JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Incoherent illumination of an optical fiber

Allan W. Snyder and Colin PaskInstitute of Advanced Studies, Department of Applied Matliematics, Australian National University, Canberra, Australia

(Received 11 August 1972; revision received 26 February 1973)

The power of the trapped modes on a semi-infinite optical fiber illuminated by an incoherent source is de-termined. All possible modes are excited, each with approximately the same power when V- -o,V= 27rp (n? -ni,) '

12 /A, where p is the fiber radius, X the wavelength of light in vacuum, and n,, n2 are the

refractive indices of the fiber and its surround, respectively. A ray-optical interpretation is given for thesummed power of the modes. For V= cc, the power corresponds to that found from classical geometricoptics, treating all rays as if they are meridional. This result is independent of the degree of coherence ofthe source. The per cent error of meridional ray optics is 100/ V when V is large. The total power withinthe fiber is the combined power of the trapped modes and the radiation field. In the limit V= -, the totalpower within the fiber at any position z along its axis is that given by classical geometric optics, i.e., thatfound by tracing all rays, skew and meridional. At the point z for arbitrary V, the total power is thatdue to the trapped modes only.

Index Headings: Fiber optics; Resonant modes; Surface-guided waves; Geometric optics.

In an earlier paper,' we found that the summed powerof the trapped modes, propagating within an opticalfiber illuminated by a coherent source in the limit X=0,is determined by tracing all rays as if they aremeridional. Since classical geometric optics containsneither the wave nor vector properties of light, it canbe strictly applied to only incoherent illumination ofthe optical fiber.2 Therefore, in this paper, we considera fiber that is illuminated by completely incoherentlight3 to investigate whether the degree of coherenceaffects our earlier conclusions.

From our previous study,' we anticipate thatgeometric optics for circular fibers will be valid only inthe limit of large V rather than large p/X, where

V=27r -[ni 2- n223I,

X(1)

p is the fiber radius, N the wavelength of light in vacuum,and n,, n2 are the refractive indices of the fiber and itssurround, respectively.

An electromagnetic analysis is required to determinethe validity of geometric optics at a particular V. Thisanalysis forms the content of the paper.

We begin with an electromagnetic analysis of lightpropagation within an optical fiber of arbitrary crosssection when illuminated by incoherent light at theentrance to the fiber. Then we specialize the results tothe circular fiber and compare them with those obtainedfrom geometric optics.

FIELDS OF THE INCOHERENT SOURCE

Figure 1 represents an incoherent, uniform, source atthe entrance to a semi-infinite optical fiber. By incoher-ence,2 -4 we mean that the emission time of the individ-ual atoms of the source is much less than the integratingtime of the detector. Each atom is assumed to have arandom orientation so that the resultant field israndomly polarized. Therefore, all the elements of

the source are statistically independent (mutuallyincoherent).

Intuitively, we expect that the time-averaged irra-diance transmitted along the fiber is found from anintegration of the radiance distribution at the source.This is in contrast to the coherent case, for which theirradiance is found from squaring the magnitude of theresult of integrating the complex amplitude at thesource. 5 Furthermore, we anticipate that as X approacheszero all modes are excited with equal powers. Thesestatements are verified later when the time-averagedpower transmitted along the optical fiber is derived.

We consider the source to be quasimonochromatic. 2' 3' 6

The transverse (x,y) vector electric field, Es, of auniform source is then

Es = 2(ji/e,)iEo Re ( cosif'(x,y,t)+ T sin4t(x,y,t))ei0(xyt)0-t), (2)

where Eo is a constant related to the source strengthand is taken as unity for algebraic simplicity. x and kare unit vectors in the x and y directions, respectively,w is the angular frequency, Re indicates the real part,and 4, hare random functions of x, y, and time. Becauseof the quasimonochromatic characteristics of the source,4) and Ak are slowly varying functions of time.2' 3

ei is the dielectric constant of the fiber and At is themagnetic permeability of vacuum. Thus, Es has bothrandom phase and polarization.

FIELDS OF AN OPTICAL FIBER

The transverse vector electric, E, and magnetic, H,fields of an optical fiber of arbitrary cross section canbe represented as a sum of modes. Thus, we have

E=Re E aPe,(xy)&-i1,1+Pz)1)

H=Re _ aph,(x,y)e-i(wt+,6Pz),p

(3a)

(3b)

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JULY 1973VOLUME 63, NUMBER 7

INCOHERENT ILLUMINATION OF AN OPTICAL FIBER

where a, is the complex pth mode amplitude and ep, h.are pth-mode time-independent transverse vectorfunctions.

Optical fibers frequently have indices of refractiononly slightly greater than those of their surrounds, i.e.,nli-n2. We assume this condition for our analysis,because it simplifies the algebra with little loss ofgenerality. It then follows that7

Se' ih,= (-)Xe (4)

where i is the unit vector in the z direction. Equation(4) is also valid, independent of ni and n2, for modesthat are far above cutoff conditions.

The modal fields are normalized so that

If eXh,*-idS= e f e,*dS=6pq, (5)

where S. indicates integration over the infinite crosssection (x, y plane),* is the complex conjugate, and 8p,is the Kronecker c symbol, which is unity when p=qand zero otherwise.

TIME-AVERAGED POWER

The source electric field Es given by Eq. (2) can beexpressed as a summation of the modes of the opticalfiber shown by Eq. (3). Then, using the orthogonalityrelations Eq. (5), we are lead to an expression for thecomplex amplitude, ap,

a,=2 ep*. {i cosip+5 sin4'}ei~dS, (6)

where SF is the fiber cross section.The instantaneous power P of the electromagnetic

fields along the fiber is

P= EXH.2dS. (7)

The time-averaged real power (P) is

1 T(P)=Re lim P dt. (8)

T -> 2T T

When Eq. (3) is substituted into Eq. (7), using Eq. (5),

(P) =4 2E (Ia.j2), (9)

which is the expression for the sum of the time-averagedpower of all of the modes.8 Because only a fraction,which we call q,, of the power of each mode is trans-mitted within the fiber, (P) does not represent thepower propagating within the fiber. Instead, (P) is the

IY His 2 41

OX 7n,1 _zFIG. 1. A semi-infinite fiber of arbitrary cross section illuminated

by an incoherent source (IS). The n's are refractive indices andIA is the magnetic permeability of vacuum.

total power propagating, both inside and outside,along the fiber. To find the time-averaged powerpropagating within the fiber (PF\, we integrate overthe fiber cross section SF in Eq. (7), leading to

(PF)== E1 (ayaq*)e-i(#v)Q)zCpq,p,q

where

0 ep. eF ds.

(10)

(1la)

C, can be identified with the fraction 7, of the powerof the pth mode within the fiber, i.e.,

CPP = liP

Power of the pth mode within the fiber

Total power of the pth mode. (1 lb)

LIt remains only to find the time average of apa,* tocalculate (PF). We have

aa* =4(lJ d dS'

Xeie(t')e,. (k cosq/+$ sin#)

Xeg*. (i cos#'+5 sin4%), (12)

where the prime notation indicates a function of x', y'rather than x, y. Unless x=x' and y=y', the timeaverage of the random phase ei(0- > is zero. Symbolic-ally, this fact is expressed as3

T T

ei(";')fQT) d1= 8(x-x')3(Y-y') f(t) dt, (13)-T -T

where T -* oo and f(t) in our case is a product of twotrigonometric functions of the random variable 4'.These integrals are given as

1 1

T _ cos21 dt= f sing dt= (14a)2T JT 2T JT

1 Tf cost/ sin4' d!= 0, (14b)2Th E

where T --> c. Using both Eqs. (13) and (14), the time

807July 1973

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A. W. SNYDER

average of Eq. (12) leads to

(aa*) =4(-)f dS

X{2(ep 2(eq)+4(ep.5)(e<- )) (15a)

= 2Cq. (15b)

used in quantum electrodynamics, that of determiningthe density of modes in a two-dimensional box. AsX -> 0, the modal fields vanish at the boundary of thefiber. It is then easily shown that the number, i, ofmodes in a differential area dk.dk, of space is9

SFn = dk, dk, X2.

(27r)2(20)

To summarize, the time-averaged power (P) of allof the modes propagating on, inside and outside, an The multiplication by 2 accounts for the two possibleoptical fiber is, from Eqs. (9), (11), and (15), independent polarization states of the modes; k, and

k, are the transverse eigenvalues of modes in the fiber.(P)= E v, (16) Integrating over all allowed eigenvalues leads to the

P total number of propagating modes,

when illuminated by a uniform incoherent sourceconfined to the fiber cross section. Thus, (P) is foundsimply by summing the fraction of the modal powerwithin the fiber of each mode given by Eq. (lib). Thetime-averaged power of all of the modes propagatingwithin the fiber (EF) is, from Eqs. (10) and (15),

(PP)== 77p2+Pcross, (17a)

where

Pcros, = E

SFN =- A (kikv) X 2,

(2r)2I(21)

where SF is the cross-sectional area of the fiber andA (kz,k,) is the two-dimensional area of k space ofallowed kr, k, values. In the next section, we determineN for a circular fiber. Here we note that, in general,A (k,,k,,) is proportional to the square of the maximumallowable lI k I, where k I is defined as

(17b) 27rn(> 2r kl2=k. +k,2= -) - .

We observe that (PF) depends on z only through theoscillating term Pcros. In the appendix, we show thatPcros is real and very small compared with E>,p 2, sothat (PF) is quite accurately given by

(17c)

The presence of the parameter qp((<Op< 1) is partic-ularly meaningful because of its physical significance.Graphs of qp are presented in Ref. 2 for the circularfiber. Letting (P0 ) represent the time-averaged powerpropagating outside the fiber, we have

(P) = (PF)+(PO). (18)

GENERAL PROPERTIES WHEN I--- 0

A general property of electromagnetic modes in anoptical fiber is that 1p, defined by Eq. (llb), approachesunity when the pth mode is far above cutoff. Anenormous number of modes satisfy the condition that77p 1 when X -O0, leading to

(P)-(Pp)_- 1 =N, (19)p

where N is the number of propagating modes.

A. Number of Propagating Modes

In order to find the number of modes that canpropagate in a fiber of arbitrary cross-sectional areaSp, we appeal to a modification of a method frequently

(22)

The maximum kI occurs when a mode is at cutoff,i.e., when k,=27rn2 /X. Thus,

2mxI 2maxlIk 1 2 = -)(n 2 -n2 2), (23)

where Xi and n2 are the refractive indices of the fiberand its surround, respectively.

B. Time-Averaged Power when X - 0

The time-averaged power (PF) of all the modes thatcan propagate within an optical fiber of arbitrarycross-sectional area SF (when A -* 0) is, from Eqs. (19)and (23),

ni)2-(n22(PF) = \2 SF, (24)

where we have neglected a constant that depends onthe source strength. The power transmitted within thefiber increases as the wavelength decreases and increasesas the index-of-refraction difference between the fiberand its surround increases.

CIRCULAR FIBER

A. Electromagnetic Results

To evaluate the power expressions, Eqs. (16) and(17), we need 77p. Simple expressions are available for

808 AND C. PASK Vol. 63

(PF)--E '7p'.

INCOHERENT ILLUMINATION OF AN OPTICAL FIBER

the circular fiber,7

2m=(Ž) {(-2W )2+ K W } (25)

71= V U1.) Kl(Wlm)K1-2(Wlm) ,(5

where 1> 0, m> 1, and we have replaced p by thedouble subscript lm. UIm and WIm are related to V by

V 2 = Utm2l+ WIm, (26)

where V is a known dimensionless parameter thatcompletely defines mode propagation and is given byEq. (1). The U's are found from the eigenvalue equation

UlmKi-i(Wim)Jl(Ulm) = WImJI-i(Ulm)KI (Wim), (27)

where the J's are Bessel functions and the K's modifiedHankel functions.

Four mode types HE1m, EH1 ,., TMoM, and TEoMpropagate on the circular optical fiber.7"l0 For eachHEIm and EHim mode there are two states of polariza-tion, so that these modes must be counted twice.This leads to

M(V) L(V)

A; qp=2 E {(1m+2 E qlim},p m=1 I 2

(28)

and similarly for Ep,,j 2 where M(V), L(V) are themaximum m and 1, respectively, allowed for a particulargiven V.

B. Asymptotic Expressions

When a mode is far above cutoff, V>>Uim and"

Uim2

Y7lm- . (29)V

3

We see that as V, defined by Eq. (1), approachesinfinity, a large number of modes satisfy 7zm1; Forthis case, the number of modes is found from Eq. (21)to be'2-'4

V2

2(30)

for the circular fiber since A(k,,k 0,)=7rmax k12 andSF=-7rp2. Table I compares the approximate to theexact N. From Eqs. (19) and (30) we have the time-averaged power of the modes transmitted along thefiber. This expression is just a constant times our generalresult Eq. (24).

A better approximation to (P) is found by includingthe second term in Eq. (29); however, we must thenknow the Uim's satisfying V>>Uim. These UIm's areapproximately 7 the zeros of Jz (X). As V-- so, thedensity of zeros in any finite interval AU is very large,so that we can think of a continuous distribution ofzeros.'5 The 1, mth mode ceases to propagate 7 whenV< Uim. Thus, for V -m we assign a Uim to each V

TABLE I. Comparison of numerical and asymptotic expressionsfor number of propagating modes N, power of the trapped modestransmitted within the fiber (PF), and percent error of meridional-ray optics for various values of the dimensionless parameter Vdefined by Eq. (1). We assume the excitation condition of Fig. 1.

Number of Error in meridional-raymodes (PF) optics (%)

Approx- Approx- Asymptoticimate imate result

V Exact (Eq. 30) (Eq. 17c) (Eq. 32b) (Eq. 40a) (Eq. 40b)

20 210 200.0 192.2 190.0 -4.06 -5.0019 188 180.5 172.1 171.0 -4.90 -5.2618 168 162.0 152.4 153.0 -6.31 -5.5617 148 144.5 134.1 136.0 -7.74 -5.8816 130 128.0 117.8 120.0 -8.67 -6.2515 122 112.5 106.4 105.0 -5.77 -6.6714 102 98.0 90.5 91.0 -8.24 -7.1413 84 84.5 75.0 78.0 -12.73 - 7.6912 76 72.0 64.9 66.0 -10.94 -8.3311 60 60.5 52.2 55.0 -15.93 -9.0910 54 50.0 45.8 45.0 -9.07 -10.009 46 40.5 37.3 36.0 -8.69 -11.118 34 32.0 27.9 28.0 -14.50 -12.507 24 24.5 19.8 21.0 -23.93 -14.296 20 18.0 14.9 15.0 -20.73 -16.675 12 12.5 9.2 10.0 -35.97 -20.004 12 8.0 6.3 6.0 -26.91 -25.003 6 4.5 3.3 3.0 -35.22 -33.332 2 2.0 1.1 1.0 -82.27 -50.001 2 0.5 0.059 0.0 - 747.6 -100.0

of Eq. (30), leading to' 3

U, .-- (2 0) i,

where t is a continuous variable (a value for each mode)that takes on all values from 0 to N. Strictly speaking,we should start the mode counting at t = 1, but asymp-totically it makes no difference. We can now approx-imate the summation over 1, m by an integral over t

when all Uim's are replaced by Eq. (31). Following thisprocedure, we have from Eqs. (16) and (17c)

(PV2 /2 V21 1

(P)-a -qQ)dS = I_ Y-t

V' 2 V' 1

(PF)-F 772()dS_ I -- l-Jo 2 VJ

(32a)

(32b)

In Table I, we compare (PF) determined numericallyfrom Eqs. (17c), (25), and (28) with the asymptoticform from Eq. (32b). Although it would be desirableto continue the comparison beyond V = 20, we arereminded that at V= 20 there are 210 modes, so thatthe numerical analysis becomes cumbersome. Ourasymptotic expression is excellent even for small valuesof V.

From Eqs. (18) and (32), we find the fraction(P0o)(P) of total light transmitted along the outsideof the fiber is

(P0 ) 1( 2V (33)

(P) 2 V

July 1973 809

(31)

A. W. SNYDER AND C. PASK

C. Absolute Power Transmittedinto the Fiber

Only a fraction of the source power is transmittedinto the fiber. In Appendix B, we show that the totaltime-averaged power (Ps) of an incoherent source isfound by summing the power radiated from eachdifferential area of the source, leading to

2 /2rpn,\2 2 V2(Ps)=- (34)

for the circular source, where 6 is the dimensionlessindex-of-refraction difference between the fiber andits surrounding medium defined as

/AI 2\6 = 1t-) .(35)

Thus, the fraction of source power transmitted alongthe fiber is found from Eqs. (32b) and (34) to be

(Pi) / I1-/-__1 I' ) (36)(Ps)

Since 3<<1, only a small amount of the source power istransmitted into the fiber, the remainder is radiated. Inorder to provide a ray-optical interpretation of thisresult, we next derive an expression for (Pp)/(Ps) usingonly classical geometric optics.

D. Geometric-Optics Power

We use the equations of Potter"6 to find the sourcepower transmitted into the fiber. Calculation of(EXH*) shows that for the source used in this paperthe angular distribution function" I(a) is Io(1+cos26),where Io is a constant and 0 is the angle between a rayand the axis or a line parallel to the axis of the fiber. InPotter's theory,' 6 two classes of rays are distinguished:One class has 0< 0, (where 0, is the critical angle -V8),while the other class has 0> 0,. The contribution fromclass one rays is called meridional-ray optics (MRO),because, although most of these rays are skew, they allcontribute to the power as if they were meridional.Some of the (skew) rays in class two strike the fiber wallat an angle to the local normal that is less than O0, eventhough 0> 0,; according to geometric optics, they areinternally reflected along the fiber. We call the con-tribution of these rays the skew ray correction (SRC).Summing all rays, we find that the geometric-optics(GO) power is

(P) - ) + (-) (37a)

(Ps) GO (PS) MRO (PS) SRC

382- ' (37b)2

which is half that found from electromagnetic theory.Our determination of (Ps) assumes that 8«1. For 8«1,

((PF)' ((PF)\

(Ps) MRO \(P S) /SRC

(38)

i.e., the contribution to the total power of both raytypes is equal, so that from Eq. (36)

((PF)\ I (PF)

(Ps)}V 2 (Ps) I GO

(39)

In an earlier paper, we concluded that for coherentillumination of the optical fiber only the MRO rayscontribute to (PF). This conclusion is also consistentwith our present results. Therefore, when V= oo, thesummed power of the trapped modes is determined bytracing all rays as if they are meridional, independentof the degree of coherence of the source. It is thereforeof interest to know the departure of (PF) from MROfor a finite V.

HOW GOOD IS MERIDIONAL-RAY OPTICS?

Here we determine the validity of using MRO foranalyzing trapped mode transmission in optical fibers.The error in MRO for finite V is defined as

Error in MRO

(40a)

(40b)

r((P.) {(P )] /rP )>

L(Ps) MRO \(Ps)/ (Ps)/

1

V

We use Eqs. (17c) and (34) to calculate (PF)/(PS) forarbitrary V. The results are presented in Fig. 2. Thefluctuations above and below the smooth (dashed)curve are associated with resonance effects of modes,i.e., the maxima of the solid curve occur at V equal tothe values for cutoff of the modes (V=0, 2.405, 3.832,5.135, etc.'). Because classical geometric optics neglectsthe wave nature of light, we would expect the maximumerrors in its use to occur at the V's associated with themodes cutoff, where the electromagnetic field displaysits most aberrant behavior.

CONCLUSIONS

We have determined the summed power of thetrapped modes on a semi-infinite fiber illuminated byan incoherent source. Our results, together with thosefrom a similar study' using coherent illumination, leadus to the conclusion that when V= oo the summedmodal power is found from ray tracing, assuming allrays are meridional. This simple result is independentof the degree of coherence of the source. The percenterror of meridional-ray optics MRO used in determiningthe time-averaged power transmitted within a circularoptical fiber illuminated as in Fig. 1 is shown by Fig. 2.

810 Vol. 63

INCOHERENT ILLUMINATION OF AN OPTICAL FIBER

0-,0-

8 10 12 14 16 18 20

V = (2Trp/A) - n2]

FIG. 2. The percent error of meridional-ray optics, for use indetermining the summed time-averaged power of the trappedmodes transmitted within a circular optical fiber illuminated as inFig. 1. The solid curve is determined numerically from Eq. (40a),the dashed curve is the asymptotic analysis Eq. (40b). p is thefiber radius, X the wavelength in vacuum, and ni, n2 the refractiveindices of the fiber and its surround, respectively. The maximaof the solid curve occur at V equal to the cutoff values of themodes, i.e., V=O, 2.405, 3.832, 5.135, etc. At V=2.405, the erroris approximately 110% and as V approaches zero the percenterror approaches infinity.

MRO is valid for large V rather than large p/X. AsI}l--> n 2 , an arbitrarily large p/X is associated with asmall V. The percent error of geometric optics for large

V is 100/ V.The total power within the fiber is the combined

power of the trapped modes and the radiation field. In

the limit V= cc, the total power is given by classicalgeometric optics, i.e., by tracing all rays, meridionaland skew.

When the index of refraction of the fiber is onlyslightly greater than that of its surround, the trappedmodes account for half the total power. At the point

z = x, for arbitrary V, the total power is that due to thetrapped modes only. This is because the radiation fielddecreases as the distance from the source increases.

It would be interesting to determine the degree towhich the power of the trapped modes approximatesthe total power for arbitrary z and V. Fibers used foroptical communication have a length to diameter ratiogreater than 10', so that their transmission propertiescan be approximated by trapped modes. However,many fibers, e.g., those used for image transmission,'9

have a small length-to-diameter ratio, compared withcommunication fibers, so that the power of the radiationfield is significant. The light transmission properties of

fibers in that last category can usually be approximatedby tracing all rays, both skew and meridional, 6 since

their V is large. In the transition region, when neither

MRO nor GO applies, an electromagnetic analysis thatincludes the radiation field is required. The radiationfield within the fiber is found by integrating over allcontinuous modes or approximated by leaky (complex)

modes. We anticipate that leaky modes correspond tothe second class of rays discussed in this paper, i.e., therays that geometric optics predicts are trapped butwhich electromagnetic theory predicts will radiate.

ACKNOWLEDGMENTS

We thank the Australian post office for financialsupport of this study. One of us (A. W. Snyder) isgrateful to G. Kidd of the Australian post office fornumerous discussions on this subject. We are extremelygrateful to the referees for providing significant addi-tional insight, which led to improvements of this paper.

APPENDIX A: STUDY OF C1, AND Poross

In this section, we present equations for the calcula-tion of C,, and Pcross for circular fibers. We also showthat there is less than 1% error in Eq. (17c) for V< 20.

Substituting the fields of the modes for n1n2 givenin Appendix A of Ref. 17 into Eq. (11a), we find thatCr9=O unless modes p and q both have the samepolarization, the same azimuthal angular dependance,and the same symmetry with the x axis. For nonzeroCP,,6

where

2I U, U,Cpq = )§ 2

1 /UpJl(Up) J,(Uq)I= . J.s 1

(Up 2 - Uq2) \Jl(Up) f 1 ,(U )I

All quantities are defined in Ref. 7.When V>>U, Cpq is given approximately as

C"q- U, Uq/V,3

(Al)

(A2)

(A3)i.e., C,,<<1.

Calculation of the power within the fiber given byEq. (17a) requires the evaluation of Poros, defined byEq. (17b). Because Cp,,= C6 p, we have

X#6qPcross=2

' Cp q2 COS(Gp-3-q)zy

Pq(A4)

where the prime on the summation means that onlyone of the pairs (p,q) and (q,p) is to be counted. Thus,

V #6

| PcrssJ I 2 Z Cpq,2

P,2

Our numerical calculations show that for V, 20,ProssI is less than 1% of Lpn 2, SO that Eq. (17c) is

an excellent approximation.

APPENDIX B: DERIVATION OFSOURCE POWER

The exact determination of the time-averaged power(Ps) radiated by the source in the presence of the fiberis complicated. A portion of the source power goes intothe modes, the remainder is radiated. When nl-n2 ,

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I I I - I

A. W. SNYDER AND C. PASK

the summed power of the modes is only a small fractionof (Ps). Then we can approximate KPs) by consideringthe source to be in an infinite homogeneous medium.This calculation follows below.

In the far field of a source of SF cross-sectional area,the electric vector is given as"'

jeikcr

E(r) =lkX zXEs(x')e-ikx'dS, (B 1)27rr J F

where r is the radius vector from the origin of thesource to the observation point, k=ki, k=27rn,/X,i =r/jr , IrI =r, 2 is a unit vector in the z direction,and the prime notation refers to the source coordinates;ES is given by Eq. (2).

The time-averaged power (Ps) radiated for z>O isdetermined as

(Ps)=- Ref (E(r)XH*(r)) .dS,

where S,, is an infinite cross section perpendicular tothe z axis far from the source and

kx(r) - kXE(r). (B3)

Using the time-averaging procedure discussed in thepaper, we find that

2n. 2(PS) = 237rSF .(B4a)

For circular fibers, SF =rp2 and (Ps) can be written as

2 V2

(Ps) =--. (B4b)38

REFERENCES'A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am.

63, 59 (1973).2M. Born and E. Wolf, Principles of Optics (Pergamon, New

York, 1965), pp. 119 and 491.3

M. Beran and G. B. Parrent, Jr., Theory of Coherence(Prentice-Hall, Englewood Cliffs, N. J., 1964), pp. 53 and57.

'H. H. Hopkins, in Advanced Optical Techniques, edited by A.C. S. van Heel (North-Holland, Amsterdam, 1967), p. 189.

5A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138(1969).

6Although the field is quasimonochromatic, it is still assumed tobe incoherent (Refs. 2 and 3).

'A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130(1969).

8These are the surface or discrete modes that propagate alongthe fiber and not the continuous modes that account forradiation.

9R. P. Feynman, R. B. Leighton, and M. Sands, Tire FeynmanLectures on Physics (Addison-Wesley, New York, 1965),pp. 4-10.

'0E. Snitzer, J. Opt. Soc. Am. 51, 1122 (1961).

"The approximation 7 - I - (U / V)'2 V 2 - U 2 1 F1/2) used inRef. 13 leads to the incorrect conclusion that the total power<P > is less than the power within the fiber <P F >. Thus, asis often the case, the first term of an asymptotic expansion ismore uniformly valid than the series with several terms.

'2This result can be derived by considering the resolution of anaperture of the fiber diameter and the acceptance angle ofthe fiber based on meridional rays (Refs. 13 and 14).

3D. Gloge, Appl. Opt. 10, 2252 (1971).4

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969)."G. N. Watson, Theory of Bessel Functions (Cambridge, U.P.,

Cambridge, England, 1922), p. 477.16R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).

17A. W. S. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998(1972).

"8R. F. Harrington, Time-Harmonic Electromagnetic Fields(McGraw-Hill, New York, 1961).

9N. S. Kapany, Fiber Optics (Academic, New York, 1967).

812 Vol. 63