Paraxial ray analysis of a cat's-eyeretroreflector

J. J. Snyder

The cat's-eye retroreflector is a passive optical system consisting of a secondary mirror placed at the focal

point of a primary lens. We analyze the cat's eye using the paraxial ray matrix approach. The position of

the equivalent reflecting surface and the angular field of view of a realizable cat's eye are functions of the

radius of curvature of the secondary mirror. The field of view is maximum for a secondary mirror with a

concave radius of curvature equal to the focal length of the primary lens. We further derive the general

dependence of retroreflection errors on misadjustment of the secondary mirror.

Introduction

An ideal retroreflector is a passive optical systemthat returns each incident light ray at an angle of re-flection exactly opposite the angle of incidence.Practical realizations of retroreflectors can be eitherthe corner cube or the focusing system commonlycalled a cat's eye: a primary lens (or mirror) with asecondary mirror located at the focus of the primary(see Fig. 1). Of the two types of retroreflectors, thecorner cube appears to be more common and moreextensively studiedlA although for some applica-tions the cat's eye offers certain advantages. For ex-ample, the ability of a properly designed cat's eye toreflect polarized light without modifying its polariza-tion state preserves the effectiveness of optical isola-tors in preventing laser feedback.

Our approach is to describe a general cat's-eye sys-tem using the paraxial ray matrix method5 and to de-velop the equations showing the retroreflection errorsinduced by misadjustment of the secondary mirror.From the analysis we find that there is an optimumclass of cat's-eye configurations for which the angularfield of view is maximum.

Paraxial Ray Matrix

The paraxial ray matrix approach is a convenienttool for describing linear combinations of simple op-tical elements. An arbitrary, axially symmetric opti-cal system may be described by the matrix

The author is with the Joint Institute for Laboratory Astrophys-ics, National Bureau of Standards and University of Colorado,Boulder, Colorado 80302.

Received 14 February 1975.

r B.

CD J

The output ray vector is related to the input ray vec-tor by

ro'I cD I r l (1)

where r and r' are the (radial) height and slope ofthe input ray at the entrance plane, and ro and ro' arethe height and slope of the output ray at the exitplane. The same matrix correctly describes thetransformation of a Gaussian beam by the opticalsystem according to the ABCD law6:

Aq + BCq + D' (2)

where q, the complex curvature of the Gaussianbeam, is related to the radius of curvature R and sizew of the beam by

1 1 . Xq = R- ¶W. (3)

The spot size co is the radius for which the electro-magnetic field drops to l/e of the peak amplitude.

Useful matrices are those describing propagationthrough a distance d,

[ d],1o 1J

and through a thin lens of (positive) focal length f,

Mirror reflections are treated by unfolding the opti-cal ray about the mirror surface so a spherical mirror

August 1975 / Vol. 14, No. 8 / APPLIED OPTICS 1825

A= 2d 2d 2d2

R f fR

B = 2d - 2d

C = _ 2 4d + 2df R R T

D=1-2d 2d+ 2d2

fR f R

2d2 'isRK

(5)

The optical system fulfills the definition of a retro-reflector,

r = -r , (6)

if and only if the latter two elements of the matrixare

(7)c=1 lD = -1

The solution of Eqs. (5) and (7) requires the mirrorplaced at the focus of the lens [see Fig. 2(b)],

d = ,

(c )

Fig. 1. Retroreflectors: (a) corner cube; (b) cat's eye with pri-mary lens; (c) cat's eye with primary mirror. (b) and (c) are for-

mally equivalent in the paraxial approximation.

(8)

so the general matrix for the cat's-eye retroreflectoris

A B J= 1, 2d ] 2d2R . (9)

with radius R (R > 0 for concave) is described by thesame matrix as a thin lens of focal length R/2,

F 1 0 1]L-(2/R) 1J

A plane mirror is described by the unit matrix

[tn.l 1-

Cat's Eye

We consider an optical system consisting of a(thin) lens of focal length f and a mirror of radius Rat a distance d from the lens [see Fig. 2(a)]. A rayentering the primary lens with (radial) height r andslope r' is reflected by the secondary mirror. Thereflected ray exits the primary lens with height roand slope r. If the entrance and exit surfaces aretaken to be coincident with the lens, the ray matrixfor the system is the product of the matrices of theseparate components:

[A B [10 [dl F 0 F dl F'01.IC DJ l-/ ii Lo 1 l-2/R 11 Lo I l1/ f 1J

(4)

The elements of the matrix for the optical system arefound to be

rTO{< I /OPTICAXIS

R

(a)

d fI

Iriwl I

f R

(b)

Fig. 2. Optical system: (a) The primary lens of focal length f isseparated from the secondary mirror of radius R by a distance d.At the lens surface the input ray of height r and slope r' is trans-formed by the system into the output ray of height r and slope r'.(b) The cat's-eye configuration. The mirror is located at a dis-

tance d = f from the primary lens.

1826 APPLIED OPTICS / Vol. 14, No. 8 / August 1975

(a)

( b)

I _

= = -_ I�rlj' ,

1 1

I'I

2a

Fig. 3. The effect of tilt of the secondary mirror.the intermediate ray

r'o = -2 (1 - ) + C + Dri',

where C and D are the matrix elements given in Eqs.(5).

The sensitivity of the output-ray slope to a smallaxial displacement of the secondary mirror from thefocal position (focusing error) may be found by dif-ferentiating Eq. (12) with respect to the mirror posi-tion d and evaluating for d = f.

ad -2 a + + ( - ) -

The slope of

(13)

The total angular error in an improperly adjustedcat's eye is, therefore,

Ir I

is changed by 2a (dashed line) if the mirror is tilted by a. Forsimplicity, a plane secondary mirror is shown.

A retroreflector appears to each incident ray as aplane, perpendicular, reflecting (and inverting) sur-face. The distance of the apparent reflecting surfaceof a cat's eye, measured from the primary lens, is

B d2

-2 =? R- d. (10)

For example, a plane secondary mirror in a cat's eyeproduces an apparent reflecting surface at a distanced = f in front of the primary lens, whereas a cat's eyewith a concave secondary mirror of radius R = f hasan apparent reflecting surface coincident with theprimary lens.

Alignment Errors

In our model, the two possible types of misadjust-ment of the cat's eye are displacement of the secon-dary mirror along the optical axis away from thefocus position and tilt of the secondary mirror. Dis-placement of a spherical secondary mirror transverseto the optical axis is equivalent to tilt.

In order to analyze the effects of secondary mirrortilt, we consider the general lens-mirror optical sys-tem described by the matrix of Eqs. (4) and (5). Inthe absence of tilt the intermediate ray,

A'VOtId,,~f [ + 2 + 2 - )]Ad. (14)

The first term in the brackets in Eq. (14) intro-duces a wedge between the input and output rays.The effect of this term may be minimized in practiceby adjusting the tilt of the secondary mirror to besmall compared with the ratio of the primary lens ap-erture to the focal length.

The second term has a lenslike effect, since it isproportional to the input ray height r. This termcan be thought of as due to a finite radius of curva-ture of the equivalent reflecting surface of the cat'seye. Its effect may be reduced only be restricting theprimary-lens aperture (increasing the f number).

The final term, which is proportional to the inputray slope ri', will limit the angular field of view of thecat's eye. However, because of the dependence ofthis term on the radius of curvature R of the secon-dary mirror, we may construct a cat's eye for whichthe field of view is insensitive in first order to axialpositioning errors of the secondary mirror. For thisoptimized cat's eye the mirror radius is equal to thefocal length of the primary lens. The ray matrix forthe properly focused cat's eye then has the simpleform

and the total error term for a slightly defocused cat'seye is, from Eq. (14),

Ar, dRf ( a + 2ri) Ad.

(see Fig. 3), reflected from the secondary mirror, istransformed into the output ray according to

r =[ 1, d I r (11Iroh -1/, 1 - d/f] r/ (11)

A tilt 'of the secondary mirror by a small angle achanges the slope r' of the intermediate ray by 2a, sothe total slope of the output ray becomes, from Eqs.(1) and (11),

(16)

Example

As a numerical example we consider a primary lensof 5-cm diam and 20-cm focal length. If we assumethe retroreflected wavefront spherical error due tothe second term in Eq. (14) may be adjusted to be ofthe order X/10, the secondary mirror position will becorrect to within

I AdI 3.84 Mim. (17)

If we (arbitrarily) require all other error terms to be'

August 1975 / Vol. 14, No. 8 / APPLIED OPTICS 1827

-(12)

less than 1% of the sphericity,' the tilt of the secon-dary mirror must be less than

I °lax I._ - 1.25 x 10-3. (18)

We optimize this cat's eye by choosing the radius ofcurvature of the concave secondary mirror to beequal to the 20-cm focal length of the primary lens.The equivalent surface of reflection of the optimizedcat's eye is coincident with the primary lens, and thefield of view, in our model, is unlimited.7 However,if the secondary mirror were, for example, plane, theequivalent reflecting surface would be located 20 cmin front of the primary lens, and the field of viewwould be only

Iima A1.25 x 10-3. (19)

Conclusion

We have derived the paraxial ray matrix for thegeneral form of a cat's eye: a secondary mirror locat-ed at the focal surface of a primary lens or mirror.The errors in the angle of the retroreflected ray due

to small misadjustments of the secondary mirrorwere analyzed. We found that the field of view ismaximized for a cat's eye having a concave secondarymirror with radius of curvature equal to the focallength of the primary lens.

The author is grateful to John Hall for many usefulconversations. This work was supported in part bythe National Science Foundation through grant39308X to the University of Colorado. The author isan NRC-NBS postdoctoral research associate.

References

1. R. F. Chang, D. G. Currie, C. 0. Alley, and M. E. Pittman, J.Opt. Soc. Am. 61, 431 (1971).

2. H. D. Eckhardt, Appl. Opt. 10, 1559 (1971).3. F. Stenman, Comment. Physico-Mathematicae 42, 39 (1972).4. R. Beer and D. Marjaniemi, Appl. Opt. 5, 1191 (1966).5. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart

and Winston, New York, 1971), pp. 18-49 and references there-in.

6. H. Kogelnik, Appl. Opt. 4, 1562 (1965).7. In fact, lens aberration (which we do not consider here) or the

secondary mirror diameter would limit the effective field ofview of the optimized cat's eye.

Comment on filler in AO

Awards available here

Hellmut Hanle

Sonderprogramm zur Frderung der fachbezogenen Zu-sammenarbeit zwischen Forschungsinstituten in derBundesrepublik Deutschland und in den VereinigtenStaaten von Amerika, Sekretariat der Alexander vonHumboldt-Stiftung, D-53 Bonn-Bad Godesberg, Schil-lerstrasse 12, Germany.

The notice in the February issue [Appl. Opt. 14, 484(1975)] has resulted in numerous applications being madeby American scientists for a Senior U.S. Scientist Award.Under this U.S. Special Program self-applications are notpossible.

The basic idea of the program is to distinguish outstand-ing American scientists with a so-called Senior U.S. Scien-tist Award in recognition of their past accomplishments inresearch and teaching. This award entitles the recipient tostay for an extended period at a research institute in theFederal Republic of Germany carrying out research of hisown choice.

We would be very much obliged to you if you could pos-sibly print a correction to this notice in one of the futureeditions of Applied Optics, pointing out the basic idea ofthe U.S. Special Program.

1828 APPLIED OPTICS / Vol. 14, No. 8 / August 1975