Generalized parameters for tunneling ray attenuation in optical fibers

Colin PaskDepartment of Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra, A.C.T. 2600, Australia

(Received 11 September 1976; revision received 27 December 1976)

Tunneling rays in optical fibers occupy a certain domain in ray parameter space and also have associatedattenuation coefficients. We show how to avoid the involved calculation of attenuation coefficients by using aneffective domain in which the tunneling rays may be taken to have zero attenuation as a good approximation.This effective domain depends only on one new generalized parameter embracing all relevant fiber and lightparameters. This program is carried out for step- and graded-index fibers. Examples are given of the use ofthe effective domain concept in power and pulse width calculations.

1. INTRODUCTION

Geometrical optics, the theory describing light aspropagating along well-defined lines or curves calledrays, has always had great conceptual appeal and itssimple application is sufficiently accurate in solvingmany physical problems. The development of wavetheory gave a foundation for geometrical optics interms of local plane waves and, more than that, to thegeometry could now be added the necessary amplitudeeffects embodied in Fresnel's laws, for example.'Wave theory also explained optical phenomena such asdiffraction which appeared to be beyond the realm ofgeometrical optics. However, such is the appeal ofgeometrical optics that ways have now been found2 todeal with even diffraction phenomena in terms of raysand a set of amplitude rules derived from wave theory.It therefore comes as no surprise that geometrical op-tics has always been used in the field of fiber optics. 3

The optical fiber consisting of a circular cross-sec-tion dielectric rod of uniform refractive index sur-rounded by an infinite uniform cladding of lower refrac-tive index provides a wave theory problem which isamenable to solution and thus is a system for which theapplicability of geometrical optics may be ascertained.Calculations of power propagating along this fiber forcoherent4 and incoherent5 excitation showed that geo-metrical optics did not agree with wave theory in themultimode, asymptotic limit as expected. We nowknow that a further wave theory result must be builtinto geometrical optics to account for reflections at acurved surface. Rays cannot be classed as refractingor totally internally reflecting (trapped); the trappedrays must be subdivided into bound rays and tunnelingrays, the latter requiring a wave theory transmissioncoefficient to account for their only partial reflectionsat the core-cladding boundary. 7

The tunneling rays correspond to part of the radiationfield or continuous spectrum of the dielectric rod. Thesolution of physical problems in terms of the wavescorresponding to the continuous spectrum is extremelyinvolved8 and the utility of the ray technique cannot beoveremphasized. The equivalence of the two ap-proaches is now firmly established in terms of analyt-ical formulation of attenuation coefficients9"10 and powercalculations 11"12 The use of these concepts has beenreadily extended to graded-index fibers.

Although the step from continuous-spectrum, wave

theory contributions to tunneling rays with an accom-panying attenuation coefficient is an enormous sim-plification, calculations with those rays can still bevery involved. The purpose of this paper is to effect afurther simplification. A tunneling ray traveling awayfrom the source attenuates with a coefficient a whichdepends in a complicated manner on the ray parametersand the light and fiber properties. We show that, to avery good approximation, tunneling rays may be dividedinto two classes in one of which the rays effectivelyhave a= 0, while a-' -o for the remainder. The divisiondepends on a single generalized parameter which em-braces all relevant details, i. e., the distance from thesource, the wavelength of the light, and the fiber coreradius and refractive indices. The calculation of rap-idly varying a's is thus avoided and at each stage theimportant tunneling rays are identified.

In Sec. II we expand these introductory remarks inorder to define and motivate our objectives more pre-cisely. We give the broad details of our approach inSec. II[. This theory is given a sound mathematicalbasis in Sec. IV and some illustrative applications fol-low in Sec. V. A summary and conclusion forms Sec.VI.

11. OVERVIEW

Before beginning in earnest, it may be appropriateto express the ideas in the Introduction in more detailand to nominate references which extensively reviewthe basis of the problem.

Our basic concern in fiber optics is with the powertransported a distance z along a fiber away from thesource. Finding the total power entails the evaluationof3,13

Ptotal(Z) = f PW(e)) e d, (1)

which represents summation over all rays labeled by 4and belonging to the total domain T. Here 4 is writtensymbolically for one or several parameters needed inpractice to fully label the rays. We give details in thenext section and refer the reader to the substantialliterature on this problem, for example, to Refs. 3, 7,13, and 14 for step-index fibers and Refs. 15-17 forgraded-index fibers. Pi, is the initial distribution ofrays and is determined by the source used to excite thefiber. A ray labeled by 4 in the fiber attenuates ac-

110 J. Opt. Soc. Am., Vol. 68, No. 1, January 1978 0030-3941/78/6801-0110$00.50 � 1978 Optical Society of America 110

cording to the constant a(o,). This loss coefficient de-pends on the fiber properties. Generally, ca will de-pend on absorption and scattering processes, and onthe behavior of the ray each time it strikes the core-cladding boundary in a step-index (e. g., whether itrefracts out), 13,14 or each time the ray passes throughthe radial extremes in its path in a graded-index fi-ber. 17,18

Equation (1) represents the central problem. Otherquantities may be derived from it. For example,Pin(4)) exp(- oaz) represents the distribution of rays withrespect to 4 and it may be used to calculate various ex-pectation values or moments. A conversion from 4 tothe transit-time coordinate would reveal the pulseshape.

According to geometrical optics3 we can write thedomain as a sum of trapped and refracting ray terms,

-P = Ttrap +Tref

We further know that the trapped rays must be dividedinto bound and tunneling rays, 14"17

Strap = 4bound + Stun ,

Let us consider Eq. (1) for an ideal fiber so that adepends only on wave propagation effects such as re-fraction. We know that

d!Y= O I CJ (i bound

01 - a, (E~ 'kref

and so

Ptotal(Z) = Iun Pin(p) dof + I.tu Pin(Y)) e-a (O°z digo

The integral over Aref has been avoided (being zero forall practical purposes) and the first integral is simpleto evaluate, usually analytically in closed form. Thesecond integral is very difficult to deal with because a!is a complicated function. It has been evaluated in afew cases using numerical techniques. However, astudy of these attenuation coefficients shows that the in-teraction of wave and curved interface properties pro-duces very rapidly varying functions (see Figs. 2 and3 in Ref. 13, for example). This result suggests that wetry to obtain a new domain 'tun such that

az- 0 , s Ttun

cYzo x T I E tun - Stun

so that Eq. (1) finally reduces to the simple integral

Ptotal(Z) = bf P in(Y) doA~bound +

4'turn

The principal aim of this paper is to produce such aresult. There is, however, a further gain in simplici-ty. We shall demonstrate that 'tun depends on only onegeneralized parameter, so that a single curve for Ptota1will cover all distances and fibers of a particular type.(The reader might compare the many curves presentedin Ref. 13. )

Before proceeding in detail, one last technical matterrequires mention. We deal with step- and graded-in-

111 J. Opt. Soc. Am., Vol. 68, No. 1, January 1978

dex fibers separately, rather than treating the step-index profile as a limiting case of the latter. The rea-son for this is that different mathematical techniquesare used to derive the tunneling ray attenuation coef-ficients in the two cases. For step-index fibers a gen-eralized Fresnel law approach is available, 14 whileWKB theory is used'8 for graded-index fibers wherenear parabolic profiles are of central interest. TheWKB techniques cannot be used in highly graded sys-tems and care must be taken when using them in lim-iting cases.

III. NOMENCLATURE AND BASIC RESULTS

In this section we explain the philosophy of our ap-proach and then lead on to the more mathematical justi-fication and results in subsequent sections.

A. Fiber parameters

We consider fibers with a core of circular cross-section radius p, surrounded by an infinite uniformcladding with refractive index n, 1. The refractive indexalong the fiber axis is no and we allow two types of pro-file for the core refractive index n,,:

step index: n0 0(r)=no , r p (2)graded index: n00 (r)=no_[ll- '(r/p)1" , r p (3)

where r is the radial variable in the fiber cross section,q is a positive constant, and

SC 1 - (n,, (4)

Then, for light of wavelength X, we have the waveguidedimensionless parameter

V= (2fpno/X)0, . (5)

We note that for practical multimode fibers, V is large(> 40 say) and Oc small (of order 0. 1), We take the fiberto be aligned with the positive z axis so that rays orig-inate from the exciting source at z = 0. The generalizedlength or parameter L in terms of which we seek tospecify power attenuation is now defined by

step index: L3e =(220 )

Dt= InLst ,

graded index: Lr(q)= c Z)

Dgr(q)= lnLgr(q),

where

[1 + (q12)]" a

(6a)

(6b)

(7a)

(7c)

It is often convenient to use the related generalizedparameters D as defined above. We note that for para-bolic-index fibers, q = 2, N(q)= 1. The importance andmathematical origin of these parameters will becomeapparent in the following sections.

B. Tunneling ray attenuation: step index

Rays in a step-index fiber may be characterized bythe two parameters 9 " 3

Colin Pask 111

R = (sin0,)/0 0 X

t=sin00

(8)

(9)

where 0, is the inclination of the ray to the fiber axis,and 0, is the angle between the ray projection onto thecross-sectional plane containing the point of incidenceat the fiber boundary and the tangent at the point ofincidence in the plane (see Fig. 1). The angles 0, and00 are preserved as each reflection so that R and t areray-path constants which may be used to completelylabel the ray. Tunneling rays have the (R, t) space do-main 9 "13

{tunneling rays:l:t-R-1 5

(10)

As a tunneling ray propagates along the core, R andt remain constant, but the power is decreased by radia-tion at each reflection at the core-cladding boundary.We define the attenuation coefficient cx by9

,1 4

P() = P(O) e-c"IP . (11)

The attenuation factor may be written in the form

(12)az/p = ht(R, t)[ f3 t(R, t)L3 t ]V

where h8 t is independent of V and

h8,t- 1 (13)

for most rays of interest. (We deal with hIzt and f8 tmore precisely in Sec. IV.) Now recall that we areconsidering large V fibers, so that

(ftLt)v 0 and e 'caz 1 for fstLit< 1 ,

(fatLat)v co- and e-"1' 0 for ftL,,t > 1

(14)

(15)

We thus reach the result that the tunneling rays may bedivided into two classes according to whether f3 tLt isless than or greater than one; in the latter case therays are greatly attenuated and may be ignored, whilein the former case the rays may be used with no atten-uation at all. This means that for a given Lt we mayignore attenuation coefficients if we replace Eq. (10) byan effective domain, and this program is carried out indetail in Sec. IVA.

C. Tunneling ray attenuation: graded index

We now repeat the above procedure for rays in a fiberwith a variable core refractive index. If nc0 only de-pends on the radial variable r, rays in the fiber may becharacterized by two path invariants' 5"16 :

,3=n(r) coso,(r), (16)

I = (rlp)n(r) sinO,(r) cosk(r) . (17)

The ray angles are defined by: 0, = inclination of ray tothe z direction;, = angle between ray projection onto thethe fiber cross section and the azimuthal unit vector atthe point (see Fig. 1). The angles 0, and t vary alongthe ray path, but 3 and I remain constant and hence theirvalues completely specify the ray. (See Ref. 16 for afull discussion.)

We now concentrate on n,-0(r) given by the profile inEq. (3) and introduce the new parameters y and p by

P0 fljr)/ /,i,'// / //,,,,1 /J,/

p nO

FIG. 1. Ray paths in optical fibers of radius p, aligned withthe z axis and with cladding refractive index n,. Lower draw-ing: step-index fiber, core refractive index no. Rays makean angle 6O with the z direction. Rays projected onto the fibercross section make an angle 0. with the tangent at each reflec-tion. Upper drawing: graded-index fiber, core refractive in-dex n. 0(r) depends on the radial variable r. At a given positionof the ray path the ray direction makes an angle O,0=G2 (r) withthe z direction. In the fiber cross section, the projected raydirection makes an angle d = 4(r) with the azimuthal direction.

y=(n., - C2/ nb

and

p = (7 /n 00,) (2/q)' 2 .

(18)

(19)

These allow us to specify the tunneling ray domain16"17

in terms of dimensionless numbers as follows:

O _•y _ 1

tunneling rays: .(20)

The tunneling ray power attenuation is given by Eq.(11) and, as we prove in Sec. IV, we can write

aZ/p = hgr(y, P) [fAr(Y, p)Lg.~(q)]V

with hgr independent of V and

hgr(YP)' 1 -

(21)

(22)

The argument is then the same as that used above forthe step-index fiber leading to the result that tunnelingrays in a graded-index fiber may be divided into twoclasses depending on whether (fgrLgr) < 1 or > 1. Thelatter condition specifies rays greatly attenuated andthe former gives those rays belonging to a modified(y, p) domain in which the attenuation coefficient maybe set equal to zero as a good approximation. This isthe program to be followed in Sece. IVB.

IV. CALCULATION OF EFFECTIVE RAY DOMAINS

In this section we see the mathematical origin of gen-eralized parameters, justify the claims made in theprevious section, and calculate explicitly the effectivedomains for tunneling rays. We shall see that theserequire only knowledge of the generalized parameterD3 t or Dgr for their specification.

112 J. Opt. Soc. Am., Vol. 68, No. 1, January 1978 Colin Pask 112

1

t 0 ,

o.2 \o' "0'o

1 2 3 4 5 6R

FIG. 2. Tunneling ray domains in (R, t) space for a step-indexfiber. The curves are labeled by the generalized parameterD~t [Eq. (6b)], and those tunneling rays which have not beeneffectively attenuated have (R, t) values in the region borderedby the R axis, the t axis, and the appropriate curve.

A. Effective ray domain: step index

For the tunneling rays of interest in fiber optics theattenuation factor may be writtenl0,13

az 2z01 R2(1 - t2R2)"2e / 2V(1 -R2t2)3/2

P = (1-R ) l / 2 exp3- 3 (R2 -) )(23)

This expression can be written directly in the requiredform, Eq. (12), with L, as already defined in Eq. (6)and

fst(R, t)L~t = 1 . (26)

We then note from Eq. (25) that those values of R and tsmaller than the values given by Eq. (26) satisfy ftD 3 t< 1 and thus fall into the set satisfying Eq. (14). Equa-tions (25) and (26) define the curve

t= R -('(1=L,,)(R

or

t = 1 [1-(2 D~t(R -1)) 3 ] 2 , (27)

where we have changed from L5 t to D3 t using Eq. (6b).Since t is real, Eq. (27) also gives us an upper limitRb on R:

Rb(DSt) = [1 + (2/3D3 t)]'/' * (28)

The boundary curve Eq. (27) for t can be written

t= tb(R) = 1 (1 - ) ]3/21/2 (29)

We can now replace Eq. (10) by the effective domain

(effective 1 ' R < Rb (Itunneling rays: 0 < t tb 3

Plots of the effective tunneling ray domain are givenin Fig. 2 for various values of Dt of interest. For ex-ample, note that fibers with VŽ 40, OS _ 0.3, and-lengths zip < 10' have D8 t < 0. 45. We demonstrate theuse of these domains in Sec. V.

B. Effective ray domain: graded index

The general philosophyl0 "l8 for calculating tunnelingray attenuation coefficients leads to the expression

COZ/p = Tz/zp , (31)

where T is the transmission coefficient and zp is thedistance between consecutive ray-path outer caustics.An adequate description of T is given by WKB theoryleading to an expression's which can be written

We immediately see the mathematical appearance ofLt-

We can check the validity of Eq. (13) by examiningthe expression (24) for various tunneling rays. Al-though ht can become large, it does so only for a smallnumber of rays. For example, when the rays are ex-cited by a diffuse, Lambertian source and Ice 0. 1,over 60% of the tunneling rays have R < 2 and ht < 4.Over 90% of tunneling rays have R < 5 and ht < 29, thevast majority having ht << 29. Only rays with verylarge R values and t- 0 have large h8 t values and theyare not of major importance. For a more highly di-rected source the rays are even more concentrated inthe small R region. On this basis and the evidencefrom calculations using this approach, we have con-cluded that Eq. (13) is a good assumption.

We can now explicitly determine the effective tunnel-ing ray domain in (R, t) space by finding those rayssatisfying Eq. (14). To do this we must solve

113 J. Opt. Soc. Am., Vol. 68, No. 1, January 1978

T=exp[- VF(y, p)], (32)

where F is a function only of the ray-path parametersy and p [Eqs. (18) and (19)]. In general, it is difficultto calculate zP exactly, but we can write'9

Z=p [r_0~( qy 1121 (I Y) (1/q)- (1 /2) Iq 330':i 1ok( 2 )] (+ 2)Iq).(3

In Eq. (33) I(q) is an integral' 9 which when evaluatedtakes the form

I(q) = 7rN(q) g (q , y.p) ,

where N(q) is defined by Eq. (7c) and

g(q, y, p) 1 .

(34)

(35)

For the parabolic-index fiber, g(2, y,p) is exactly 1for all values of y and p. For other practical valuesof q, Eq. (35) is certainly valid; when 1 4 q u 8, forexample, 0.95 <g< 1,3.

The above considerations give us exactly the required

Colin Pask 113

(24)

(25)

h,(R, t) = R' /(1 - R 2t2) 1/2M - R?-02,) )

f.t(R, t) = exp 2 (1 - R 2t2)3 / 2( - _� (R 2 _ 1) J I

p

0' I0 0.2 0.4 0.6 0.8

yFIG. 3. Tunneling ray domains in (y,p) space for a parabolic-index profile fiber. The effective domain for a given value ofthe generalized parameter D,,(2) [Eq. (7b)] is the area en-closed by the p axis, the line p = (1 +y)/2, and the appropriatecurves [Eq. (39)1 as shown. The initial domain correspondingto rays just at the source is bordered by the curve p 4y.

form for az/p, Eq. (21), with

fg,( y, p) = exp[-e F( y, p)]

and

(1 + q=/2)() 1/2 ('1

/ Q

(36a)

(36b)

Again the natural appearance of the generalized param-eter has been demonstrated.

The only outstanding requirement for applying the ef-fective domain argument in Sec. IIIC is the establish-ment of the validity of Eq. (22), i. e., to show that hIi-1 We have seen that g- 1, and since 9, is small thewhole denominator in Eq. (36b) is close to unity. Thenumerator is also close to one (varying from 0. 82 to1. 83 for 1 - q - 8) so that Eq. (36b) does indeed givehgr. 1.

The explicit form for the effective tunneling ray do-main is now obtained by following the procedure usedabove for the step-index fiber. We next give the detailsfor one specific, important graded-index fiber.

C. Effective ray domain: parabolic-index profile

The parabolic-index profile, q = 2, is of particularinterest because of its dispersion characteristics and Tand zP in Eq. (31) may be obtained exactly. An approx-imate analytical form for T which gives good accuracyleads to 20

f (Y P) =(-p+ ,2 ) exp[2(p2 -y)"/2 ],

To obtain the effective domain we must solve

fr( Y, p)Lgr(2) = 1

or, using Eq. (7b),

lnfgr( y,p) + Dgr(2) = 0 .

( 37)

(38a)

(38b)

Thus the boundary curve Pb( y) is given by the solutionof

Pnly - 2pIn[p+ (p2 - y)1 2 ]+ 2(p2 y)1 /2+Dg(2) 0(39)

114 J. Opt. Soc. Am., Vol. 68, No. 1, January 1978

Since the original domain requires p - (1+ y)/2 [Eq. (20)with q= 2], Eq. (39) gives us an upper limit Yb for ywhen p is set equal to (1 + y)/2. Thus

Yb 2 (1 + Yb)i Inyl = 1 + Dgr(2) .The effective domain is defined by

{ effective 0 - Y - Yb

I tunneling rays: Pb P (1+y)/2 |

(40)

(41)

Plots of the effective tunneling ray domain are givenin Fig. 3 for values of Dr(2) of practical interest.

V. APPLICATIONS OF THE EFFECTIVE DOMAINCONCEPT

We now give some examples of the use of the effec-tive domain concept. It lets us avoid the calculation ofthe attenuation coefficient a and the difficulties arisingfrom its rapidly varying behavior. 7 Instead we use adomain which shrinks as the generalized parameter,D8 t or Dgr, increases. The role of the source is toprovide a weighting of the rays over the initial domainas explained in Sec. II.

A. Step-index fiber

We consider a multimode fiber excited by a Lamber-tian source. A full discussion is given in Ref. 13, lead-ing to the following expression for the power Ptr(z) oftunneling rays at distance z from the source:

Ptr(Z) 8 0 R- R dRf0

(42)We have taken 0. small and ignored terms of order OCin the constant factor multiplying the integrals. We nowdispense with a but change the ranges of integrationaccording to Eq. (30). The power is seen to be a func-tion of Dt alone. We can carry out one of the integra-tions, leading to

Ptr(Dst) b8 R tb

Ptr (initial) - 7r 1 0j ((43)

=([sin- ltb -tb(l -t' I/ 2]R dR , (44)

where Rb=Rb(DSt) and tb= tb(R) are defined by Eqs. (28)and (29). Various approximations can be made in Eq.(44), but it is a trivial integral to evaluate numerically.

We compare results calculated from Eq. (44) with theexact results13 using integrations including attenuationcoefficient a as in Eq. (42). To do this we find the to-tal power P as the sum of Ptr and the bound ray power.Results for P using many combinations of practical val-ues for V, 0,, z/p do indeed fall on or very close to asingle curve when P vs D3 t is plotted13 ,

21 thus confirm-

ing the validity of the generalized parameter concept.That curve is the dashed line in Fig. 4. The solidcurve is given using Eq. (44) and the accuracy and re-liability of the effective domain approach is apparent.

The approach developed in this paper may be appliedto pulse response calculations. As an example, let us

Colin Pask 114

(1 _ t2)-1/2 t2 e-01 (R, t)zlpdt .

0.91

0.8

0.6

0.5

0 I I I01l 0.3 0.4 0.5Dt

FIG. 4. Total power P in a step-index optical fiber excited bya Lambertian source vs generalized parameter Dgt [Eq. (6b)J.The solid curve is given by the tunneling ray effective domainapproach, while the broken curve is based onexactcalculationsusing attenuation coefficients.

calculate an effective pulse width. The transit time T

for a ray propagating with angle 0, to the z axis is

T(O) = Zno/C COS0, , (45)

where n0 is the core refractive index and c is the speedof light in vacuum. In terms of R [Eq. (8)] we have

T(R) = T(O) (1 -R 2 (46)

The bound rays have 0 S R > 1 and tunneling rays have1 R Q 01, so that for a diffuse source exciting all raysthe transit times range from T(0) to . However, weknow that there is an effective upper limit on R givenby Rb, Eq. (28). Thus an effective pulse width W isgiven by

W(Dt) = r(Rb) - T(O)

= r(0)[(l -R202)-" 2 - 1]

= 7(0)2 o0[l+ (2/3Dst)] .

(47)

(48)

We have assumed RbBC is small as is the case for prac-tical situations. Equation (48) gives a simple indicationof the effects of tunneling rays on pulse widths in step-index fibers. For a prescribed critical angle Oc, theeffect of all other parameters is contained in the gener-alized parameter Dt.

B. Parabolic-index fiber

As a final example we calculate the power in a para-bolic-index fiber excited by a Lambertian source. Thetunneling ray power Ptr is given by standard acceptancetheory 16 which in terms of our variables y and p [Eqs.(18) and (19] gives

exp[- a(y,p)z/p] dp . (49)______ = 124 dy j(l1v)/2

Changing to the generalized parameter Dg,(2) (we dropthe argument in this example) and using the effectivedomain Eq. (41) instead of the full attenuation factorleads to

Ptr(initial) 1 Dyfdp (50)

= 12 yb( +y -Pb)dY

= 6Yb+ 3Yb- 12 Pbdy .ba (51)

Equations (39) and (40) give Yb = Yb(Dg) and Pb=P,(Y). Wemay now proceed numerically or establish approximateanalytic forms for Pb and Yb, For example, a good ap-proximation for Pb is

2b(Y)Y +2 ( ylb/)2 (y/yb)1/4

Using this in Eq. (51) produces

Ptrt(D )- _ 9Yb+ 8y'/2 ) .

One simple approximation for Yb is

Yb=exp(- 3. 25Dgw - 1.1),

(52)

(53)

(54)

a formula which is particularly good for 0. 2SDgr S0.5.(Note that this covers most of the practical range; e.g.,for V=50, 0,=0.1, p=80p, wefind 100m z 10kmcorresponds to 0.211 Dg. S 0. 304. ) Equations (53) and(54) give a good approximation for Ptr, the error being•5% for 0.2•Dr •0.5.

The effective domain technique gives very good re-sults for the universal power curve shown in Ref. 21.The exact results obtained using the full attenuationcoefficient analysis are reproduced with an error oforder 1% or 2% for practical values of Dgr.

VI. SUMMARY AND CONCLUSION

The parameters V and 0Q [Eqs. (4) and (5)] have longbeen used to characterize optical fibers. This paperdemonstrates the appearance and role of the new gen-eralized parameters Dt and Dgr [Eqs. (6b) and (7b)].Tunneling ray attenuation is simply taken into accountby using the effective ray domains, Eqs. (30) and (41)which require only knowledge of D~t or D1,, for theirspecification. The calculation of rapidly varying at-tenuation coefficients is thus avoided. The effective do-main concept allows us to identify exactly which tunnel-ing rays are important in a given situation and the de-pendence on a single parameter embracing all fiberparameters enables simple comparisons to be made be-tween different fibers, For example, the curve in Fig.4 may be used to see the effect on total power attenua-tion in the fiber of varying one or more of the parame-ters V, Q,, z, and p. Other factors such as core orcladding absorption losses may be described in the usualmanner; the effective domain merely indicates whichrays should be treated in the calculations.

ACKNOWLEDGMENTS

The generalized parameter for the step-index fiberwas discovered by my colleague John Love. I am in-debted to him for drawing my attention to this work andfor many helpful discussions. Telecom (Australia)provided financial assistance.

115 J. Opt. Soc. Am., Vol. 68, No. 1, January 1978 Colin Pask 115

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Spatial summation effects on two-component grating thresholds

R. Frank Quick, Jr. and W. W. MullinsDepartment of Electrical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

T. A. ReichertBiomedical Engineering Program, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

(Received 2 June 1977)

Some experimental studies of subthreshold summation between sinusoidal grating components havebeen interpreted as showing very narrow channel bandwidths in human visions. This paper discusses

an alternative interpretation of these experiments based on consideration of probability-summation ef-fects among spatially distributed detectors. We conclude that frequency-selective channels must stillbe hypothesized in order to fit the data, but the channel bandwidth may be much wider than earlier inter-

pretations suggest.

Since the multiple-channel theory of contrast detectionwas proposed' there has been considerable controversy overthe spatial-frequency bandwidths of the hypothetical chan-nels. Experiments with adaptation 2 and masking3 suggestbandwidths of the order of one octave. Subthreshold sum-mation experiments,46 however, have consistently producedmuch narrower curves which have been interpreted by someresearchers as resulting from narrow channel bandwidths. Allof these studies, however, have interpreted the experimentalresults through the use of models which essentially neglect theeffects of interactions between spatially distributed detectionunits. In this paper we provide an alternative interpretationof the subthreshold summation experiments of Quick andReichert 6 which shows that a one-octave bandwidth may beconsistent with these experimental results.

As originally formulated, the multiple-channel theory as-serts that grating frequency components are processed by

separate channels when the component frequencies are morethan one octave apart. Sachs, Nachmias, and Robson 4 offereda quantitative description of the probability of detection whentwo widely separated frequency components are presentedsimultaneously. They showed that there is some interactioneven between widely separated components, which can beexplained by assuming that each component has a probabilityof being detected which is independent of the probability ofdetection of the other component. This type of interactionhas come to be known as probability summation (as distin-guished from direct summation of the component amplitudeswithin a single channel). Quick7 showed that an approxi-mation can be made which allows a simple calculation of theeffects of probability summation among a large number ofchannels. The most recent development in multiple-channeltheories is the extension of the channel concept to includedetectors which are localized in the visual field, as well as beingselectively responsive to restricted ranges of spatial

116 J. Opt. Soc. Am., Vol. 68, No. 1, January 1978 � 1978 Optical Society of America 1160030-3941/78/6801-0116$00.50