Perfect geodesic lens designing

Perfect geodesic lens designing

Jacek Sochacki

A new treatment for the problem of perfectly imaging geodesic lenses is presented that provides a uniform

mathematical description for full aperture as well as for nonfull aperture media. A family of the existing

particular solutions is surveyed and completed with novel results. The proposed formulation is in someaspects similar to the mathematical description of the Luneburg lens, which may be of some convenience for

those familiar with the Luneburg lens theory.

1. Introduction

Geodesic lenses initially studied for application tomicrowave antennasl- 3 have recently been successfullyused in waveguide optics and now are recognized as amost important passive component of integrated opti-cal circuits.4" The fundamental advantages of theselenses over the variable-index planar Luneburg lensesconsist in their potential applicability to all substratematerials (even with high refractive indices) and intheir facility to focus equally well the light provided bydifferent waveguide modes. This is the main reasongeodesic lenses lend themselves extremely well to theintegrated-optic spectrum analyzers and planar Fouri-er transform devices.' 2 However, for these devices it isessential that the lenses used be as free as possiblefrom aberrations.

The working principle of geodesic lenses is that thepropagation of light occurs along a curved surface in a2-D Riemann space. Within the geometrical opticsapproximation the ray paths have to follow the geode-sics of the surface of revolution according to Fermat'sprinciple. So the solution to the problem of perfectimaging resolves itself to finding the appropriate de-pression (or protrusion) profiles that will yield theexcellent focal properties of geodesic lenses. (A sepa-rate problem is connected with an accurate fabricationof such profiles.) In this regard two relatively differ-ent treatments might be pointed out.

The first of these treatments originates from thework of Rinehart,' who proposed (for microwave appli-cations) a radially symmetric geodesic profile closelyrelated to the refractive-index distribution of the vari-able-index perfectly focusing lens described earlier byLuneburg.'3 The problem of the equivalence betweengeodesic and refractive-index profiles was studied lat-

The author is with Central Laboratory of Optics, ul. Kamion-kowska 18, 03-805 Warsaw, Poland.

Received 8 July 1985.0003-6935/86/020235-09$02.00/0.© 1986 Optical Society of America.

er by Kunz 2 and subsequently by Southwell, 6 who de-scribed the numerical procedure for calculating themeridional profiles for so-called Rinehart-Luneburgand generalized Rinehart-Luneburg geodesic lenses.All of them are capable of perfect imaging between twoconjugate concentric circles of arbitrary radii by anappropriate declination of any ray entering the lensdepression area. Unfortunately such depressions turnout to have extremely sharp edges, which lead to severeradiation losses by the guided modes.

The second possible approach is to design the geode-sic profiles with the preliminarily assumed smoothtransition between flat and curved sections of thewaveguide. First proposed and discussed by Toraldodi Francia3 this concept has been subsequently devel-oped by Righini et al.5810 who derived general equa-tions for the whole family of smooth-transition geode-sic lenses with arbitrary focal lengths.9 In principlethese lenses are aimed to perfectly focus only thoserays that enter the central part of the geodesic depres-sion. Nonetheless, the small reduction of the workingaperture is sufficiently compensated by the considera-bly low level of reflection and scattering losses, and,therefore, this solution is well suited to waveguideoptics needs.

It might seem at first that the full-aperture andnonfull-aperture (or smooth-transition) geodesiclenses need two distinct theoretical descriptions; a re-view of the cited works would tend to confirm such astatement. It turns out, however, that the problem ofperfectly focusing geodesic lenses can be treated quiteuniformly as a whole, and this unique formulation mayprovide a variety of particular solutions in a conciseway. The aim of this paper is to give such a completedescription. It is worthwhile to note that the present-ed formulation is in some aspects similar to the mathe-matical treatment of the variable-index Luneburg lensproblem. Such unification may be of considerableinterest for all those who would like to utilize theirexperience in the design of Luneburg lenses while en-tering the geodesic lens area.

The plan of this paper is as follows: in Sec. II thetheory of full-aperture (FA) geodesic lenses will bepresented. Following Clairaut's theorem a general

15 January 1986 / Vol. 25, No. 2 / APPLIED OPTICS 235

equation for the light rays configuration will be de-rived, and formulas for the family of the Rinehart-Luneburg geodesic profiles will be evaluated. In Sec.III some modifications will be introduced into the fun-damental integral ray equation to provide the nonfull-aperture (NFA) solutions. The formal derivation willbe illustrated by a novel meridional profile for NFAgeodesic lenses. The possibility of designing othergeodesic components will be discussed in Sec. IV, andfinally in the Appendix some useful relations will beevaluated.

II. Full-Aperture Lenses

Consider a flat guide of surface SI with a radiallysymmetric section of surface of revolution S2, as shownin Fig. 1(a). Let us assume that the surface S2 isuniquely determined by the generating meridionalcurve 1(r), being a continuously decreasing invertiblefunction of the radial position r. According to theconfiguration sketched in Fig. 1(a)

z'(r) = [l'(r)2- 1]1/2,

Iz(r)

r

P

(1)

[the prime Ad/(dr)] whence the local depth z(r) of thelens depression may be expressed as

z(r) = J ['(r)2 - 1]/2 dr.r

(2)

Our object is to ascertain the functional dependence I= (r) [or z = z(r)] for the lens that perfectly images twoconcentric circles on each other [see Fig. (b)]. With-out loss of generality we can assume that the radius ofthe lens depression equals unity, and thus the radii ofthe conjugated circles f, and 2 (ff2 > 1) are normal-ized to unity at the lens boundary.

The traces of all the rays originating from the pointPI that are to be sharply focused at point P2 [Fig. 1(b)]can be described by polar coordinates r and 0. Recall-ing the well-known Clairaut theorem (see Ref. 10, forexample), the differential equation for them takes thefollowing form:

dO = kl'(r)dr 3r(r2

- k2 )1/2

where the ambiguous sign changes as the ray passesthrough the turning point characterized by the mini-mum radial distance r* and the angle 0 = 7r/2. Here kis the specified constant for each ray determined by itsinclination at the initial point P, and in accordancewith Clairaut's theorem

k A r sin4, (4)

where '1' is the angle between the geodesic (ray path)and the meridian at r. At the turning point = * =7r/2, and simultaneously 'I = 7r/2; therefore, k = r*, andnow the physical meaning of the constant k is evidentas the shortest distance between the geodesic path andthe axis of rotation. Since the radius of the lens is

Fig. 1. Configuration of the full-aperture (FA) geodesic lens: (a)cross section; (b) top view. The radially symmetric surface of revo-lution S2 is described uniquely by the generating curve (r), where r =

(x2 + y2 )1/2.

assumed to be unity, it is easy to see that the rays whichmeet the lens must have 0 k 1.

The condition that every ray which starts from Pl(r= fl, 0 = 7r) and enters the lens depression is bent topass through P2 (r = f2, 0 = 0) may be mathematicallyexpressed, using Eq. (3), in the form

r= frr=k kl)(r)dr fr kl'(r)drfi r(r 2 -k2)1/2 fr.=k r(r2 - 2)1/2 '

0<k 1. (5)

Taking into account that l'(r) = -1 outside the lensarea (r > 1), integrating separately over the portions ofthe ray path inside and outside the lens depression andrearranging, we derive the integral equation

| I kl'(r)dr = arcsin (KQ + I arcsin (TkJk r (r2

- k2)1 2 V 2 f2)

+ arccos(k), 0 k 1. (6)

This fundamental ray equation may be solved in dif-ferent ways. A simple change of variables reduces Eq.(6) to Abel's integral equation, the solution of which isknown to exist in very general conditions; however, wecan deal with this problem directly. Setting on theright arcos(k) = r/2-arcsin(k), replacing the dummyvariable on the left by iq, multiplying both sides by (k2

- r2)-1/ 2, and integrating over k from r to 1, afterinterchanging the order of integration on the left weobtain

236 APPLIED OPTICS / Vol. 25, No. 2 / 15 January 1986

Si

a.

ts P 2~~~~~~~~~~~~~~~I V'2~~~~~~~~~~~~

Fig. 2. Equality of light paths in the perfectly focusing FA geodesic

lens: s,+s2+S3=S 4 +S5+S6+S 7 ,wheresl=fl-1,s2=21(0),S3=f2

- 1, S4 = (fr2 - 1)1/2, S5 = arcsin(1/fi), S6 = arcsin(1/f2), and S7 = (f2 -

1)1/2. These relations lead directly to Eq. (15).

I1 L'( ' w(rfl) + .(rf2) - 2w(r,1)

+ ln[1 + (1 - r2)1/2] - ln(r),

0 r<1, (7)

where the function is defined as

_(r f) A 1 arcsin(k/f)dk 0 < r < 1. (8)

Functions of this type were introduced by Luneburg.'3

They played an important role in the theory of spheri-cally symmetric gradient-index lenses, and now theyturn out to be useful in describing geodesic lenses. Forcompleteness some elementary discussion of theirproperties is given in the Appendix; we will often referto those results. First let us utilize the relation [seeEq. (A2)]

2w(r,1) = ln[l + (1 -r2)1/2. (9)

In view of this equality, Eq. (7) takes the followingsimple form:

1 1'(n)d =(rfj) + w(r,2 ) - ln(r), 0 < r < 1. (10)

Differentiating both sides over r and rearranging weobtain

I'(r) = r d [w(r,f1) + w(rf)] -1, 0 < r < 1, (11)

and according to analytical formulas for d w/dr evaluat-ed in the Appendix [Eq. (A6)] we finally derive anexplicit prescription for l'(r) as follows:

I(r) =-1 [arccos I - r2) + arccos ( -r 2 1

arcsin(l/fl) + arcsin(l/f2)

(1-r 2)"2 '

0 r 1. (12)

As we can see, for r - 1, the term 1'(r) tends to beinfinite, and, in view of Eq. (1), this provides also aninfinite value for the derivative of the depth profilefunction z'(r). On the other hand, for r = 0 we have1'(0) = -1 and z'(0) = 0. Physically it means that in ageneral case, the lens surface of revolution has a hori-zontal tangent plane at r = 0 and a vertical tangentplane at r = 1. This last property will provide anextremely abrupt refraction, of light at the lens bound-ary independently of values for f, and f2. [One excep-tion takes place, however, when f1,f2 - -; then l'(r) =

-1, but in such a case we merely have a uniform flatguide.] This is the price for the requirement that thelenses be the full-aperture ones.

Now we are in position to ascertain the functionaldependence for the generating meridional curve 1(r).Fortunately the direct integration of Eq. (12) works ina general case, and finally we have

1(r) = f l'(r)dr = 1 +2 arcsin + 2arcsin (9-)

+ 1 {[(f12 1)1/2 + (f22 - 1)1/2]

X arcsin(1 - r2 ) 2 - r[arccos ( 2 2 )

+ arccos (1-r21

-arcsin(r) rcsin + arcsin-

[ar f (2)]

- / [ arcsin fi 2 r2) ]

- f2 arcsin[t2 (1r 2) 1/2]}

0 r 1. (13)

Let us note that the reduced formula for the full lengthof the generating curve,

1(0) = 1 + arcsin + arcsin (-)

+ (f 2 1)1/2 + (f22 - 1)1/2 - f, - f2], (14)

is in perfect agreement with the purely geometricalrelation expressing the equality of the optical paths ofthe central (meridional) and the grazing extremal rays(see Fig. 2)

21(0) + fl + f2 - 2 = arcsin (9-) + arcsin (9-)

+ (f, - 1)1/2 + (f22

- 1)1/2. (15)

The general formula (13) preserves the correct re-sults for two well-known particular cases. For theclassical Rinehart-Luneburg lens (f, f , /2 = 1) wehave


I(r) = 1 [1 + arccos(r) - rj2

I (1 + 7) - [r + arcsin(r)],

0 r 1. (16)

This expression coincides very well with the relationderived by Rinehart'; the only difference is that in thepresent formulation the length of the generating curvel(r) is measured from the lens boundary to its center.For the second particular case, when f/ = 2 = 1, weobtain

1(r) = arccos(r) = - arcsin(r).2 (17)

Such an expression was previously derived by Kunz.2The analytical formula (13) for the generating me-

ridional curve l(r) does not seem to be the most conve-nient tool in geodesic lens design. Of considerablymore interest would be the possibility of evaluating aclose-form expression for the lens depression profilez(r) rather than for l(r). According to Eqs. (2) and (12)we have

1 Jr ([ ( 1-r2 \1/2 (1-r

2 1/2z(r) =- J jarccos ~ 2 ) + arccos 2-r)arcsin(l/f,) + arcsin(1/f2)

2 _ 72} 1/2+ (1 -r

2)1"2 J ~ 2 / dr,

0<r 1. (18)

Unfortunately this integral may be closely evaluatedonly for the two particular cases mentioned. One canshow that for the classical Rinehart-Luneburg lens (fA

- ~,2 = 1)'

ztr) J0 {[2~ + 2(1r2)1/2] 1 dr

= W-[1--r2 + (1-r2)"/2] 1/24

(3 2 1/2 1/2 {1 2 1/2 } 1/2)n [1+ (1 -r2 1 1} + [3(1 -r )" +1 11 2

0 r 1. (19)The integral (18) may be easily evaluated also for thecase f = f2 = 1. We have then

z(r) . _ rd 1 - (1r-r2)/, 0 < r 1. (20)

As was expected, Eq. (20) describes a hemisphere de-pression which is the geodesic equivalent for the vari-able-index Maxwell's fish-eye. All the remainingcases, that is the so-called generalized Rinehart-Lune-burg lenses (1 < f < and/or 1 < f2 < ), need thenumerical methods to be used in evaluation of thegeneral Eq. (18).Ill. Nonfull-Aperture Lenses

As noted in Sec. II, the depression geometry of FAgeodesic lenses provides an extremely abrupt refrac-tion of light at their boundaries. This situation ishighly undesirable especially in integrated-optic cir-

b.

pi

Fig. 3. Configuration of the nonfull-aperture (NFA) geodesic lens:(a) cross section; (b) top view. The smooth-transition section of theradially symmetric lens depression surface S2 is determined by thegenerating curve L(R), where a < R = (x2 + y 2)1/2 < 1. The innerpart of the depression is described by the generating curve 1(r),

where 0 r = (x2 + y 2)1/2 < a.

cuits for several obvious, mostly energetic, reasons.Thus it is not at all strange that theoretical and subse-quently experimental investigations have tended to-ward a new family of so-called smooth-transition solu-tions, such solutions having emerged toward the end ofthe 1970s. The most significant contributions to thisarea are those cited in Sec. I, the work of Italian re-searchers Righini et al. who gave comprehensive andmathematically elegant descriptions of the problem ofsmooth-transition or, in other words, NFA geodesiclenses. It is not our intention to duplicate that materi-al here; however, our formulation provides in effectquite similar results. Our objective is to show how topass in several steps from the theory of FA to thetheory of NFA geodesic lenses and to prove the utilityof the already developed tools in handling the reformu-lated problem.

First, let us look at the geometry of a NFA geodesiclens and explain briefly how it works. According tothe configuration sketched in Fig. 3, the radially sym-metric lens depression (or protrusion) is divided intotwo parts. The external section or outer-shell deter-mined by the generating meridional curve L(R) (a < R< 1) is assumed to be a smooth interface between theflat guide S, and the inner surface of revolution S2.Let the latter be described by the generating meridio-nal curve l(r) (0 < r < a). In general, the functionalform for L(R) may be chosen freely, but we cannotsimultaneously put any requirement on the configura-tion of rays that will cross exclusively the external lens


a.

z(r)

r, R

l II l

area (a < R < 1). On the other hand, we require thatall the ways entering the inner part of the lens (0 _ r _a) image sharply between two conjugated circles ofradii f, and 2 [Fig. 3(b)]. This last condition willdetermine a functional dependence for 1(r).

According to the earlier explained physical meaningof the constant k, all the rays which meet the inner partof a NFA geodesic lens will be specified by 0 < k < a.Therefore, in view of Eq. (6), the condition of sharpimaging will take in the present case the followingform:

Ja kl'(r)dr (1 kL'(R)dR

-k r(r 2 - k2) 1/2 a R(R2 - k2)1/2

_1 arsn - arcsin k+ arccos(k), 0 _< k _< a, (21)

2 f,1 2 (f2)

where the integral has been written in two parts todistinguish between the trace of the rays in the innerpart and in the outer shell of the lens and to emphasizethe functional difference between l(r) and L(R). (Thereason for which we distinguish between r and R isalready the same.) It is clear that the value of adetermines the size of the working aperture of a NFAgeodesic lens; for a - 1 our present equation willreduce to Eq. (6) which describes the FA case.

Assuming that L'(R) [or L(R)] is a known function ofthe radial position R, Eq. (21) may be solved in asimilar way as Eq. (6); multiplying both sides by (k2 -r2)-/ 2 and integrating them over k from r to a, aftersome simple conversions we derive

_ = .(r,f 1 ,a) + .(r,f 2 ,a)

-2 w(r,1,a) + ln [a + (a' -r2)1/2]

-I~)+2 J: [Jal L'(R)dR 1I kdkr +- [ R(R2 - k2)1/21 (k_-r2)1/2 -

0<r a, (22)

where (see Appendix)

(r,f,a) a arcsin(k/f)dk = r f rV i(kr- r (a a) (23)

some restrictions on A'. The first one results from theobvious condition that at R = 1 the geodesic profilemust be tangential to the external flat surface. Ac-cording to Eq. (1) this yields

L'(R = 1) = -1,

and as a consequence

(1) = 0.

(26)

(27)

The second restriction is strongly connected with therequirement that the extremal perfect ray, specified byk = a, must just only touch the inner region of the lens.Following Eqs. (21) and (25) this condition can bemathematically expressed as

r1 adR [1a+(R)dR0+1+Ja R(R2-a2) 1 2 Ja R(R 2_a

2)1/2

=2 arcsin (f-) + 2 arcsin (-) + arccos(a). (28)

The first integral may be easily evaluated in terms ofreverse trigonometric functions, and finally we have

1 a(R)dR 1 . a 1 (a)na R(R -a2)1/2 2 V1+ 2 f

(29)

This condition gives a smooth transition at R - r = a.The last requirement is that 41 must be a continuousand monotonic function of R.

Having established the general conditions for thefunction VI(), we now can put Eq. (25) in Eq. (24), andafter some calculations we derive

1 la 2_ -r2_1_l'(r = - - ~accos

'V V (, r)a a2 r2 _ 2 1/2

+ arccos f2 - r /

a[arcsin(a/fl) + arcsin(a/f2)]l(a2

- r2)1/

2 J

_2r d 1ff 1 (R)dR I kdk7r dr r La R(R2

- k2)F/2 (k2

Differentiation of Eq. (22) over r leads to:

I'(r) = r d (.(rfia) + .(rf 2,a) - 2w(r,1,a)

+ 1n{1 + [, (r) 2]1})

1 [J r L'(R)dR kdk7r dr t R L~ (R2-_ k2)1/22 TP _ r2) 1/2J

0<r a. (24)

Equation (24) may be greatly simplified if we assumesome special form for L'(R). Let us set

(25)

where 4'(R) is an unspecified function of the radialposition R. However, to assure a really smooth transi-tion in the outer-shell area (a < R < 1), we have to put

0<r a, (30)

where the relations (A4) and (A7) (see Appendix) havebeen used. For a - 1 Eq. (30) reduces to Eq. (12)evaluated for FA geodesic lenses.

Now we are finally in a position to construct thegeneral relations for the depression profiles for theouter shell and inner part of NFA geodesic lens. Inaccordance with Eqs. (1), (25), and (30) we obtain

1 fr a2 -_r2\1 /2 /a2 r21/2z - Larccos 2 + arccos _r2a7 r csi0 a f1 -arsinfa/f 2

+a[arcsin(a/fl) + arcsin(a/f2)](a2

- 2)1/2

2 01/2+ Q(r,a) _-1r2 dr, O < r < a, (31)


L'(R = -[1 +,P(R)], a < R _- 1,

z(r) = z(a) + Ir [t2 (R) + 2t(R)]"/ 2 dR, a < r < 1, (32)

where

Q(ra d r aJ (R)dR 1 kdk= dr l La R(R2

-k2 ) 2 J (k2 - r2)1/21

d2r+ [ (R) arcsin a2-r2)/2dR] (33)

Equations (31)-(33) fully describe the depression pro-file of NFA geodesic lenses with arbitrarily chosen f/and f2, once the value of a and functional form of A'(R)are preliminary specified, and Eqs. (27) and (29) aretaken into account. Since we have a relatively greatflexibility in choosing the V'(R) functions, a variety ofparticular solutions may be achieved. Sottini et al.'0considered the case for which ip(R) can be written as

1-Rconst. (34)

Another possibility is to employ the formula alreadyused in the NFA Luneburg lens description.' 4 Howev-er, to examine the preceding formal derivation, let uspropose in this place a novel solution.

A. Particular Solution

Let us introduce the function

-(R) = 1-R a < R < 1, (35)

where h is the real parameter. It is easy to see thatM(R) is the continuous monotonic function of R, and

O(1) = 0. This assures a smooth transition for a < R <1. For simplicity we limit ourselves to the case whenfi-o and 2 f > 1. In other words, we will design afamily of lenses being NFA perfect focusers with thearbitrary focal length f > 1. Then, according to Eq.(29), the parameter h is specified as follows:

2[arccos(a) - an (1 a)]h 2a= a

arcsin(a/f)

Fig. 4. Multiple-foci geodesic lenses: (a) double-focus lens; (b)triple focus lens.

z(r) = z(a) + 12 J [(1 - R)2 + 2h2 (1- R)]112 dR

= z(a) + 1 h2In [(1 -r) 2 + 2h2(1 -r)]' + 1 + h2 - r2 ! [(1-a) 2 + 2h2(1- a)]1 /2 + 1 + h2 - a

+ h2 r [(1-r)2 +2h2(1-r)]/ 2 + h2

(38)

where the value for z(a) results from Eq. (37). Thefundamental advantage of the solution presented isthe analytical formula (38) for the outer-shell depres-sion profile.

(36)

If this relation is fulfilled, the smooth transition atR -r = a is automatically achieved. Now let us put Eq.(35) into Eq. (33). Integration may be performed interms of elementary functions, and after substitutingthis result into Eq. (31) we derive finally

z(r)= 2 (a r in a

+ r arccos [1(c 2 -r 2 )1/2]

h2 1a2 -r 2 \1/2l 2- arccos 2 2-r

IV. Other Geodesic Components

The formulation presented of the geodesic lensproblem is flexible enough to allow one to design manyother geodesic components providing interesting con-figurations from both theoretical and experimentalpoints of view. Detailed analyses of all the possiblesolutions would take too much time so let us reviewsome of them.-arccos (a2-.2.

t- 2 dr, ° O< r < a.

The depression profile in the outer shell is described byEq. (32). With the help of Eq. (35) and some calcula-tions, we obtain

A. Multiple-Foci Lenses

A problem of the double-focus or of the triple-focusgeodesic lens (see Fig. 4) may be formulated in a simi-lar way to that of its variable-index equivalent.'5

Following the fundamental Eq. (6) the configurationof rays for the double-focus lens [Fig. 4(a)] may bedetermined as follows:


a.

Id

X [(1 - a)2 + 2h 2 (I - a)] 1121, a < r _< 1,

J1 L'(R)dR 1 . kI - ~~~ ac.- + arcos(k),ik R(R2 - k2)1/2 2 MF)

a < k 1, (39a)

ra kl'(r)dr _' kL'(R)dR 1a. k

k r(r2-k

2 )1/2 Ja R(R2 - k2)l /2 =2 M + arccos(k),

0 < k < a, (39b)

where L'(R) is the derivative of the meridional curve ofthe outer-shell section (a < R < 1), which is responsiblefor focusing at the longer distance F, and l'(r) describesthe inner part of the lens (0 < r < a). Equation (39a)can be easily solved with the help of methods displayedin Sec. III, and then a formula for L'(R) will be ob-tained. This formula will make a solution to Eq. (39b)possible by a similar method. Having establishedfunctional forms for L'(R) and l'(r) we will be able to

determine the lens depression profile z(r), according toEq. (1).

On the other hand, the ray configuration equationsfor the triple-focus lens [Fig. 4(b)] will take the follow-ing form:

rlkL'(R)dR 1 lk\ 1 /d\ k - kL'(R2)1 2 arCSin - 2 arcsin (•') + arccos(k),k R(R2 -k) 1 2 f/ 2 f

a < k < 1, (40a)

(a kl'(r)dr [1 kL'(R)dR 1 ik\

Jk r(r2 - k2)1/2 la R(R 2 - k2)1/

2 2 a + arccos(k),

0<k a, (40b)

where d denotes the distance between the central andthe side foci [see Fig. 4(b)]. The way to solve the set ofequations (40) is as in the former case. Following Ref.15 it is worthwhile to point out that the triple-focuslens may be used as the perfect collimator and, simul-taneously, beam splitter.

J1 kl'(r)dr - 1 arcsin - - arcsin (kk r(r2 - k 2 )'

1 2 2 kf'+ 2 a fmV

+ arccos(k) + Nr,0<k-1, (41)

where N is the natural number which counts the loopsmade by every ray. For N = 0 we are faced with directimaging [see Eq. (6)]. For the general case N > 0,according to the former derivation (Sec. II), we have

l'(r) =- [arccos ( )2 + arccos (12 r2 )

arcsin(1/f1 ) + arcsin(1/f2) + 2N7r

(1 - r2)1/2 ]0<r 1, (42)

and then, following Eq. (2), the z(r) depression profilecan be obtained.

To solve the NFA case, it is enough to replace theintegral on the left-hand side of Eq. (41) by

J a kl'(r)dr f' kL'(R)dR

k r(r2-k2)1/2 Ja R (R2 - k2)1 2 (43)

and to follow the methods described in Sec. III.

C. Beam-Steering Components

Finally, it is worthwhile to point out another inter-esting use of geodesic profiles in integrated-optic cir-cuits: the radially symmetric waveguide depressions(or protrusions) may be utilized not only as perfectlenses but also as beam-steering components. In par-ticular, it is possible to design geodesic corner reflec-tors or beam deflectors working in configurationsshown in Fig. 5.

The fundamental ray equation for the full-aperturecorner reflector [Fig. 5(a)] is in principle similar to Eq.(41) describing the beam-loop lens. The only differ-ence is that both arcsin functions responsible for thefocusing effect have to vanish (fA = f2 - -), and theterm N7r must be replaced by (1/2 + N)r, where N stillcounts the number of loops. For the simple case N = 0we have

Jk r(r2 - k 2)1/2 2

and after solving

l'(r) = -1 -(1 - r2)-1/2, 0 < r 1.

(44)

(45)

According to Eq. (2) this provides the expression Forz(r):

B. Beam-Loop Lenses

Geodesic profiles of lenses which perfectly imagetwo concentric circles on each other may be modifiedby the requirement that every ray has to encircle com-pletely the lens origin before it attains the focus.(Such a modification was introduced to the Luneburglens theory by Stettler in 1955.16) For FA geodesiclenses this requirement may be mathematically ex-pressed by:

z(r) = J [(1 - r'Y) + 2(1 - r') 1 '] 2 dr,f,

0<r 1, (46)

which can be evaluated in terms of elementary func-tions.

The FA geodesic beam deflector shown in Fig. 5(b)may be described similarly:

l k'(r2)r2 = arccos(k) + + IY'V,krr2 - k2)l2 2 ) 0 < k <1, (47)


where a is the deflection angle. Equation (47) will alsoprovide an analytical prescription for the z(r) depres-sion profile.

The NFA solutions for both cases may be obtained ifwe replace the integral

' kl'(r)drik r(r2 - k2 )1/2

in Eqs. (44) and (47) by the combinationfn kl'(r)dr -' kL'(R)drk r(r2

- k2)1/2 JaR(R2 -k 2)

12

Setting in both cases

N = , L'(R) = 1 + a < R ' 1,

ae= 7r/2

(48)

(49)

a)

(50) 4 3 2 1 1 2 3 4

(51)

in Eq. (47), one can derive particular solutions that arein perfect agreement with those presented earlier bySottini et al.17 It is evident, however, that one can usemany other forms for L'(R).

V. Conclusions and Summary

All the presented formulation of the geodesic lensesproblem might be summarized in the following generalway: according to Clairaut's theorem the requiredconfiguration of rays may be expressed in the form ofan integral equation similar to the relation known fromthe Luneburg lens theory [Eq. (6)]. This equationmay be modified to provide various solutions subject toparticular conditions put either on the character of thegeodesic profiles (FA or NFA solutions) or on the be-havior of the light rays (focusing and/or beam steer-ing). A family of the resulting equations can be solvedby the same method, which in several cases leads to theanalytical prescription for the depression (or protru-sion) profiles or at least to the well-defined integralsfor them.

The presented derivation has been limited to thecases when the lenses have been imaging between twoconjugated circles settled outside the depression (pro-trusion) area, i.e., for flf2 1. However, the problemof the NFA lenses with one external and one internalfocus (f1 > 1, f < 1) may be treated similarily: it issufficient to replace Eq. (6) by

[a kl'(r)dr _ ff kL'(R)dR 1 kL'(R)dRJk r(r2 k2)1/2 Ia R(R2 k2 )1/2 2 J2 R(R 2 -k 2 ) 1 2

= 2 [ir + arcsin () + arcsin(k)], 0 < k a f2 , (52)

and to choose the function L'(R) [or L(R)] suitable forthe smooth-transition solutions. A detailed analysisof such lenses has been omitted due to their lack ofimportance in integrated-optic circuits.

b)Fig. 5. Full-aperture(FA) beam-steeringcomponents: (a) cor-ner reflector; (b) beam

deflector.

,4

1 2 3 4

spherically symmetric, perfectly focusing variable-in-dex lenses. In a particular case when f = 1

w(r,l) = ln[I + (1- r2)'12 J, 0 < r 1.2

(A2)

This result can be proved either formally, as was doneby Luneburg,13 or in a more physical way as shown inRef. 18. For f> 1 the function c(r,f) may be expressedexclusively in the form of an infinite power series.19-22

The same refers to the modified Luneburg function:o(r,f,a) 1 a arcsin(k/f)dk =/ , ) 0 r a, ()

i(~~a r (k 2 _ r2)1/2 ta a) I ,(3

introduced in 1983 by Doric and Munro.23 When f = a

co(r,a,a) = 2 In{ + [1 - () ] } 0 r a. (A4)

Appendix: Derivatives of the Function

As pointed out in Sec. II, the function

w(rf) 1 arcsin(k/f)dk 0 r 1, (Al)'Vs Id(k2 - r2)1/2

was introduced in 1944 by Luneburg' 3 to describe

However, what we really need in a mathematical de-scription of geodesic lenses are the derivatives of the cofunctions rather than their values. Fortunately, theanalytical expression for d/dr may be obtained in thegeneral case f 1. Integrating Eq. (Al) by parts wehave


w(r f) - arcsin(1/f) ln[1 + (1 _ r2)'12] - arcsin(r/f) ln(r)'Vr -'nlV 1-r )/i nr

1 1 ln[k + (k2 - r 2)1 2 ] dk,

and after differentiation

d [a(rjc)sn=cn1t. r /f)d r [ ( j I a s .= '

-r[1 + (1 - r2)1/

2 1 (1

arcsin(r/f)'rr

ln(r) ln(r)(f2 r2)1/2 r(f

2- r2)/2

rdk(f

2- k2)1/2 [k + (k2

-r2)1/2]

= -[arcsin(1/f) - arcsin(r/f7rr

1 J) kdkr (f2 - k2

)1/2 (k2 r2)1/

- Jr (1 -1 dk2)l = 1 [arc:7r r (f2 _ 2)1/2 irr

- acinlf) 0 -1r -1.(1-: r2)l/ J

This result agrees with those deriStettler 16 and by Doric and Munro2 4 byIn a similar way we can obtain

d [w(rf~a)I_. = [arcsin (a 2 - r2 _ aardr a-..sL 7rr L f2 r2

/ (a 2

when f = 1 and/or f = a we have

dr 2)] = 1 (1 r2)1/2]

and, respectively,

dr [@r~a~)] =2r a2 _ r2) 2]dr2 'a

It is also possible to evaluate direct]formula for derivatives of the functi(length If. One can easily prove that

df~~ ~~ -((e)]~os - arcsin 12r i)1 d 1 arc Ia2

-r2N/

- [W(rfa)]1ronst. - arcsin I 21df aconst 'r f r2

These last two relations have not bipreceding general derivation. Nevertbe helpful in evaluation of the multillens problem.

1. R. F. Rinehart, "A Solution of the Problem of Rapid Scanning1 (A5) for Radar Antennae," J. Appl. Phys. 19, 860 (1948).

2. K. S. Kunz, "Propagation of Microwaves between a Parallel Pairof Doubly Curved Conducting Surfaces," J. Appl. Phys. 25, 642(1954).

3. G. Toraldo di Francia, "Un problema sulle geodetiche deller

2)1/

2supervici di rotazione che si presenta nella tecnica delle mi-

- ) croonde," Atti Fond. Ronchi 12, 151 (1957).4. G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia,

"Thin Film Geodesic Lens," Appl. Opt. 11, 1442 (1972).5. G. R. Richini, V. Russo, S. Sottini, and G. Toraldo di Francia,

"Geodesic Lenses for Guided Optical Waves," Appl. Opt. 12,1 ' 1477 (1973).'V L 6. W. H. Southwell, "Geodesic Optical Waveguide Lens Analysis,"

J. Opt. Soc. Am. 67, 1293 (1977). See also the comment by E.Marom and 0. G. Ramer, "Geodesic Optical Waveguide Lens

(k22- r2

)1/2 Analysis: Comment," J. Opt. Soc. Am. 69, 791 (1979), and the

author's reply: W. H. Southwell, "Geodesic Profiles for Equiva-lent Luneburg Lenses," J. Opt. Soc. Am. 69, 792 (1979).

2 arcsin(1/f) 7. G. E. Betts and G. E. Marx, "Spherical Aberration Correction,V(f2 - r2 )1 2 and Fabrication Tolerances in Geodesic Lenses," Appl. Opt. 17,

3969 (1978).8. G. C. Righini, V. Russo, and S. Sottini, "A Family of Perfect

2 Aspherical Geodesic Lenses for Integrated Optical Circuits," J.Quantum. Electron. QE-15, 1 (1979).

l 2 1/2 9. It is also worthwhile to point out the numerical treatment of thesin - 2 problem given by D. Kassai and E. Marom, "Aberration-Cor-

f2-r rected Rounded-Edge Geodesic Lenses," J. Opt. Soc. Am. 69,1242 (1979).

(A6) 10. S. Sottini, V. Russo, and G. C. Righini, "General Solution of theProblem of Perfect Geodesic Lenses for Integrated Optics," J.

ved earlier by Opt. Soc. Am. 69,1248 (1979).vedth er method. 1. D. W. Vahey, R. P. Kenan, and W. K. Burns, "Effects of Aniso-other methods. tropic and Curvature Losses on he Operation of Geodesic Lenses

in Ti:LiNbO 3 Waveguides," Appl. Opt. 19, 270 (1980).csin/affi 12. D. Mergerian, E. C. Malarkey, R. P. Pautienus, J. C. Bradley, G.

csmra f) B E. Marx, L. D. Hutcheson, and A. L. Kellner, "OperationalIntegrated Optical R.F. Spectrum Analyzer," Appl. Opt. 19,

0 r a. (A7) 3033 (1980).13. R. K. Luneburg, Mathematical Theory of Optics (Brown U.,

Providence, R.I., 1944), pp. 189-213.14. J. Sochacki and C. G6mez-Reino, "Nonfull-Aperture Luneburg

0 r < 1, (A8) Lenses: A Novel Solution," Appl. Opt. 24, 1371 (1985).15. J. Sochacki, "Multiple-Foci Luneburg Lenses," Appl. Opt. 23,

4444 (1984).16. R. Stettler, "Uber die optische Abbildung von Flachen und

0 _< r < a. (A9) Raumen," Optik 12, 529 (1955).17. S. Sottini, V. Russo, and G. C. Righini, "Geodesic Optics: New

Components," J. Opt. Soc. Am. 70, 1230 (1980).ly the analytical 18. J. Sochacki, "Simplified Way to Derive the Luneburg Equa-n over the focal tion," J. Opt. Soc. Am. Al, 1202 (1984).

19. A. Fletcher, T. Murphy, and A. Young, "Solutions of Two Opti-cal Problems," Proc. R. Soc. London Ser. A 223, 216 (1954).

0 r < 1, (A10) 20. E. Colombini, "Index Profile Computation for the GeneralizedLuneburg Lens," J. Opt. Soc. Am. 71, 1403 (1981).

21. J. Sochacki, "Exact Analytical Solution of the Generalized Lu-neburg Lens Problem," J. Opt. Soc. Am. 73, 789 (1983).

22. J. Sochacki, "Generalized Luneburg Lens Problem Solution: AComment," J. Opt. Soc. Am. 73, 1839 (1983).

0 < r a. (All) 23. S. Doric and E. Munro, "General Solution of the Non-Full-Aperture Luneburg Lens Problem," J. Opt. Soc. Am. 73, 1083

en used in the (1983).

heless they may 24. S. Doric and E. Munro, "Improvements of the Ray Tracele-foci geodesic through the Generalized Luneburg Lens," Appl. Opt. 22, 443

(1983).


References

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Perfect geodesic lens designing