Attenuation of an optical fibre immersedin a high-index surrounding medium

J.M. Arnold

Indexing terms: Attenuation, Fibre optics, Optical materials, Optical waveguides

Abstract: A theoretical description is given of the propagation characteristics, in particular the attenuation,of an optical fibre immersed in a medium whose refractive index is larger than that of the core medium. Thecomplete range of possible V values is considered, both above and below core-mode cutoff, and approximateanalytic expressions are derived for various regions of interest within this range. A variety of analytical tech-niques are used to obtain solutions even where conventional perturbation techniques fail.

1 Introduction

In the past, a number of mechanisms causing loss of powercarried by a mode propagating down an optical waveguidehave been studied. Among these, one might refer to bulk-glass attenuation,1 bending loss,2'3 attenuation due to im-perfections of the boundary between core and claddingglasses, scattering by small particles in the fibre4's andleaky mode attenuation in core-infinite cladding wave-guide.6 All these attenuation mechanisms may be consideredas intrinsic in either the fibre or the process by which it ismade. In addition to these intrinisc mechanisms, it is poss-ible to induce attenuation in a waveguide in a controlledmanner by modifying the properties of the region directlysurrounding the waveguide. Of this latter class, two methodsmay be referred to. First, the region directly outside theguide may be made of a lossy material. Theoretical calcu-lations by Clarricoats and Chan7 have shown that higher-order modes can be made to suffer much greater loss thanthe dominant mode, and, if the lossy region extends overmany wavelengths of the waveguide, its effect is to reducethe higher-order modes to negligible magnitudes at the ex-pense of only a small loss of power in the dominant mode.An alternative method of introducing a controlled attenu-ation is achieved by immersing the fibre in a medium whoserefractive index is slightly greater than that of the coreglass.8 It is the theoretical description of this type of attenu-ation with which this paper is concerned.

There is an important difference between this type ofattenuation mechanism and the type previously described.If the outermost region is lossy, the power lost from a modein propagating down the guide is completely dispersed intothe medium. On the other hand, when the outer mediumis a high-index dielectric, the power is lost from the modeas electromagnetic radiation which propagates freely (andessentially without further loss) through the outer mediumand away from the fibre axis. This energy need not be lostin the medium, and is accessible to measuring instruments.This idea forms the basis of a series of experiments by Mid-winter and Reeve,8 described in detail elsewhere. It is ofsome interest, therefore, to know what the electromagneticproperties of such a configuration are, and, in particular, todetermine the attenuation as a function of frequency.

Paper TSS M, first received 6th October 1976Dr. Arnold is with the Department of Electrical & Electronic Engin-eering, Queen Mary College, Mile End Road, London El 4NS,England

2 Physical considerations

Before proceeding to the mathematics, it is instructive toelucidate some aspects of the physical mechanisms involved.We consider a waveguide formed of three media, of refrac-tive indices nx, n2 and «3; their interfaces are of radius rx

and r2, and we use cylindrical polar co-ordinates (r, 6, z).The wayenumber is k, and the time dependence is assumedto be e~lwt. The refractive indices satisfy the inequality

«3 >nx >n2 (1)

A well guided mode in an open cylindrical structure mustsatisfy the requirement that the axial propagation constant0 be greater than the wavenumber for the outermost region;i.e.

0>n3k (2a)

This condition ensures evanescent decay of the fields inthe outermost region. On the other hand, to obtain aguiding action by the core-cladding interface the propagationconstant should satisfy

0<nxk (2b)

It is obvious that, if «3 >nx, eqns. 2a and 2b cannot besimultaneously satisfied, and therefore the mode cannot beperfectly guided. For a mode to exist at all in such a struc-ture, 0 must be complex:

0 = Refl?) + *Im(j3)

= 0R + i0i

(3a)

Ob)

and the wave is attenuated as it passes along the fibre. Sincewe have assumed that all media are lossless, this loss ofpower from the mode must appear as radiation away fromthe fibre axis.

Considering first the unperturbed waveguide formed bythe boundary at r = rx, with the boundary at r = r2 removed,it is well known that 0 is a solution of the characteristicequation10' n

(4)UJm(U) WKm(W)

with

U2 = (n\k2-02)r2

W2 = -(n\e-0*)r\

(5a)

(5b)

MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3 93

u2 + w2 = v2

= k2r\(n\-n\)

(5c)

When V is greater than a fixed positive number Vc, a num-ber of unattenuated modes (with 0 real) can exist. Let usconcentrate attention on one such mode whose cutofffrequency is Vc. When a new boundary is introduced at r =r2 such that n3(r > r2) is greater than nx, the fields are dis-turbed, and radiation occurs. For the purpose required here,we assume r2 > rx, and we consider what happens as V isdecreased from a very large value. When K> Vc, we knowthat the power carried by the unperturbed mode in theregion r>r2 is very small because the mode field has astrong evanescent decay away from the core-cladding inter-face. Consequently, we would expect the introduction of adiscontinuity at r = r2 to have little significant effect onthe unperturbed mode, and we conjecture that, although 0 iscomplex, its imaginary part is very small. If 0 = 0# + /0j weexpress this conjecture as

and V> Vc (6)

Since the rate of field decay of the unperturbed mode inr>rx becomes less as V-* Vc, we expect the attenuationto rise smoothly as V decreases towards Vc. Even when V =Vc (core-mode cutoff) the field is evanescent across r > rx,although its decay is now algebraic rather than exponential,and the rate of decay is dependent on the azimuthal modenumber m being faster as m increases. Therefore a secondconjecture would be that, at V= Vc, 0/ is still small and isa decreasing function of m; i.e.

0/ -• 0 as m -*• °° and V = Vc (7)

When V passes below Vc, the concept of a leaky modemust be introduced, as this is the normal propagation con-dition of the unperturbed mode.6 According to the theoryof leaky modes, this is the state of an ordinary mode whenV<VC, and it has a transition radius at r = rt associatedwith it, where

m -m2r\

Re{nlk2-p2}(8)

The fields are evanescent in r < rt and propagating waves inr>rt, and we find that a different view must be adoptedaccording to whether rt >r2 or rt < r2. At V = Vc, the un-perturbed mode has & = n2k, and consequently rt is infinite.As V is reduced below Vc, rt moves towards the boundaryr = r2. As long as rt>r2, the field at this boundary is stillevanescent, and the arguments advanced previously are stillvalid. As rt -+r2 the fields here increase as the leaky-modeeffect becomes stronger, giving rise to higher attenuationbecause the influence of the boundary at r = r2 is detectedmore strongly by the wave. In addition, the energy reflectedback into the core becomes significant, and, as a conse-quence, the phase coefficient 0« must be adjusted in orderthat the incoming reflected wave interferes correctly withother fields in the core for modal behaviour of the wave.More precisely, the wave reflected from the outer boundaryplus the wave reflected into the core from the inner bound-ary must exactly cancel the outgoing wave in the core toproduce pure stationary waves (with no singularities) on theaxis of the waveguide. This condition requires a perturbationon the real part of 0.

Finally, when rt <€r2, the discontinuity at r = r2 is in aregion of oscillatory wavelike behaviour of the fields, andthe resulting attenuation may be expected to be large.Because the leaky-mode field spreads into a substantial partof the cladding, the energy reflected back towards the coreis large, and this makes the necessary approach to the calcu-lation of 0 quite different to that adopted in other regions.In fact, the guiding mechanism is now the reflection ofenergy at the boundary at r = r2, modified by the loss ofenergy into the outer medium caused by refraction (sincen3>n2).

This picture broadly describes the physical features ofpropagation in this type of waveguide. The only significantexception to this behaviour may be expected in the HEXn

modes near to core-mode cutoff. For these modes, the fieldis not evanescent in the cladding, but almost like a planewave propagating axially, and one would consequently ex-pect the coupling between the two boundaries at r = rx andr = r2 to be strong This turns out to be the case and wehave to treat the analysis of these modes differently. Never-theless, the broad descriptive features of this physicalanalysis are verified subsequently by mathematical tech-niques.

3 Notation

Although the system of symbolic notation in use in thetheory of optical waveguides has become standardised overrecent years,10'12 some variation in methods of calculatingparameters exists, depending on what one takes to be fun-damental variables. Typically, one looks for relations be-tween 0 and V, or U and V. This is not always the best wayof going about it, however, and in this paper we shall use aslightly different set of variables. The characteristic equationfor the 2-layer guide is given by eqns. 4 and 5. In eqn. 4 theupper (lower) sign corresponds to HE (EH) modes. Here,we shall consider only EH modes except for the single caseof HEm modes, using the degeneracy of the two modegroups for all other values of m.

Each EH mode has, for a given m, a cutoff V, V— Vn,which identifies a particular root of eqn. 4. This equationhas, in general, an infinite number of roots (considered asan equation in W). Each root depends on the value of V,thus defining a many-valued function W of V. We selectbranches of this function so that, as V-*- Vn, W-+ 0 and Vn

is the «th zero ofJm(Vn) = 0. We now define a parameter

A2 = V^- (9)

such that W2 -> 0 when A2 -> 0, and W2 -> °° when A2 -> °°.This then gives us a natural mapping between A2 and W2

both on the positive half of the real line, as opposed tomappings between different spaces, e.g. between W and U orU and V, as is conventional. The quadratic form of eqn. 9 isretained wherever possible as it simplifies analysis, often lead-ing to the elimination of factors of 2 or 1 /2 which would bepresent if the more obvious A = V— Vn had been used.The parameters W and U almost invariably appear as thesquares W2 and U2 in analysis, and it is best to regard thepowers as fundamental. Thus, we regard A2 as a funda-mental variable, and W2 as a function of A2.

We then define 0 and Kby

V2 = K + A 2

(10)

(11)

94 MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3

From eqn. 10

0 L W2

n2k= 1 - n\k2r\

1/2

i

W7

V2

(since n\ —n\ is small). Therefore,

2 K2

«2A:

Thus, to find a we need to find Im(W2). Also, we have

U2 = V2-W2 = J ^ + ( A 2 - W 2 )

(12)

(13)

(14)

(15)

(16)

(17)

(18)

The characteristic equation for the 3-layer configurationcan be found by standard methods based on the solution ofMaxwell's equations accompanied by the requisite boundaryconditions, and it is found to be of the form

XX(WX)X2(WX) = F(WX)YX(WX)Y2(WX)

where

-/m + l(^l) Hml\(Q\)UXJm(Ux) QlHm\Ql)

J - (U ) Hi2l (Q )Yi(Wl) UxJm{Ux) QXH%\QX)

H%lx(Q2) H%1X(XQX)Q2H{

r}l\Q2) xQiH%\xQi)

#m*i«22) H%\x(xQx)Q2^m\Q2) XQlHm\xQl)

Ux — (nxk — p )vx

\cl — ^ / » 2 f P ) ' l — " 1

Ci2 == (jl'ik jj )V2

X = r2lrx

(19)

(20)

(21)

(22)

(23)

(24)

(25a)

(256)

(25c)

(26)

Fig. 1 Geometric configuration

MICROWA VES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3

An alternative form uses the modified Bessel functions

X (W) = y " * ' ( ^ ) + M ! l )

_Jm + x(Ux) ,WxIm(Wx)

(&F /-li/ \ _ ^ m + 1 ((?2 ) . A

Y2(WX) =

w2 = -Q]

F(WX)

(27)

(28)

(29)

(30)

(31)

(32)

A characteristic equation in connection with a 3-layer prob-lem has been derived by Kawakami and Nishida,9 and isidentical to that given here. To obtain these equations, wemust assume that nx ~ «3 — n2 and 0 ~ n2k.

The degeneracy existing between EHm+xn— HEm.xn

modes can be proved by using the recurrence relations forBessel functions (see Appendix 8.2). Here we use the sub-scripts to distinguish the parameters in the 3-layer problemfrom those in the 2-layer problem. Thus

A2 = V2 - VI

lf\ = v2 -W\ = V2 +<2?

and

Q\ =

where

X = - and a2 = -f \rx n\ -n\

This completes the fundamental parameter definitions.

4 Approximate solution of the characteristic equation

To study the approximate solution of the characteristicequation, we define the following three separate regions:

(i) V>Vc(A2>0)

(ii) V- VC(A2 - 0 )

There is a region between (ii) and (iii) in which A2 is nega-tive but has moderate magnitudes, being neither large norsmall; here, it is extremely difficult to obtain approximatesolutions. We exclude this region for the present analysis,and we discuss the problems which arise briefly in a sub-sequent Section. To proceed, we consider region (i) above.The characteristic equation is

XXX2 = FYXY2 (33)

and we note immediately that the characteristic equation ofthe unperturbed 2-layer problem is simply

Xx = 0 (34)

95

Since we expect the new solution to be a small perturbationon the 2-layer solutions when A2 > 0, we rewrite eqn. 33 as

_ FYXY2 _x\ ~ ~J— ~ e (35)

and we look for a solution when |e| is small. Indeed, we findthat, for large A2, V is large, and so the unperturbed valueof W is large. Therefore the perturbed value Wx is also large,and in this case the factor F is

F(Wx)^e~2{x~l)Wl (36)

which is easily established by using the asymptotic form ofthe modified Bessel functions in eqn. 32. Consequently, weshould always be able to find a Wx sufficiently large that\F(WX)\, and therefore |e|, be arbitrarily small. This estab-lishes the perturbation hypothesis. We now solve eqn. 35 bylst-order perturbation theory. We let

W2 = W2 +82 (37)

where W is a solution of the unperturbed equation Xx (W) =0, and 6 is a small increment. Then

S2 =e(W)

X[(W)

where e(W) is the value of e when W= Wx and

dXxX[(W) =d(W2)

(38)

(39)Wx = ]

Differentiating eqn 27 with respect to Wx and setting W =Wx, we obtain, after some tedious algebra,

(40)

(41)

X[(W)= ± 4 — - ! ( l - 2 ( m

. V2

UJm{U)

U2W2 1 + 2(m + 1)

where W and U have their unperturbed values. The twoforms are identical by virtue of the characteristic equation,but eqn. 41 is better for A2 > 0. By using the asymptoticform for the Bessel functions, we obtain, for eqn. 41, asW

V2

U2W2

V2

U2W2

W

(42)

Again using the asymptotic forms for the Bessel functions11

in eqns. 28-30, we find that, with Wx = W> 0

and

Also, when Wx = W and Ux = U,

Y (W) = / w T 1 ^ 1

(43)

(44)

(45)

UJm(U) WKm{W)

Km-+x{W)

WIm(W) WKm(W)(46)

Using the characteristic equation ^ 1 ( ^ = 0 to eliminatethe first bracket, and the Wronskian11 to simplify thesecond, we obtain

YX(W) = ±1

W2Im(W)Km(W)

1— \y —>• oo

w

(47)

(48)

Hence, collecting all the approximate forms together toevaluate e, we have

e(W) ^ ±Q2

and, using eqns 42 and 49 in eqn. 38, we obtain

c 2 2U2W2(ixW-Q2)e-2(x~l)W

V2 W

(49)

(50)

This is the amount by which W2 is perturbed. Conse-quently,

V2w

and

' • « « > -

(51)

(52)

If Q2 is expressed in terms of V, U and W, we obtain

Ql = X 2 { ( ^ 2 - 1 ) ^ + ^ 2 } (53)

Q2+X2W2 = o2x2V2 (54)

where a and x are defined in Section 3. With these para-meters, Im(W2) becomes

and, by eqns. 16 and 17,

a 2(ri-nl)U2W2

(56)

Some simplification to this result can be effected if we con-sider the limit F->°°. If a2=£l , the term (a2 — l))^2

eventually dominates U2, since12 U-+Un where/ m + 1((/n) = 0. Also, W->V, and we have the crudestasymptotic result:

" x

where

C, =

(57)

(58)

96 MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3

In the special case «3 = nt, then a2 = 1, and we have

a e-2(x-i)v

n2k K4

where

2 -»v_2{n\-nl)

(59)

(60)

Note the different power of Kin the denominator.Equation 56 is the complete approximate expression for

a, although eqns. 57 and 59 may be used for extremelylarge values of V. We observe at once from these expressionsthat the dominant term is the exponential. When r2 ^/̂ (X— 10, for example) a fairly small change in K(or W)results in changes of several orders of magnitude in a, andthis phenomenon clearly validates the hypothesis advancedin Section 2 concerning the physical nature of the attenu-ation when V is large. Fig. 2 shows some numerical resultsfor a few low-order modes for a moderate and a large valueof r2 /r , .

The approximate expressions derived here are valid forall modes, where EH or HE types.

4.1 Region (ii) V * VC(A2 * 0)

When the case A2 ^ 0 is considered, we have to distinguishbetween various mode groups because the behaviour of theBessel functions K0(Wi) and HQ^{QX) is different fromthat of Km(W) and H$(Q) (m > 1) when Wx and Qx aresmall. Thus, the HE on — EHon — HE2n mode group and theHEin group need to be considered separately from theremaining EHmn — HEm+2, n(m > 1) group. We considerthis latter group first

(a)EHmn modes (m > 1): We assume essentially that A2

is so small that both Wx and x^i are small. We shall seethat in a small neighbourhood of A2 ^ 0 this can be ensuredfor all values of x- Thus we now have the assumptions\A2\<\ and |K>2x2l^l, but \Q2 \ > 1 still holds. Byassuming that \W2\ is small when |A2| is small, even whenthe waveguide is perturbed, we imply the physical hypoth-esis discussed in the Section 2. This enables us to use thesmall-argument approximations for the Bessel functionsKm{Wx) and Km(xWx). We retain the characteristicequation

FYYY2X, = = e

used in the Section 3, but this time, mainly for variety, weuse a direct algebraic approach to find its approximate sol-ution.

For small Wi and x^i > we find that-2m 2m

(61)

We note again that this factor can be made arbitrarily small,this time by making x or m arbitrarily large.

We have, for EH modes,

For small

2m(62)

and (see Appendix 8.1)

(i) - 2

- 2A2 -W\

and therefore

Similarly,

(63)

(64)

(65)

(66)

(67)

(68)

and, substituting these expressions into the characteristicequation, we obtain the approximate equation

- 2 2m 2 X ' 2 m 2m ( i 1

-2

A2 -Wi

Q2 2{m

i , 2m

A2 - W\ W\ A2 - Wi x2 W2 y Q2 2(m +(69)

where, in eqn. 68, the first term on the right is negligible.This equation reduces directly to a linear equation in W2

and A2 whose solution is

mA2

m + 1 m +

2m ir \ 2 ( m + i ) / ./ \( )/ .

l\r2/ \Ga

12(m + 1)

Now the unperturbed value W2 is, for small A2,

mA2

W2 *m+ 1

(70)

(71)

and, since Q2 > 2(m + 1), the binomial theorem may beused in the last term, and so eqn. 70 simplifies to

( \ 2(m

Clearly, since m>l, the second term on the right can bemade to vanish by allowing r2 -*• °°, thus recovering the un-perturbed value W2 = W2, as is necessary. Taking real andimaginary parts, we obtainand so , , 2(m + o

Re(W2)^W2 - 4 m l - I (73)V2/

\2(m + l)

(74)

and so

0: 4(ri\ —n2) m(m + 1) /V,~ " njV2 Q2 \72

2(m + 1)

At cutoff V = Vn and A2 = 0 and so

Q2 ^ xaV = XaVn

(75)

(76)

MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. l,No. 3 97

and so we have the final result for A2 = 0

n2k n\o V*(77)

This expression enables the attenuation at V— Vn to becalculated, and the variation around this value as A2 variesaround zero can be estimated approximately from eqn. 75.The major observation to be made here is the very strongdependence of a on the azimuthal mode number m whenr2 > /-j through the factor (r{ /r2 ) 2 m + 3 .

1 r

10"'

10'

10

Fig. 2A A ttenuation as a function of V

/?, = 1-515; n2 = 1-50725; n3 = 1-538; r2/r1 - 2

-5010

15

Fig. 2B A ttenuation as a function of V

«, = 1-515; «2 = 1 -50725; n3 = 1-538; r2/rl - 9

Values computed from eqn. 77 are also plotted on Fig. 2as small circles at the points V= Vn. The EH-HE modedegeneracy for these modes permits the entire group to becontained under this heading.

(b) EHOn modes: These modes are degenerate with theHEon —HE-m modes. Again, using small argument approx-imations, we have, when m = 0,

- 2 1

-2

In

(78)

(79)

(80)

(81)

(82)In

Ignoring relatively small quantities where necessary, theapproximate characteristic equation is

2 1—: ; +

{Q2 2

This equation can be arranged into the form

W\ 1 - In - 6

where

XM<2

- 1

(83)

(84)

(85)

(86)

Inspection of eqn. 84 shows that, if it is assumed that \W]\is small, A2 — 52 must also be small. Accordingly, insteadof inquiring into the state of affairs when A2 = 0 (as in theprevious case), we make

A2 = Re(52) = -5X

(87)

This choice makes the right-hand side of eqn. 84 as small aspossible, and therefore makes \W2\ as small as possible. Thisis also a necessary procedure, as the choice A2 = 0 leads tovalues of W2 that violate previous assumptions leading tothe conclusion that the perturbation method fails here.Proceeding with the analysis after making the choiceprescribed in eqn 87, we obtain

8/1 — In (88)

98 MICROWA VES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3

This equation is solved by direct recursion of Wx to give

8/

where

1 — In

8/

X2(22ln

Hence, we obtain

2P (89)

(90)

(91)

We thus have the final result that, when A2 = -^,

ocn2k (92)

X2Ga

where we have used V ^ Vn, and Q2 is again given by eqn.76.

(c)HEXn modes: This mode group requires more carefulanalysis than the previous case. If one assumes that A2 andW>\ are small simultaneously, the perturbation methods donot work, and we must reject this hypothesis. The methodwe use here is to turn the procedure around, and find avalue of A2 for which W\ is actually small, a techniquesuggested by the previous case we have studied. Accordingly,we assume that W\ and x2 W2 are small but that A2 is notknown.

We have

XUiJm(Ux) WxKm(Wx)

JO(UX) K0(WX)

UXJX(UX) WXKX(WX)

UXJX{UX) 2

Yi(Wx) = rr°T I + ——

(100)

(101)

(102)

WXIX(WX)

= Z + lnTWX

(103)

(104)

(105)

(106)

(107)

(108)

= - + l n X

.̂ —— +

2

(108a)

(109)

(110)

(111)

(112)

(113)

Combining these expressions into the approximate charac-teristic equation, we obtain

Xx *. Z + lnTWX\ W\

•Xx (114)

We now have to search for a consistent hypothesis concern-ing the roots of this equation. Let us suppose that, near a

root, we can say that

approximately

Xx ^ Z + In ——

1. Then eqn. 114 would be

and, from eqn. 101, this clearly reduces to

(115)

(116)

We must now check back to make sure that this result isconsistent with the original hypothesis; it is evident thatW2 W] FWX

~^XX is of the order —In-—— for small Wx and is thereforenegligible with respect to unity. The original assumption istherefore correct.

Eqn. 116 now permits us to make an estimate of thevalue of A2. We note that, from eqn. 108a, Z is large ifx(= /"2Ai) is large. We deduce therefore that Ux must beclose to a zero of JX(UX). Therefore, since this is also thestate of affairs close to core-mode cutoff in the 2-layer fibre,when Jx (JJ) ?L 0, we deduce that A2 may still be small. Wehave

2

v\2 +

2

J

ux

A2

2(UX)

2

Hence

A2 -

A2 -W\

I

(117)

(118)

(119)

(120)

MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3 99

or

Thus

W\ =, A2 -

If we take

A2 = -±-lnx

then

(22lnx

(121)

(122)

(123)

(124)

(125)

(126)

Clearly, when this choice is made, \W]\ is very small andmuch less than A2. The assumptions made to obtain this re-sult can be retraced and evaluated in the light of eqn. 126,and we find that we have obtained an answer that justifiesthese assumptions. Thus, when A2 is given, eqn. 124, thenWf is given by eqns. 125 and 126. Effectively, we havefound a value of A2 which permits convenient assumptionsto be made, and we have calculated W\ at this value of A2.We have also seen how this procedure places the region ofcalculation close to core-mode cutoff. In fact, we havefound a modified cutoff, since, within the order of approxi-mations made, Re(W2) is zero, and this implies, in turn,that the real part of the propagation coefficient 0 is exactlyequal to n2k, the coefficient of a plane wave in the claddingregion.

The preceeding analysis is not relevant for HEn modes,since, for these modes, the points A2 — 0 correspondto V ^ 0, and are not therefore in the optical region.

4.2 Region (Hi) A2 < 0

When A2 -4 0, we conjectured earlier that the solutions ofthe characteristic equation would be perturbations onthose appropriate to the boundary atr = r2. The canonicalproblem, of propagation in a 2-dielectric medium with theouter refractive index higher than the inner, has been studiedpreviously by Marcatili and Schmeltzer,14 but it turns outthat the effect of the perturbing boundary at r = rx has asurprising influence on the solutions, which we can onlybriefly outline here; precise analysis is rather complicated,and will form the basis of a separate study. Marcatili andSchmeltzer have derived expressions for the propagationconstant of a hollow waveguide, and, in notation appropriateto this study, we have

X2 = 0

where X2 is given by eqn. 22.

1/2

(128)

(129a)

(1296)

To investigate perturbations on these solutions, we setthe characteristic equation in the form of eqn. 19. Now wenote that, for the terms Xx and X2 on the left of thisequation, the following behaviour is evident. Roots of Xx =0 correspond to the (unperturbed) leaky modes of theboundary at r = rx. These roots have Re(W2) increasingwithout bound as A2 decreases below zero. On the otherhand, roots of X2 = 0 correspond to the modes of thewaveguide at r = r2, and are given by the above eqns. 129.Evidently, for these roots — Re(H/2) is constant (within theorder of approximation). Consequently, there must bepoints at which the loci of the roots of Xx = 0 cross theloci of the roots of X2 = 0 (on a graph of—Re(W2) against—A2). At such points, the roots of Xx = 0 and X2 = 0 haveidentical real parts. In the neighbourhood of such points,zeros of the left side of eqn. 19 are not simple but ofsecond order, and calculation of the perturbations mustcorrectly account for this behaviour. This necessity increasesthe complexity of the analysis considerably.

Direct numerical solution of the characteristic equationreveals the way in which this anomaly is exhibited. Fig. 3depicts the dependence of Re(W2) on A2 in this domain fora low-order mode group (EHln), showing behaviour whichis typical for all modes. We see that over most of this

EH 1 4

- 4

Fig 3. Transitions behaviour ofRe(W2) for several EHin modes inthe vicinity ofEHxx cutoff

A2 = V2 — V%; Vc = 3 - 8 3 1 7 1 ; « , = 1-515; n2 = 1-50725, n3 = 1-S38;rjrt =9

100 MICROWAVES, OPTICS AND ACOUSTICS, APRIL, Vol. 1, No. 3

region, the locus of Re(W2) is indeed as predicted by eqn.129a; it is constant as A2 (or V2) varies. However, as itapproaches a transition region, it is deflected away fromthis constant level and settles again at a different levelappropriate to the next value in the sequence {Un}, whichappears in eqn. 129. Thus, as V decreases from Vc for theparticular mode, we find that —Re(W2) passes through aseries of constant levels corresponding to successive mem-bers of the set {£/„}. The transition zones are centredaround the points of confluence of the roots of Xx = 0 andX2 = 0. A further feature which emerges from numericalsolution of the characteristic equation is severe degradationof the accuracy of computed values of attenuation in thetransition zones. This is directly attributable to the doubleroot of the characteristic equation.

The remarks made in this Section indicate that a simpleperturbation technique is not appropriate in the regionA2 < 0, and more refined techniques are necessary.

Table 1 Calculated values of attenuation a for various modes nearcutoff: parameters as in Fig. 2B

V a/n2k a

dB/kmEHU 3-83171 141 (-8) 7-55(2)EHI2 701559 2-02(-9) 1-97(2)EHl3 10-17347 6-63(-10) 9-39(1)EHtl 5-13562 1-91 (-10) 1-36(1)EHJ2 8-41724 4-34(-11) 5-10(0)EH3i 6-38016 246(-12) 2 2 ( - 1 )EH32 9-76102 6-86(-13) 90(-2)fW41 7-58834 3-01 ( -14 ) 3-2(-3)EHsl 8-77148 3-60(-16) 4-4(-5)EH0l 2-40483 9-35( 8) 3-124(3)EH02 5-52008 6-16(-9) 4-73(2)EHQ3 8-65373 1-53(-9) 1-84(2)HEX2 3-83171 2-32(-6) 1-20(5)HEl3 7 01559 3-53(-7) 3-45(4)

EH modes: A2 = 0(m * 0)A2 = 4/x

2 (m = 0)HE modes: A2 = — 2/lnx = -0-9102

5 Conclusions

This paper has dealt at length with the theoretical deter-mination of attenuation coefficients and propagation con-stants of an optical fibre immersed in a medium whoserefractive index is higher than that of the core glass («3 >n\ >«2)- Both physical and mathematical approaches havebeen considered, and it has been shown that elementaryapproximations can be obtained for the parameters of in-terest for various ranges of normalised frequency V.

The results of this research may be applicable to a widevariety of practical problems: the cutoff experiment ofMidwinter and Reeve,8 the technique of cladding-modestripping with index matching liquid, and the coating offibres with nylon or plastic sheathing are all experimentaltechniques in which a detailed knowledge of the attenuationcharacteristics of this three layer configuration are desiredto be known.

6 Acknowledgments

The research described in this paper was supported by theBritish Post Office Research Centre, Martlesham. Discussionswith Prof. P.J.B. Clarricoats, J.E. Midwinter and M.H. Reevehave been invaluable, and their helpful interest is gratefullyacknowledged.

8 Appendixes

8.1 Approximations for Bessel functions

A number of approximations for Bessel functions have beeninvoked in the text without derivation or proof. These canbe found in standard sources.11'13 The principal approxi-mations used are as follows:

(i) When Wis large,

(130)

(ii) When W is small

(lnr = 0-57722 . . .)

(lnr = 0-57722 . . .)

1 W\™

(131)

(132)

(133)

(134)

(135)

(136)

(137)

where the {Vj} are the zeros of Jm(Vj) = 0. If

U2 = VZ+(A2-W2) (138)

where Vn is also a zero, then

2 _ ̂ i

UJm(U) A2 - W2 fa (A2 - W2) + (V2 - V?)

(139)If U^ Vn, then A2 - W2 is small and

_1V2 (140)

This approximation is extremely useful in the analysis ofoptical-fibre properties,

(iv) Wronskian equation:

Im+dW)Km(W)+Im(W)Km+1(W) = ± (141)

8.2 Degeneracy of mode pairs

It is a very well known property of 2-layer dielectric wave-guide that, when the approximation n{ ^ n2 is made, thecharacteristic equations for EHm>n and HEm+2,n modes

MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3 101

become identical.10' 12 It is of interest to inquire whetherthis is so in the 3-layer case when n j ^ n2 — «3.

The characteristic equation for EH modes is, rearrangingeqn. 19,

X2

with the lower sign in eqns. 20—24. We have

UxJm{Ux)

and so, dropping the arguments,

X, =J miim I U 1 \/_\

Use of the recurrence formula gives

X\ =

(142)

(143)

(144)

Qx' m + l •'m +2

1 1/ / / " ' / H (1)

J

Restoring the arguments, we obtain

(145)

(146)

(147)

Similarly,

Yx =

(148)

X, = -

(149)

r, = -

(150)

By inserting these expressions into the characteristicequation it is a trivial matter to show that the resultingequation is indeed the characteristic equation for theHEm+2,n modes. The degeneracy EH^— HEtherefore demonstrated.

We can also demonstrate the degeneracy EHOn

very simply, since

m+2^n

-HEQn

J-\(UX) = - / i ( t / i ) (151a)

^-i(Gi) = ~ ^ i ((?i) (1516)

and so on. Thus the //£On characteristic equation convertsdirectly to the EHOn characteristic equation.

7 References

1 KAPRON, F.P., KECK, D.B., and MAURER, R.D.: 'Radiationlosses in glass optical waveguides', Appl. Phys. Lett., 1970, 17,p. 423,

2 GLOGE, D.: 'Bending loss in multimode fibres with graded andungraded core index', Appl. Opt., 1972, 11, p. 2506

3 SNYDER, A.W., and MITCHELL, D.J.: 'Bending losses of multi-mode optical fibres', Electron. Lett., 1974, 10, p. 11

4 DYOTT, R.B., and STERN, J.R.: 'Effects of multiple scatteringin optical-fibre transmission line', ibid., 1971, 7, p. 624

5 SNYDER, A.W.: 'Excitation and scattering of modes on a dielec-tric or optical fibre', IEEE Trans., 1969, MTT-17, p. 1138

6 SNYDER, A.W.: 'Leaky ray theory of optical waveguides ofcircular cross section', Appl. Phys., 1974, 4, p. 273

7 CLARRICOATS, P.J.B., and CHAN, K.B.: 'Propagation behav-iour of cylindrical dielectric-rod waveguides', Proc. IEE, 1973,120,p. 1371

8 MIDWINTER, J.E., and REEVE, M.: 'A technique for the Studyof mode cutoffs in multimode optical fibres', Opto Electron.,1974,6,p.411

9 KAWAKAMI, S., and NISHIDA, S.: 'Perturbation theory of adoubly-clad optical fibre with a low-index inner cladding', IEEEJ. Quantum. Electron, 1975, QE-11, p. 130

10 GLOGE, D.: 'Weakly guiding fibres', Appl. Opt., 1971, 10, p.2252.

11 ABRAMOWITZ, M., and STEGUN, I.A.: 'Handbook of Math-ematical functions' (Dover, New York, 1971)

12 SNYDER, A.W.: 'Asymptotic expressions for eigenfunctions andeigenvalues of a dielectric or optical waveguide', IEEE Trans.,1969, MTT-17, p. 1130

13 WATSON, G.N.: 'A treatise on the theory of Bessel functions'(Cambridge University Press, 1968)

14 MARCATILL, E.A.J., and SCHMELTZER' R.A.: 'Hollow met-allic and dielectric waveguides for long distance optical trans-mission and lasers', Bell Syst. Tech. J. 1964, 43, p. 1783

102 MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3