture characteristics of mirrors that are used in thelaser and the laser system. Here, the He-Ne laser isemployed first to attain correct mirror alignment ina plane and second to determine simultaneously theoff-axis distance and focal length by analyzing laserreflections in three dimensions. By Eq. (5) one re-alizes that proper mirror alignment yields a straightline plot of q(w) and vs(w) and in addition gives esti-mates of the off-axis distance and focal length fromthe intercept and slope. The equations derived inSec. II.B apply only after correct mirror alignmentis attained. In this section, the radius vector R, thereflected vector V, and the indicent vector I havebeen formulated. The two-dimensional figure

Table I. Measured Values of Vx and Vy forSettings from 50 to 350

0 V. Vy

50 8.08 cm -94.34 cm10 17.78 -94.1315 28.91 -91.4420 38.10 -88.9023 43.18 -86.0625 46.99 -84.7928 52.07 -81.6130 55.58 -79.7133 60.48 -76.2035 63.83 -71.76

K 419.1 cm.B = - 1.88 cm.D = 13.28 cm.

February 1973 / Vol. 12, No. 2 / APPLIED OPTICS 215

-is. 0 -0. 00 -S.00vY

Fig. 6. Calculated plots of V vs VY for B values ranging from0.0 cm to 6.2 cm in increments of 1.2 cm (B = 0.0 on right).

formed on the plane Z = K has been found, andthese coordinates V, and Vy are given in terms of theparameter 0. Because it is known that8 the inci-dent, normal, and reflected rays must fall in thesame plane, the equation of this plane was deter-mined, and all vector point functions for R, V, and Ido lie in this plane.

Experimentally, the best value of A and F ob-tained were A = 6.17 cm and F = 49.9 cm which arewithin 2-3% of the known values for the mirror.Randomly simulating ten new sets of data yieldsranges of 5.89 A 6.20 cm and 47.8 F ' 49.0cm.

Looking at the slope and intercept of the straightline plots for curves A and C in Fig. 5 indicates thatthe off-axis distance is very sensitive to mirror align-ment while the focal length is not so sensitive. Thecurves in Fig. 5 are supposed to show the linear con-ditions holding with curve B and not holding withcurves A and C. Instead, all three curves appear tobe nearly straight lines. This is due to the fact thatthe present experimental setup allows only a maxi-mum misalignment of 2 in one direction and 4 inthe opposite direction. For this reason, curve Awhich was misaligned 4 shows more deviation thandoes curve C which was misaligned only 2. Forlarge misalignments curves A and C would have agreater degree of upward and downward concavity,respectively.

The quantity B has been designed into the experi-mental setup for the purpose of allowing the investi-gation of essentially every point on the mirror sur-face. When B = 0, the closed figure formed on theplane Z = K is a circle. This condition is necessary

but insufficient for proper alignment and is used as acheck when proper alignment is attained.

Some initial experimental difficulty was encoun-tered in attaining the correct mirror alignment.This was due to the fact that the back side of themirror surface does not lie in the X-Y plane (asshown in Fig. 2). Instead, it was found from discus-sion with Ferson Optics that the back side of thismirror was ground approximately perpendicular tothe radius vector originating from the center of themirror surface. This eliminates one from aligning offthe back side of the mirror and then flipping themirror 1800 about the Y axis. If the method of find-ing the straight line plot is used for mirror align-ment, one need not worry about the relationship be-tween the front and back surface at all. However, ifthe rotating method is used, it would be nice to havea mirror whose back face does lie in the X- Y plane.

The parabolic mirror has a black arrow scribed onits edge that is parallel to the Z axis, lies in the Z-Yplane, and points in the +Z direction. The purposeof this arrow is to show the direction of the focal axisrelative to the mirror. The location of this arrowwithin the cylindrical mirror holder (shown in Fig. 4)is important when using the straight line plottingmethod for alignment. If a straight line plot cannotbe obtained after experimentally searching throughthe range of alignment, it will be necessary to loosenthe screws in the cylindrical mirror mount and movethe mirror to another position before repeating theprocedure. If the mirror is rotated within the cylin-drical mirror holder by small incremental anglesthrough 3600, two best straight line plots will be ob-tained when this arrow falls in the Z-Y plane at thepoints A and A + D on the Y axis.

In conclusion, this paper describes two simpletechniques. The first allows the investigator to at-tain the correct alignment for an off-axis parabolicmirror using only a He-Ne laser rather than an opti-cal collimator. In addition, this alignment proce-dure also yields estimates of the off-axis distance Aand the focal length F from the slope and interceptof the straight line plot. The second technique, alsoby using only a He-Ne laser, allows one simulta-neously to determine estimates of the values A andF. In this case, the methods of vector analysis andray tracing were applied to the parabolic surface.This technique differs from the first in that it in-volves a three-dimensional rather than a two-dimen-sional analysis. Also, this latter technique to deter-mine A and F can be applied only after the mirror isproperly aligned by some other method.

The plots shown in Fig. 6 show the dependence ofthe shape of the closed figure as B is varied from 0 toD/2.

I thank Tom Hewlett of the Air Force AvionicsLaboratory and Angie Lombardo of the System Re-search Laboratories for stimulating discussions per-taining to the alignment problems; Richard A. Za-charias of the Air Force Avionics Laboratories forprompting my attention to the alignment problemsassociated with off-axis parabolic mirrors and for the

216 APPLIED OPTICS / Vol. 12, No. 2 / February 1973

loan of the mirror used in this investigation; andlast, Robert Jurick of the Computing InformationSystem Branch, Scientific System Analysis Section,Wright-Patterson AFB, Ohio for providing invalu-able computing assistance throughout all phases oftheir investigation.References1. W. L. Wolfe, Ed., Handbook of Military Infrared Technology

(Office of Naval Research, Washington, D.C., 1965), p. 440.2. By the term alignment, I refer to the condition where the in-

coming rays are parallel to the focal axis. Using a He-Ne

laser with a small beam diameter and a small beam diver-gence, one may assume to a first approximation that the inci-dent light striking the mirror is collimated.

3. R. R. Middlemiss, Differential and Integral Calculus(McGraw-Hill, New York, 1946), p. 341.

4. J. D. Evans, Appl. Opt. 11, 712 (1972).5. M. R. Spiegel, Vector Analysis (McGraw-Hill, New York,

1959), p. 49.6. J.D. Evans, Appl. Opt. 10, 995 (1971).7. J.D. Evans, Appl. Opt. 11, 945 (1972).8. D. Halliday and R. Resnick, Physics (Wiley, New York, 1963),

p. 922.

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February 1973 / Vol. 12, No. 2 / APPLIED OPTICS 217