Ray launching and observation in graded-index optical fibersKevin F. Barrell and Colin Pask

Department of Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT, Australia(Received 11 July 1977)

General mathematical and numerical results are given for graded-index optical fibers excited byparallel beams. The excitation of bound and tunneling rays and their influence on power transmissionand impulse response are described. The relevance of the results to other problems such as fiberexcitation by partially coherent sources and the appearance of fiber output faces is discussed. Thegraded-index fiber equivlent of the black-band phenomena in step-index fibers is described and anapplication to profile determination is considered.

1. INTRODUCTION

In this paper we examine some aspects of the behaviorof rays in graded-index optical fibers. We include both boundand tunneling rays. Bound rays remain trapped within anideal fiber, but tunneling rays, although predicted to betrapped by geometrical optics, attenuate as they propagatealong the fiber away from the source.1,2 One of our aims is todescribe tunneling ray attenuation in terms of the generalizedparameter technique recently reported.3

For the purpose of this paper we consider fibers with cir-cular cross section, radius p, and surrounded by an infiniteuniform cladding with refractive index ncl. The fiber coreindex nc. is a function of the radial variable r in the crosssection and we define

nco = nco(r), r U p (1)

nc0(O) = no, (2)

nco(p) = ncip (3)

and a critical angle Oc,

cos O, = nci/no. (4)

The core index can be expressed in terms of a general gradingfunction G(r):

n2'(r) = n' - n2 sin2OG(r), 0 < r < p.

We shall assume G (r) to be a monotonically increasing func-tion of r in most cases so that

0<G(r) 1. (6)

For detailed discussions it is useful to introduce the power lawprofile given by

G(r) = (r/p)q, (7)

where q is a positive constant. When q - we approach thestep-index profile.

In order to introduce the subject matter of this paper let usnow consider a parallel beam of rays incident upon the fiberentrance face at z = 0 (fiber axis aligned with the z axis) as inFig. 1. A convenient parameter for describing the incidenceis

a = ni sinOi/n 0 sinO,, (8)

where Oi is the beam incidence angle in the medium with re-fractive index ni and O, is defined by (4). The resulting poweracceptance curves are shown in Fig. 2 with G given by (7). In

the step-index case bound rays are excited for 0 < a < 1, whilefor all a > 1 some tunneling rays are excited. This lack of asharp cutoff at a = 1 appearing in asymptotic wave theory4

but not in geometrical optics was important in the identifi-cation and classification of leaky modes and tunneling rays.5

For a graded profile we find the following features, which wewill develop in detail in the later sections. Again bound raysare excited only by beams with a < 1; however, some tunnelingrays are also excited when a < 1. In fact, tunneling rays areexcited for 0 < a < amax and for the power law profile

amax = (q/2)112, q > 2

= 1, q < 2. (9)

These general properties of fiber excitation by a collimatedbeam also have importance in other contexts. Most othersources may be considered as a superposition of parallel beamsand then the results in Fig. 2 may be used to discuss fiber ex-citation by different source types. For example, sources withcomponent beams having a < 1 excite bound and tunnelingrays. If the angular distribution of components is increasedto a > 1, no further bound rays are excited, but more tunnelingrays may be excited. Increasing the spread of component a'sbeyond amax will produce no change in the power accepted bythe fiber. Such arguments as these are used in Ref. 6 to dealwith fiber excitation by partially coherent sources and thispaper provides the justification for the acceptance propertiesused there.

A second important context relates to observation of fiberoutputs. Considering the ray directions are reversed, theacceptance properties may now be used to discover the outputwhen observed at angle Oi to the fiber axis. The possible ex-istence of output rays can be deduced, but of course their in-tensity will depend on source intensity, fiber absorptionproperties, and tunneling ray attenuation due to radiationlosses. In the step-index fiber, the tunneling rays are seen atangles 0, such that a > 1 and lead to what is known as theblack-band phenomenon,7' 8 which was soon recognized as apowerful argument in favor of the existence of tunneling rays.5

In this paper we examine the black-band phenomenon forgraded fibers and comment on its possible use for profile de-termination.

Thus we proceed as follows: Section II is devoted to thegeneral formulas and results for power acceptance. In Sec.III we discuss power propagation using the generalized pa-rameter technique.3 We also evaluate the accuracy of thattechnique and apply it to impulse response descriptions for

294 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979 0030-3941/79/020294-07$00.50 C 1979 Optical Society of America 294

ni/ ///

ncI

e

FIG. 1. Geometry for beam in medium with refractive index n, incidentat angle O0 on the entrance face of a fiber with radius p, core and claddingrefractive indices nc0(r) and ncl. The fiber is aligned with the z axis andr is the radial variable. Inside the fiber the ray makes an angle O(r) with thez direction and by Snell's law sin0(r) = n1 sinO,/nco(r).

parallel beam excitation. The output properties of gradedfibers are described in Sec. IV.

II. DERIVATION OF POWER ACCEPTANCEPROPERTIES

A. General principlesRays in a graded-index fiber with profiles only depending

on the radial variable r are completely described by the twoparameters 9

0 = nco(r) cos0(r),

and

We note that

I = (r/p)n,0 (r) sinO(r) cosk(r).

12 = (rip)2 cos 2.k(n 0(r) - ,2).

(10)

(Ila)

(lib)

These two parameters remain constant along the ray path andmay be determined by specifying the ray angles 0,0 at anypoint on the path as shown in Fig. 3. The rays are then class-

nco(r)

FIG. 3. Geometry of ray path at radial position r in a fiber with graded indexnco(r) and aligned with the z axis. The ray makes an angle 0 = 0(r) withthe z direction. In the fiber cross section, the projected ray direction makesand angle ' = 0(r) with the azimuthal direction.

ified according to their :,[ values by9

bound rays: ni _< 1 < no, (12a)

and any I consistent with (i1);

tunneling rays: 0 < a < ncl,

I >_ (n 2 - 32)1/2 (13a)

and consistent with (11). For the power law profile (7) theconsistency requirement may be evaluated, leading to

bound rays: nl < < no,,

0 - I < Imax, (12b)

and

tunneling rays: [max(O,nl - (q/2)no sin2 )]<2 n cl,

(n 1 - p2)1/2 < I < lmax (13b)

where

Tflax = q[(n2 - 32 )/(q + 2)]l+2/q[2/n2 sin20,] 2/q. (14)

The excitation problem requires us to consider rays enteringthe fiber as shown in Figs. 1 and 4 and to identify a particularray by calculating its 3,I values and then applying (12) and

y

-> x

0 05 1 0 15 2-0 25

FIG. 2. Power P(a) accepted by an optical fiber when excited by a parallelbeam as shown in Fig. 1 with the angular incidence parameter a definedby (8). Pinc is the total power incident on the fiber entrance face. Thecurves are labeled by q, the power law profile parameter (7). For thestep-index fiber, only bound rays are excited for a < 1. For q > 2, P(a)= 0 when a >_ (q/2)112.

295 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

FIG. 4. Projection onto fiber face of a ray incident at the point 0 = Q(r,4).Note that 0, the polar coordinate, also gives the initial ray angle 0 as definedin Fig. 3. The incident beam in Fig. 1 is taken as being perpendicular tothe x direction.

Kevin F. Barrell and Colin Pask 295

. I I i

(13). Applying Snell's law as in Fig. 1 gives

2 = n20 (r) - (ni sinOj) 2

= nc 0(r) - n. sin2 Oa 2. (15)

Thus from (12) we find that bound rays are launched when

n20(r) - (no sinO,) 2a

2 > n 2

which on substituting (5) requires

G(r) 1 - a2.

(16a)

(16b)

For a monotonic grading function we found the limits in (6),i.e., G (r) > 0, so that (16) confirms our statement that boundrays are launched only when a < 1. Furthermore, as G(r) ismonotonic increasing, we see that for a given a, rays incidenton the fiber face are only accepted as bound rays if they fallon a disk with radius rmax given by

G(rmax) = 1 - a2, a < 1

rmax O= a, a> 1.

For the power-law profile [Eq. (7)],

rmax = p(l 7a 2 ) 11q, a K 1

=0, a >1.

(17a)

(17b)

(17c)

A similar application of the limits for tunneling rays (13)leads to the condition for tunneling ray launching:

(1 - a2) < G(r) < (1 - a2) + a2(r/p)2 cos20. (18)

We observe that for a given a this defines an area on the fiberface, in terms of (ro) values (see Fig. 4), over which tunnelingrays are accepted. Comparing (17) and (18) we see that thisarea lies outside the bound ray acceptance area. We returnto these areas in detail in Sec. IV in connection with theblack-band phenomenon. When a > 1, (18) requires onlythat

(19)1 - G(r)1 - (rip)2 COS20 -

This condition may limit the range of a's which give rise totunneling ray excitation. For the power law profile (7) ap-plication of (19) reveals that only beams with a < max launchtunneling rays, with a°max as in (9).

The above results embody the general properties of theexcitation process as regards selection of rays excited. Toobtain quantitative results for total power excited we simplyintegrate over the areas of fiber face identified above as ac-cepting the rays. Thus, assuming a uniform beam, if PR is thepower in ray type R,

PR =Io f J' rdrdk, (20)

where A'~R is the area of fiber face which accepts rays of typeR, and Io is a source strength constant. For example, thebound ray power Pbr is

Pbr(a) = Ia frma. r d f 2r do (21)

where rmax is given by (17).

B. Detailed resultsThe above formulas are easily evaluated for bound rays. If

296 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

Prian

Inc

0 05 1.0 1 5 20

FIG. 5. Tunneling ray power Ptr launched by a beam with power Pin, versusincidence parameter a, (8). The curves are for a graded-index profile ofpower law type and are labeled by the parameter q, (7). When q < 2, Pg(a)= 0 for a >_ 1. When q > 2, Pt(a) = O for a > (q/2)112.

we let the total power incident on the fiber be

Pinc = Io7rp2

,

we obtain

Pbr(a)/Pinc = rmax(a)/P2

For power law profiles (7), substitution from (17) gives

Pbr(U)/Pinc = (1 -a2)2/q,

(22)

(23)

a < 1

= 0, a> 1. (24)We note that the bound ray power increases as q increases.

The necessary area to be used in (20) when tunneling raypower is required is given in (18). In general it is difficult toobtain simple analytical results, but for the parabolic indexprofile (q = 2) we find

Ptr(a)/Pinc = (1 - a2 )1/2 - (1 -2), a 1

-0, > 1. (25)

The total power P = Ptr + Pbr has been given in Fig. 1. Thevariation Of Ptr with a and q is displayed in Fig. 5. Notice thatas q increases beyond 2, beams with a < 1 become less efficientfor launching tunneling rays and Ptr approaches the step-index result: Ptr(a) = 0 for a < 1. Thinking of a generalsource as a mixture of beams, we see that for graded-indexfibers it is almost inevitable that whenever bound rays arelaunched some tunneling rays must also be excited.

Ill. POWER PROPAGATION

A. Tunneling ray attenuationEven in an ideal fiber tunneling rays attenuate as they

propagate away from the source (at z = 0) due to radiationlosses each time they pass through their outer turning pointor caustic.10 Since rays are fully parameterized by , and T wemay write

Ptr = f f J(3,T)eY10(1 )z/P dT d3, (26)

where y is the appropriate attenuation coefficient. Thesource determines the form of J and the integration limits aregiven by (13). According to geometrical optics, y = 0 andtunneling rays are indefinitely trapped in the same manner

Kevin F. Barrell and Colin Pask 296

.J22

00

n1n no

FIG. 6. The A, Idomains for a graded-index fiber. The curves are correctfor q = 2, but the results are similar and the diagram is schematically correctfor other values of q, (7). The fiber has refractive index no on axis, claddingindex nc0 and cosO0 = nc 0no. The limiting curve /= /a,, is defined by (14)and the calculation of /Low and mLow is explained in the Appendix. Whenthe generalized parameter D, (28) = -X all tunneling rays are present and

' = /LOW = (n2l - /2)1/2 As D increases the I = /Low curve moves pro-

gressively from coincidence with the I = (n,21 - 32)112 curve to finallycoincide with the / = n01 line when D = oo. The curve I = 'LOw as shownis an intermediate case corresponding to D = 0.1 for q = 2.

as bound rays. However, electromagnetic theory gives -ynonzero. The wave theory nature of this result means thatoy depends on X the wavelength of light (in vacuum) and weintroduce X via the usual waveguide parameter

V = 27rpn0 sinO,/X. (27)

The evaluation of (26) is difficult because -y is a complicatedfunction of a and 1. Furthermore, Ptr now depends on anarray of fiber parameters-V, 0,, and p. These difficultiesare alleviated by using the generalized parameter formalism3

in which the rays are divided into two domains in one of which,yz/p - 0, while yz/p -X in the other. The division dependson one generalized parameter, which for power law profilefibers with q not too large is3

V (7rN(q) p),with

N(q) = 2[1 + (q/2)]-1/qq-1/ 2.

For the step-index profile,

D(step) = 1/V ln [20, (zlp)].

tr 0.4 -

0*-2

0~0 02 04 06 02 0-4 06

D

FIG. 7. Behavior of tunneling ray power Ptr in a parabolic index, q = 2,graded-index fiber excited by a parallel beam with incidence parameter a,(8). The generalized parameter D is defined in (28). D =-X correspondsto the initial (z = 0) tunneling ray power.

by solving a certain nonlinear equation.3 " 1 This can be donenumerically or approximate solutions can be obtained.3' 1' Forthe sake of completeness we give a few further details in theAppendix. It is now simple to evaluate (26) and obtain Ptrwhich, in addition to source properties, now depends on onlythe one parameter D for any given fiber type.

Results for tunneling ray power are given in Fig. 7 wherePtr(D)/Ptr(initial) is plotted versus D for a parabolic indexfiber (q = 2) excited by parallel beams with various incidenceparameters a. We remind the reader that these curves canbe interpreted in a variety of ways since D embodies severalphysical parameters. In particular, fixing the fiber parame-ters V, O, p we can read off the values of Ptr for a given dis-tance z from the source. Alternatively, thinking of z as fixedwe can think of variations in the fiber or light parameters, e.g.,increasing wavelength decreases V and hence increases D, andthus from Fig. 7 causes a reduction in Ptr. Similar resultsfollow for other profile parameters and in Fig. 8 we show D lo,

(28)

(29)

(30)

Note that D in the range 0 < D < 0.5 is the parameter rangeof practical interest. For example, in a parabolic index fiber(q = 2) with V = 50, 0, = 0.1, p = 80 Az, we find 100 m < z < 10km corresponds to 0.211 < D < 0.304. Then (26) is evaluatedby replacing the exponential by one and using the domain forwhich yz/p 0 which is defined by

ALow(D) <: n..I,

TLow(3,D) < 1 Tmax(73), (31)

where Tmax is defined by (14). The ray domains are showngraphically in Fig. 6. The limits AGLOW and [LOW are obtained

297 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

08

0-6

04

02

00 02 04 06 08 1:0 12

a-FIG. 8. Plot of D10, the value of the generalized parameter D, (28) for whichtunneling ray power has dropped by 10 dB, versus the indidence parametera(8) for parallel beam excitation. The curves are labeled by the power lawrefractive index profile parameter q, (7).

Kevin F. Barrell and Colin Pask 297

the value of D for which Ptr has fallen by 10 dB, as a functionof the incidence parameter a.

The generalized parameter formalism does not give an exactevaluation of (26), but its great merits are that Ptr may beobtained approximately with ease, a multiparameter problemis reduced to a single-parameter problem, and the importantrays which are little attenuated for a given D value are readilyidentified by the conditions (31) (see also Fig. 6). This lastpoint is of great value when discussing pulse propagations(see below). For a diffuse source the approximation is accu-rate to about 5%.3 We have checked the accuracy of thegeneralized parameter formalism for parabolic index fibersexcited by a single parallel beam and found satisfactory re-sults. For example, over the whole parameter space 0.1 < Oc< 0.2, 30 < V < 70, 0.1 < a < 0.9, the value of Ptr(D) for prac-tically relevant D values is given with errors of order 10%. Inmost cases the error is less than 5% and only for some oddparameter region does it approach 20%. We conclude thatthis method provides a very useful scheme for discussing therole of tunneling rays in graded-index fibers.

B. Pulse propagationWe now turn to the behavior with respect to time of power

propagating away from the source and to do this we discussthe impulse response for graded-index fibers with the powerlaw profile (7) and excited by a parallel beam as in Fig. 1.Some results are already available,12-'4 particularly for boundrays, so here we make use of the generalized parameter tech-nique to evaluate the tunneling ray contribution.

The ray transit time T in a power law profile depends onlyon : and for small material dispersion the time to travel anaxial distance z is given by9

T = ( ) (2/3 + /n)' (32)

where c is the speed of light in vacuum and no is the on-axisrefractive index (2). [Material dispersion is easily incorpo-rated9 by changing the coefficients of / and 1/3 in (32).] Sincea and T characterize the rays we can write

To co -10 \ <- 0 2

3

00 4 2 3 5 I 6 75 10 X

lto

FIG. 9. Impulse response Q (34) for a parabolic-index fiber, q = 2, excitedby a parallel beam with incidence parameter a = 0.5. rl Is the leadingedge time given by (34) and (35) and To = zn,/c. The curves are labeledby the generalized parameter D, (28).

Using (15) and (32) produces

rl = (zn./(2 + q)c) [2(1 - a2 sin 2oc)1/2+ q(1 - a 2 sin2 Oc)-1/2]

L (zn0 /c)[1 + a sin202 (q - 2)/2(2 + q)], q > 2 (35)

where we have assumed a sinc << 1, which is usually validsince Oc is very small in practice. This also means that -1r isonly weakly dependent on a and q.

The shape of the pulse is similar to that shown in Fig. 9 forother beam incidence parameters, so we now explore the be-havior of the pulse width for excitation and fiber variations.We define the width W by

T2 - T1 = (zn./c)W, (36)

and then W will be a function of a, q, and D. Results areplotted in Figs. 10 and 11 which show that although the im-pulse response for a 5 0.5 is similar in shape to that shown inFig. 9, the trailing edge corresponding to tunneling rays mayform the major part of the pulse e.g., a = 0.8 in Fig. 10.

P = X X J(T,b) d: dl, (33)

where J is the source function and the limits enclose those A,I values which define the bound rays, (12), and the effectivetunneling rays, (31), as described in Sec. IIA. If we now in-tegrate over T and use (32) to change variables from , to T-. weobtain

9

w i05

(34)P = Q(,) d,

where Q is the fiber impulse response. We should note thatnot all possible bound and tunneling rays are excited by anindividual parallel beam [see (15)], and mathematically thismeans that J may be zero for some A, I values. Also, T, or T2

may depend on a. The impulse response for a parabolicindex, q = 2, fiber is shown in Fig. 9 for a = 0.5.

We now consider the effects of varying the incidence pa-rameter ce tsee (8)N and the fiber parameters q and D. Forillustrative purpose we consider fibers with q > 2. In this casethe pulse leading edge travels the distance z in time TJ whichis related to the largest value of S given by the exciting beam.

298 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

.0 8

I 0 1 0 0 I 05 o5-C 0 O 0-1 0-2 0-3 0-4 0-5 00

D

FIG. 10. Pulse width W [as defined by (36)] for a parabolic index, q = 2,fiber versus generalized parameter D, (28). When D =-X, all tunnelingrays are present, and D = - corresponds to total attenuation of tunnelingrays, i.e., bound rays only. The curves are labeled by the incidence pa-rameter a (8).

Kevin F. Barrell and Colin Pask 298

7

6

5

/4

W13

2

0

10 a=0 2 5A.2

*...

, 3. . . ......

I .......

' .5 I ' 'S I-co 0 01 02 03 0/4 05 o

D(q)

FIG. 11. Pulse width W [as defined by (36)] for graded-index fibers excitedby a parallel beam with incidence parameter a = 0.5, versus generalizedparameter D, (28). When D = -X, all tunneling rays are present, and D= - corresponds to total attenuation of tunneling rays, i.e., bound rays only.The curves are labeled by the power law profile parameter q (7).

IV. OUTPUT PATTERNS

We now turn to the output from a fiber excited by a diffusesource, a Lambertian for example. We then have all raysexcited, and viewing the exit face at angle Oi we will see exactlythose rays shown in Figs. 1 and 4 (but but with their directionsreversed, of course) which are classified as bound or tunnelingrays. Based on the results obtained in Sec. III we know thatthose rays will emanate from particular areas of the fiber face,thus giving rise to a pattern of light and dark areas. For thestep-index fiber this pattern takes on a simple form 7' 8: Whenthe fiber exit face is viewed at angles 6i such that a < 1, itappears bright all over; when a > 1 a dark band forms acrossthe fiber face increasing in width as a increases. This is il-lustrated in Fig. 12 along with the patterns for graded-indexfibers with the power law profile (7). (When viewed at anangle, the circular fiber face will appear to be elliptical. Ingeneral, since O, is small, Oi will usually be small; the resultingeccentricity is also small and therefore not shown in Fig.12.)

A few observations can be made from Fig. 12 and the theoryin Sec. III. For large q, the pattern approaches the step-indexfiber pattern. For q < 2, the whole face is dark when a > 1,and for q > 2 the same applies for a > (q/2)1/2 . For a < 1, agraded-index fiber with a monotonic profile will always havean output pattern consisting of a circle of radius rmax due tobound rays plus some additional bright areas exterior to thiscircle due to tunneling rays. The radius rmax is given by (17)and we note that if rmax is measured for a given a, then (17)determines the value of the profile function G. Now Eq. (18)enables us to deduce that along the y axis (q = 900) there willbe no tunneling ray contribution, and hence the bright patternonly extends out to r = rmax in that direction. Thus obser-vation along the y axis of the pattern (see Fig. 12) will give aset of a, rmax values and therefore a profile determination via(17). This profile determination requires no special sourceconditions beyond a general diffuse source and no tunnelingray corrections as are necessary in most methods, e.g., Ref.15.

We have spoken of patterns of light and dark on the exit

299 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

Z j Observationt Direction '

N1 8 n. sin G

n sin 9

"": : x Cos S~C = nc[I/ no0 I

aC = 0 4 0 8 10 11 1 6

Step -( I

10( 0¢ I )

3 ( )¢>2 ( 1 I 1

qO = * (

FIG. 12. Appearance of fiber output face when viewed at angle Oi for afiber excited by a diffuse source. The drawings are labeled by thepower-law profile parameter q, (7) and the direction parameter a. Thedashed circles contain the areas illuminated by bound rays. The slightellipticity of the output face is not indicated; this is a small effect when 0,is small.

face, but any attenuation mechanism will change the intensityof the bright portion. The patterns shown in Fig. 12 will beaccurate for short lengths of fiber, but for longer lengthstunneling ray attenuation will be important. The resultingpatterns could be obtained to a good approximation by re-peating the calculations in Sec. II using the domain (31) in-stead of (13). Note that the condition on rmax is independentof tunneling ray corrections, and even scattering process willonly produce refracting rays, which could then exit from thermax < r < p region of the y axis. Such rays leave the fiberextremely rapidly and rarely need to be considered.

V. APPENDIX

The generalized parameter formalism is outlined in detailin Ref. 3 where the variables

y = 2(n 2 - f 2 )/qn 202 (Al)

p = T(2/q)1/2/n 00C, (A2)

are introduced. Using the approximate electromagnetic at-tenuation coefficients 3 for the general power law profile (7)gives"l

Pb In [ ± + (-- 1) '2] - (p2 - y)1/2 = D(q)

where D(q) is given by (28) and Pb iS given by (A2) when T =!Low. Alternatively, substituting

coshx = Pb/l'Y,

(A3) may be written

x coshx - sinhx = D(q)/(2qy)1/2 .

(A4)

(AM)

The procedure for calculating T = lLow(f,D) is now as follows:For each $, find y from (Al), substitute into (A3) and solve theresulting equation for Pb, and then calculate T = [LOW by

Kevin F. Barrell and Colin Pask 299

I I I . . l

substitutingp =Pb in (A2). Thevalueof ,owoccurswhenthe T = Ti ow and T =T,, curves intersect (see Fig. 6). Thusto find /LOw the following steps are required: Convert Tmax(1

to T nax(Y) using (Al); find Prax(Y) by substituting I = ax in(A2); set Pb = Pmax(Y) in (A3) and solve the resulting equationfor y; substitute that solution into (Al) to obtain /3LOW. Theseare straightforward computational problems. Simple ana-lytical approximations for 3Low(D) and TLOW(G,D) are givenin Refs. 3 and 11.

'M. J. Adams, D. Payne, and F. M. E. Sladen, "Leaky rays on opticalfibres of arbitrary (circularly symmetric) index profiles," Electron.Lett. 11, 238-240 (1975).

2W. J. Stewart, "Leaky modes in graded fibers," Electron. Lett. 11,321-322 (1975).

3C. Pask, "Generalized parameters for tunnelling ray attenuation inoptical fibers," J. Opt. Soc. Am. 68, 110-116 (1978).

4A. W. Snyder, C. Pask, and D. J. Mitchell, "Light acceptance prop-erty of an optical fiber," J. Opt. Soc. Am. 63, 59-64 (1973).

5A. W. Snyder, D. J. Mitchell, and C. Pask, "Failure of geometricoptics for analysis of circular optical fibers," J. Opt. Soc. Am. 64,608-614 (1974).

6D. J. Carpenter and C. Pask, "Geometric optics approach to opticalfibre excitation by partially coherent sources," Opt. Quant. Elec-tron. 9, 3;3-382 (1977).

7R. J. Potter, "Transmission properties of optical fibers," J. Opt. Soc.Am. 51, 1079-1089 (1961).

8N. S. Kapany, Fiber Optics (Academic, New York, 1967).9A. Ankiewicz and C. Pask, "Geometric optics approach to light ac-

ceptance and propagation in graded index fibres," Opt. Quant.Electron. 9, 87-109 (1977).

10A. W. Snyder and J. D. Love, "Attenuation coefficient for tunnellingleaky rays in graded fibres," Electron. Lett. 12, 324-326 (1976).

"1A. Ankiewicz and C. Pask, "Tunnelling rays in graded index fibres,"Opt. Quant. Electron., in press (1978).

12R. A. Sammut, "Pulse dispersion in partially-excited graded-indexfibres," Opt. Quant. Electron. 9, 61-74 (1977).

13L. Jacomme, "Approximations in the evaluation of the impulseresponse of nearly parabolic fibres," Opt. and Quant. Electron. 9,197-202 (1977).

14L. Jacomme and D. Rossier, "Parallel beam impulse response ofmultimode fibres: numerical results," Opt. Quant. Electron. 9,203-208 (1977).

15M. J. Adams, D. N. Payne, and F. M. E. Sladen, "Length-dependent

effects due to leaky modes on multimode graded-index opticalfibres," Optics Commun. 17, 204-209 (1976).

Aberrations in curved graded-index fibersAruna Rohra and K. Thyagarajan

Department of Physics, Indian Institute of Technology, New Delhi 110029, India

A.K. GhatakInstitute of Advanced Studies, Department of Applied Mathematics, Australian National University, Canberra, ACT, 2600,

Australia(Received 12 September 1977)

Graded-index fibers can be efficiently used as image relays and therefore it is necessary to studythe aberrations in the images transmitted through these media. Since curvature in fibers is inevitable,aberrations in curved graded-index fibers are also important. In this paper, we have obtained explicitexpressions for the second- and third-order aberration coefficients in curved fibers which reduce tothe ones for the straight fiber as the radius of curvature of the fiber tends to infinity. It is shownby numerical calculations that the total aberrations (up to third order) in a graded-index fiber dependcritically on the values of grading parameters a and /3. It is also shown that for small values of gradingparameter, a - 2 cm-, curvature effects are significant for radii of curvatures less than - 25 cm,whereas for larger values of the grading parameter,tions for radii of curvatures less than about 10 cm.

1. INTRODUCTION

Graded-index fibers have been used as image relays aswell as light transmission guides.--8 A large number of paperson the studies of aberrations in graded-index media have beenreported but they have been restricted to straight fibers.2' 5-8

Since curvature in fibers is inevitable, aberration analysis ofthe image transmitted through curved fibers is also of con-siderable interest and forms the subject of the present paper.Paraxial propagation characteristics in curved graded-indexfibers have been studied by some authors.9-12

In this paper we have obtained explicit expressions for thesecond- and third-order aberration coefficients of a graded-index fiber bent along the arc of a circle, starting from Ham-ilton's equations.15"13' 14 It is interesting to note that since sucha system does not possess rotational symmetry, even-orderaberrations are also present. The even-order aberrations tend

a - 50 cm-', curvature affects the aberra-

to zero as the radius of curvature of the fiber tends to infinity,i.e., as the fiber becomes straight.

We have used our analysis to obtain the explicit aberrationcoefficients in a specific graded-index medium [whose re-fractive index distribution is independent of the t cordinate(see Fig. 1)] and have shown that for such a medium some ofthe second-order aberration coefficients are zero.

Using typical values of various parameters, it is shown thatthe effect of curvature (which is exhibited through second-order and part of third-order aberrations) is critically de-pendent on the values of a, /3 [a and /3 are the grading pa-rameters of the medium; see Eq. (47)]. In flexible plasticfocusing rods, recently fabricated by Ohtsuka et al.," thevalue of a is -2 cm-'. It has been shown that for such smallvalues of a, the curvature plays a significant role for radii ofcurvatures less than about 25 cm. On the other hand, in the

300 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979 0030-3941/79/020300-11$00.50 C 1979 Optical Society of America 300