Supermodes of optical fibre couplers

Optics Communications 94 (1992) 574-588 North-Holland

OPTICS COMMUNICATIONS

Full length article

Supermodes of optical fibre couplers Hagen Renner Technische Universiti~tHamburg-Harburg, Arbeitsbereich Optik und MeJ3technik, Eiflendorfer Str. 40, W-2100 Hamburg 90, Germany Received 30 April 1992; revised manuscript received 24 July 1992

A general scalar wave equation for the corrections to the supermode fields of first order in weak coupling on two identical weakly guiding optical waveguides is derived. It allows to obtain the well-known general expression for the supermode beat length, or equivalently for the coupling coefficient, without restriction to a special field trial. While the conventional method of superimposing unperturbed isolated-core mode fields is found to be correct only to zeroth order in weak coupling, i.e. only with respect to the large field contributions, the first-order wave equation derived here permits us to calculate the fields correctly up to first order, thus additionally including correct small field terms. The method is illustrated by constructing first-order field terms for parallel circular step-index fibres. The general polarisation properties induced by the first-order field corrections are discussed. Polarisation corrections to the scalar supermode propagation constants are calculated for the two circular step-index fibres, yielding the correct polarization difference of the supermode beat lengths. Scalar second-order corrections to the propagation constants are shown to be the same for both supermodes. Hence, they do not affect the rate of power exchange between the fibres. However, they may be useful for improving cutoffcalculations for the odd supermode. It is shown that on lossy fibre couplers with various transverse distributions of absorption the supermoa*esmay propagate with different attenuation rates. Near the odd-mode cutoff, the losses of the even and odd supermodes may typically differ by a factor of 1.4 to 1.7 for lossless cladding.

1. Introduction Although n u m e r o u s publications have dealt with the power exchange between parallel optical fibres (see for example refs. [ 1 - 6 ] , and for a review a n d discussion o f theories ref. [ 7 ] ), there is still a need for a thorough u n d e r s t a n d i n g o f the physics o f such couplers. In particular, it is o f interest to formulate quantitatively the consequences of incorporating into conventional scalar results b o t h the p o l a r i z a t i o n properties [ 8 - 1 3 ] a n d the effects o f loss or gain [ 14,15 ]. F u r t h e r p r o b l e m s are the scalar i m p r o v e m e n t o f results for the rate o f power exchange [ 2 8 ] a n d the transverse power d i s t r i b u t i o n [ 3,6 ]. In general, the power exchange can be described by the coupling o f the m o d e s o f the i n d i v i d u a l fibres, or by the beating o f the s u p e r m o d e s o f the complete structure [16,17]. In conventional analyses, the scalar fields o f the s u p e r m o d e s have been a p p r o x i m a t e d by merely superimposing the fields o f the u n p e r t u r b e d isolated-core m o d e s [3,6,8], i.e. the m o d e s o f the constituting i n d i v i d u a l fibres with the respective 574

other core replaced by the cladding material. A i m i n g at the d e r i v a t i o n o f the polarization difference o f the beat lengths o f the supermodes, or, equivalently, o f the difference in the rates o f power transfer between the fibres for the two orthogonally polarized states o f light, such fields have then been e m p l o y e d [ 8,10,11,17-20 ] in the usual p e r t u r b a t i o n formalism for vector corrections [21]. Recently [22,23], however, these results have been shown to be in error, as the polarization d e p e n d e n t coupling properties d e p e n d on the small field corrections not accounted for by the conventional superposition o f u n p e r t u r b e d isolated-core m o d e fields. This discrepancy m a y be explained by the fact that the unpert u r b e d field o f one core does not obey the wave equation inside the respective other core a n d hence has no physical meaning there. The field o f an isolated-core m o d e decays from its accociated fibre core towards the region occupied by the other core in the complete structure approximately by a factor T = exp ( - ~d) [ 24 ], where ~ is the decay constant o f the cladding field a n d d is the sep-

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aration of the core centres. Thus, T decreases with enhancing field confinement of the isolated-core modes and with increasing core separation. Apparently, this parameter T is a characteristic quantity of the evanescent cladding field behaviour in fibre couplers with optically well-separated cores, where T<< 1. It seems reasonable to formally expand quantities appearing in the supermode wave equation into orders of T. Then, it is possible to identify the first-order field terms to be responsible for the polarisation difference of the supermode beat lengths [22,23]. Similarly, the calculation of scalar second-order corrections to the supermode propagation constants as well as of the differential supermode attenuation requires knowledge of first-order corrected fields. However, constructing field corrections of first-order in T by spectral Taylor expansion and approximate field matching as in [22,23] is convenient mainly for circular fibres. In the present paper, a general first-order wave equation for the field corrections to the scalar supermodes is derived, which also applies to cores of non-circular cross sections. It relates to each other the isolated-core fields and the first-order corrections both for the fields and for the propagation constants of the supermodes. The present first-order wave equation permits to construct first-order fields for mirror-symmetric weakly-guiding waveguides with otherwise arbitrary refractiveindex distribution by standard methods for inhomogeneous differential equations. Here, the wellknown scalar coupling coefficient is derived without restriction to a special field trial. Supermode fields derived for a coupler composed by two circular stepindex fibres agree with those obtained in ref. [22]. The consequences for polarization behaviour and loss, as well as for scalar second-order corrections of the propagation constants and for the power excitation, are discussed in detail.

15 December 1992

distributions n, and rib, respectively, such that in the absence of the respective other core nq2= n2q(X, y ) = An2(x, y ) + n 2 for q = a , b. The structure of the coupler is invariant in the z direction. The squared index distribution of the composite waveguide structure is then n 2 ( x , y ) = A n 2 a ( X , y ) + An2(x, y ) + n ~ as shown in fig. 1. The scalar felds F -+(x, y) exp(-ifl+_z) of the propagating even ( + ) and odd ( - ) supermodes on the coupler obey the scalar wave equation [ 17 ] V2F + _+

[k2n2(x, y ) _ fl2 ]F + _ =0.

(1)

Here, fl_+ is the propagation constant of the corresponding supermode. Further, k = 2rr/2 denotes the free-space wavenumber, and 2 is the wavelength. The mode field Fq(x, y ) of waveguide q in isolation from the respective other core obeys Vt2Fq+ [kEnE(x, y ) - f l 2 ] F q = O ,

with q = a , b .

(2) The refractive-index distribution of the composite structure is assumed symmetric with respect to the yz-plane, i.e. rib(X, y) =ha( --X, y), as in fig. 1. This situation often occurs in practical couplers, such as twin-elliptical-core fibers [ 9 ]. It simplifies the analysis as the modes of both waveguides are synchronous, i.e. fla=~,=flo, and the principal optical axes [ 17,21 ] are simply determined by the cartesian coordinate axes. Further this assumption allows to normalize the isolated-core fields such that Fb(X, y ) = F a ( - x , y). I f the total field is written F+-(x,y)=F~(x,y)+Fb(X,y)+f+-(x,y)

,

(3)

w h e r e f -+ is arbitrary but everywhere finite and zero at infinity, eq. ( 1 ) becomes

nc

2. First-order wave equation for fibre couplers Two weakly-guiding optical waveguide cores a and b, embedded in a common infinite uniform cladding material of refractive index nc, are characterised by their individual two-dimensional refractive-index

d "t Fig. 1. General fibre coupler with symmetric refractive-index distribution.

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V ~ f ± + [k2n2( x, Y) - (ill + 8fl~ ) ]f + = (6//2_+ -k2n~,)Fa+_ (6//2+_-kZn~)Fb.

(4)

Here, 6//2 = f l ~ - / / I defines the difference between the propagation constants of the corresponding supermode and the isolated-core modes. The overlap terms k2An2F~ and k2An~Fb are obviously of first order in T = exp ( - yd), where, as already mentioned in the introduction, y= (f12-k2n2) 1/2 is the cladding-field decay constant [24] of the isolated-core modes, and d is the separation of the core centres, see fig. 1. If the perturbations 8fl2 and hence the products 8fl~ Fq are assumed also to be of order T ~, as will be shown in the following to be actually true, then there will be solutions f o r f -+ which are at least of first order in T. Outside both cores, the fields are purely evanescent. I f f +- is at least of first order in T throughout the infinite cross section, it decays with distance from each core in the evanescent cladding region, and may increase not stronger than to first order in T towards the respective other core again. Consequently, f +- can be split into two field contributions f ~ and f if, each concentrated near its associated core and decaying (approximately) exponentially with distance from this core: f +-(x, y) = f ~ (x, y) + f f f (x, y) .

(5)

Since each of these fields is at least of first order in T inside and near its associated core, it decays at least to second order towards the respective other core. Therefore, it may be neglected around the other core, thus sparing there any considerations of boundary conditions. Further, 8fl 2 Fa in the near of core b, and 8fl 2+ fg-+ across the infinite cross section, are at least of second order in T. The same is valid with a and b interchanged. Hence, taking only first-order contributions into account, one obtains

Mj+=Tfl~F~-T-k2AnZFb,

near core a ,

Mbf~ = ± 6//2++_Fb-k2AnZFa, near core b ,

(6a) (6b)

can be seen by inspection, the two equations (6a) (one for each sign) become identical for f + = - f 2 = f , and 8fl2+ = - 8if__ =Sfl 2, while equations (6b) go over into each other for f ~- = f ~- =fb and, again, 8//2 = -6//2_ =8//z. Now, 6//2 and one of t h e £ ' s can be determined by solving one of the following inhomogenous first-order wave equations with the conditions that the fields be of first order in T, finite, continuous, and have continuous first partial derivatives:

M.f. = 8//2F. - k 2 An2~Fb ,

(8a)

M b fb = 6//2Fo -- k 2AnZFa.

(8b)

The other first-order field follows directly from the symmetry ofeqs. (8a) and (8b), i.e.fb(x, y) =f~( --X, y). Within the constraints of being at least of first order in T there is only one unique value for 6//2 to satisfy the finite-field and boundary conditions. Hence, 8ff can be directly determined from eqs. (8a) or (8b) by constructing the fields, e.g. by standard methods for inhomogeneous partial differential equations [25]. The difference of the supermode propagation constants is then found as 26//= 8Bz/flo = (//+ - f l _ ) + O ( T 2) ,

(7)

where the upper (lower) sign refers to the even (odd) supermode. The Mq's are differential operators. As 576

(9)

correct to first order in T. It appears as the coupling coefficient C = 8 / / v i a

fl+_ =flo +_C+O( T 2) ,

(10)

when the power exchange is described in terms of the coupling of the isolated-fibre modes as discussed in sect. 6. Alternatively, one may multiply eq. (8a) by Fa and eq. (2) for q = a by f,, integrate both equations over the entire cross section, subtract one from the other, and apply Green's theorem [25] to the differential operator with the condition of fields vanishing at infinity, to obtain the well-known general expression

6//2 =Kba/Naa ,

( 11 )

where the definitions +oo

Kpq= f f k2An2qFqFpdxdy, -

Mq=Vt 2 +k 2 n q2 ( x , y ) - f l 2, q = a , b ,

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oo

+oo

(12a)

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with p, q = a, b have been introduced. The result with a and b interchanged has the same value as eq. ( 11 ), since fl~=/~, = flo. Obviously, 6fl2 and hence ~f12 are of first order in T, verifying the above assumptions to be correct. Note that similar manipulations could already have been applied to eq. (4) to yield eq. ( 11 ), assuming that the same relations between the quantities hold concerning their orders in T. Thereby, for the first time, the beat length LB = 2n/(fl+ - f l _ ) = n / ~ f l = n / C of the supermodes on two parallel weakly-guiding identical waveguides has been derived in a formally rigorous asymptotic expansion of orders in weak coupling without any initial restrictions on the fields. The supermode fields corrected up to first order in T are then

F ± = f f a + ffb,

(13a)

ffa =Fa + f b ,

(13b)

Pb =Fb +f~.

(13c)

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lated-core modes under consideration, F ± and r+ may represent any set of associated even and odd supermodes. To analyse the set of the two fundamental supermodes on a pair of identical circular step-index fibers with An2(x, y ) = n 2 --n 2, core radii p and separation d of the core centres, the unperturbed isolated-core mode field of core q with q--a, b is written in the associated polar coordinates (rq, Cq) (see fig. 2) as [17]

Fq( rq, q)q)=Ko( W)Jo( xrq) / Jo( U),

=Ko(~ro),

O<~rq <~p , p<<.rq< ~ , (14)

with U=xp and W=Tp, where/¢2=k2n2 _f12, and 7 has been defined in sect. 2. Following the steps described above, one obtains the coupling coefficient in its well-known form [6,8,17,22,26,27] C = 5f12/2flo = ( 2A )I/2U2Ko( WD ) /pV3K21( W), where

V=kp(nZ~-n2) 1/2 , The first-order fieldf~ may be considered to be composed of a contribution --Fb inside core a to compensate the second term on the r.h.s, ofeq. (8a), and of a part (~f12/2flo) (OFJOfl) at fl=flo in each of the regions separated by the core-cladding boundary. The cladding a n d / o r core parts of all solutions of the homogeneous equation (8a) may then serve to meet the condition of continuity of the field and of its first partial derivative across the core-cladding interface. The contribution of the independent complete homogeneous solution Fa to f~ has to be of first or higher order in T. For example, this condition can be met by requiring the total field to remain unchanged at some axis in core a, or, equivalently, by settingfa+Fb equal to zero there. Alternatively, the inhomogeneous equation can be solved e.g. by standard Green's function techniques, modal expansions [ 17,25 ], or numerical methods. Clearly, fb is obtained analogously with a and b interchanged, or from the symmetry relation. As the field contributions of any orders in weak coupling are evanescent in the cladding, the inclusion of the first-order fields ( 13) improves the accuracy of the representation of the supermode fields over the simple superposition of unperturbed isolated-core fields by one order in weak coupling throughout the entire coupler cross section. Since no restrictions have been made on the type of the iso-

(15)

and D = d/p is the normalized core separation. Requiting the total field to remain unchanged on the fiber core axes yields for q = a or b

fq(rq, Oq) = - K o ( WD )Io(Trq) + ~flZK°( W)rqJl

(Xrq)

2XJo (U)

+ ~ {Cos[m(~q-Oq)][AmJm(~Crq) mini

-2Kin( WD )Im(Trq) ] } ,

O<~rq<~p ,

(16a)

rb /

d Fig. 2. Coupler composed of two identical circular step-index fibres a and b, and associated polar coordinates. 577

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L(rq, ¢q)= I1 ( W)Ko( WD )go(Trq)

~fl2rqK1 (7rq)

K~ (W)

27

+ m=l ~ cos[m(Oq-Oq)]BmKm(Trq), P~<~,

(16b)

where

Am = 2K,,( WD) / [ UJm_ 1( U)gm( W) -}-WKm_IJm(U)]

Bm =Am[ Wlm_ l ( W)Jm( U)

-

-

,

(17a)

UJm- I Ir,( W) ], (lVb)

and 0q is that angle, which points to the respective other core axis. J, L and K represent Bessel functions [25]. These field corrections agree with those obtained by spectral Taylor expansion and approximate matching of the field functions [ 22 ]. The latter method, however, is convenient only for circular fibres [23 ]. Care has to be taken in constructing thd far-from-core fields of the individual-fibre modes in eqs. (8) and (12), e.g. Fb around core a, when the fibre cores are non-circular [24]. There have been several publications calculating scalar coupling coefficients of fundamental modes on parallel graded-index circular fibres from the farfrom-core part of improved isolated-core field approximations [ 10,11,28 ]. However, they require numerical integrations for the calculation of overlap integrals. Recently, a particularly simple, yet exact general expression for the coupling coefficients of synchronous modes on circular inhomogeneous fibres has been derived, which even allows to apply the usual gaussian field approximation despite its nonrealistic far-from-core behaviour [26]. (Note that, due to some cancellation, the error induced by the gaussian field approximation is even smaller for the full expression of the coupling coefficients than for the ratio R of the field intensity at the core boundary to the mode power studied in ref. [ 26 ]. ) Convenient expressions for the coupling coefficients between modes on arbitrarily-shaped parallel fibres are given in refs. [29,30].

15 December 1992

3. Polarization corrections

Polarization corrections to the scalar supermode propagation constants are obtained by applying the usual perturbation technique (sect. 32-5 of ref. [ 17 ] ) to the complete waveguide structure by use of the corrected supermode fields (13). However, often one may reduce the calculations by manipulating only those contributions which are of first order in T and simultaneously depend on the polarization direction, and hence account properly for the polarization beam splitting. Since the coupler structure is symmetric to the yz-plane, its optical axes [17,21 ] coincide with the cartesian coordinates for all supermodes, and one has

+oo

5 s+ = _

0(Pa-+;b) (Pa-+Pb) On

~-s d x d y

Os

)~(kF/c f ~ (/~a -~/~b) 2 dx dy)

,S=X,

J2.

(18)

This can be approximated accurately to first order in T by

1 (Z6

2Z~+ Z6 N,+~

\No +

No No / '

S=X, y (19)

where

(OF. F, On~

(20a)

--oo

Z~

+oo + f f '-=W - - /-/ O e a "-O.ffbx W-' Ona

+_=_

\ Os Fb +F. Os j os dxdy

+ O ( T z) ,

(20b)

+oo

(20c)

-co (20d) -oo

The subscripts on the quantities defined in eqs. (20) give their orders in T accurately to O(T~). Ob578

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viously, No is identical with Npqof sect. 2 for p = q up to order T 1. Due to the postulated symmetry of the coupler, it is sufficient in eqs. ( 2 0 a ) - ( 2 0 c ) to take the integrals only over contributions associated with the region inside and around one of the cores, such as done here about core a. Just as well one could have interchanged a and b in eqs. ( 2 0 a ) - ( 2 0 c ) . The first term in parentheses in eq. (19) is of zeroth order in T and is the same for both the even and the odd supermode, and hence does not appear in the coupling coefficients. Its difference for x and y polarized light is the averaged form birefringence of the isolated-core modes in those cases, where the optical axes of the individual fibres coincide with those of the coupler, e.g., where the individual cores have geometric symmetry axes along the cartesian coordinates. The third term is the zeroth-order polarization correction wheighted by the relative change of the supermode's total power, induced by the field deformation due to a finite distance of the cores. Finally, the second term accounts for mixed contributions. The coupling coefficient for s-polarized light is obtained as Cs= (/Y~--/Y'_ ) / 2 ,

thought in isolation from the other core, form angles of n/4 with the symmetry axes of the actual composite coupler structure. The latter situation is illustrated with two oppositely rotated rectangular waveguides in fig. 3a, where the optical axes of the individual guides are parallel to the dashed lines. When the optical axes of the individual cores coincide with those of the composite structure, as it is the case for the symmetrically arranged elliptical cores with aligned major axes shown in fig. 3b, then ( Z 6 - Z~)/kncNo is identical with the isolated-core birefringence. For the circular step-index fibres of fig. 2, one has Z~ = Z-~, and the supermode fields (13) with eqs. (14) and (16) are used in eqs. (20b) and (20c) to yield

Cx - Cy W ZUKz ( WD ) CA - V 2 J , ( U ) K o ( W D ) [ J o ( U ) - J 2 ( U ) ] . (26) This result is illustrated in fig. 4 by the solid curves for various core separations D. For D = 3, the dashed

(21)

where //~ =flo + c5,8+ ~fl~

(22)

are the propagation constants of the associated supermodes. Since N1 =N1+ = - N t _ ,

(23a)

Z] = Z ] + = - Z ] _ ,

(23b)

(a)

d

one has C , = C + (8p% - ~/~'__)/2

=C+(Z$N1-2NoZ~I)/kncN~,

s=x,y.

(24)

The normalized polarization difference of the coupling coefficients is then

Cx-Cy NI(Zr-Z~)-ZNo(Z~[-Z~;) CA kncAN ~ C

(25)

Under the assumed symmetry of the coupler, the first term containing Z r - Z ~ vanishes for all non-birefringent core geometries, such as circular (fig. 2) or square fibre cores. It also vanished for all geometrically birefringent cores, whose optical axes, if

(b)

°L5

Fig. 3. Coupler composedof (a) oppositelyrotated rectangular fibre cores, (b) elliptical fibres with aligned major axes. The optical axes of the individual fibres are parallel to their symmetry axes shown by dashed lines. The optical axes of the composite structures coincidewith the cartesian coordinateaxes. 579

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1fo-

Cx- Cy C~ 0.5~ 2

0

I

2

3

4

5

Fig. 4. Normalized polarization difference of the coupling coefficients for the coupler of fig. 2 verus Vfor various core-centre separations D (solid lines). The dashed curve shows erroneous results from analyses, which use isolated-core mode fields only [8,17]. line represents erroneous results from the analyses [ 8,17 ], in which the scalar supermode fields had been approximated by the inadequate superposition of the unperturbed isolated-core fields, being correct only to zeroth order in T. This principal error also occurs in analyses, which superimpose approximate isolated-core fields only to represent the supermode fields [ 10,11,18-20 ]. In a recent paper, Huang et al. [12] have developed a coupled-mode theory with vector corrections, but their approximation of the scalar supermode fields by superimposing unperturbed isolated-core mode fields only has also lead to erronous results, close to those of refs. [ 8,17]. A more detailed discussion of correct results for the case of two circular step-index fibres is given elsewhere [ 22 ]. It is only noted here, that the polarisation difference (26) passes through zero at V= Vo=3.454, contrary to previous results [ 8,10-12,17,19 ], which have no finite zero. Around Vo, fibre couplers would have to be extremely long to exploit form birefringence for polarization beam splitting. On the other hand, polarisation-independent coupling, such as exists at Vo, should be useful, among others, in long nonlinear couplers. (In this special case of two circular step-index fibres, Iio is beyond the single-mode range of the individual fibres, so that the set of the corresponding LP~ modes should be able to propagate. Accordingly, one would have to avoid excitation of the parasitic higher modes, or to suppress them by appropriate loss mechanisms.) Even at the upper limit of the single-mode region V= 2.4, the results of refs. [8,17] differ by a factor 1.41 from eq. (26) in580

15 December 1992

dependently o f D [ 22 ]. Thus, the difference between eq. (26) and previous analyses may be quite significant. To obtain reliable results for the form birefringence and the polarisation beam splitting in the case of arbitrary (including elliptical [9] ) refractive-index distributions of the fibre cores, one has to apply corrected first-order fields obeying eqs. (8) and ( 13 ). This need becomes particularly clear, as then Vo and the higher-mode cutoffs may vary in opposite senses, and Vo may possibly penetrate into the single-mode region of greatest interest. An algorithm for analysing couplers composed of circular fibre cores with inhomogeneous refractive-index profiles is given elsewhere [23]. Alternatively, inserting transverse fields of orders A° and A~ and longitudinal fields of order J~/2 [ 17,21 ] into the vector formulation [ 31 ] of the coupling coefficients, one obtains polarisation properties of the coupler accurate to first order both in A and in T. For two circular step-index fibres, the results agree with eq. (26).

4. Scalar second-order corrections

Various investigations (see ref. [ 7 ] ) have aimed at the derivation of correcting terms for the supermode propagation constants of higher than first order in weak coupling, such as from so-called "full perturbation theory" [ 8 ], variational methods [46 ], or from reciprocity relations [ 5 ]. These analyses effectively rely on a field representation by unperturbed mode fields of the individual fibres only. Here it is demonstrated, that, in the sense of a formal expansion by orders in weak coupling, second-order corrections in general can only be calculated from first-order corrected fields. Accordingly, correcting terms of still higher order should require knowledge of higher field corrections. However, it should be emphasized that these conclusions are based on the assumption that T<< 1, where such an asymptotic expansion is reliable. On multiplying eq. ( 1 ) by/;'~ and eq. (2) for q = a by F -+= Jea_+Fb + O ( T 2), subtracting each equation from the other, integrating over the full cross area, and making use of Green's theorem with the fields vanishing at infinity, one obtains

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___

= o ,

(27)

where +oo

(28a) -boo

gpq-~ f f k2mtl2FqFpdXdy, oo

(28b)

+oo

(28c) Here, 8fl~ is defined as the difference r 2 _ po2 correct at least to order T 2. As only terms up to second order in T are to be accounted for, terms of higher than second order may be neglected. From eqs. (28b), (28c), (12) and (13), one has +oo

--co

(29a) +oo

(29b) -oo

where Kab=O(T 1) and N a a = O ( T ° ) . The integral expression in eq. (29a) is of third order in T, as/7, a n d f , are of order T Oand T 1, respectively, and each of them still decays by one order in T towards core b. The double integral in eq. (29b) is of second order in T, asfb is of first order around core b and decays to second order near core a, while Fa is of zeroth order there, but has dropped to first order in T near core b. Therefore, one may approximate/(',u ~ K,b and N,a~Naa in eq. (27). The scalar first-order correction to the supermode propagation constants is defined as 6fl~l)_+ = + Kab/Na,( = + 5fl2 of eq. ( 11 ) ), since it has been obtained already from the zeroth-order fields. With eqs. (29) one finds

Mab+6fl~,)±Na.--6fl~N.~T-6fl~Nb.=O.

(30)

This equation is correct to order T z. The remaining difference 5fl~2) _+= 8P2+ - 6fl~ ) ± is then found to be of second order in T,

~5fl~2)_+ =

UQaDNa,--KabND,)/N2,.

15 December 1992 (31)

Just this term has still been neglected in eq. (30) in connection with the first-order term ND, in the last summand on the 1.h.s. The result with a and b interchanged is equivalent. In the sense of the formal expansion by orders in weak coupling, eqs. (31 ), (28) and (13) point out that corrections to the supermode propagation constants of higher than first order in weak coupling can be deduced only from fields which are also corrected. From this point of view, second-order or even higher corrections derived from the isolated-core mode fields do not systematically improve the first-order result, although such calculations have occasionally been closer to the exact results for weak optical core separation [8], apparently due to reasons not relying on the weakness of the coupling. One could argue that variational techniques such as that of ref. [4] are known to provide best accuracy within the range of the variable parameters, leading to the question whether second-order terms obtained in ref. [4 ] are more reliable. The method [4] employs unperturbed isolated-core mode fields with undetermined amplitude coefficients to represent the trial functions for the supermode fields. In the case of non-synchronous (fla ~ fib) isolated-core modes, the variational principle then determines the amplitude coefficients and the supermode propagation constants. Such results should be expected to be more accurate than those obtained from other methods, which use the same unperturbed isolated-core mode fields to represent the supermode fileds, but determine the amplitude coefficients via different principles, such as the conventional coupled-mode theory [ 1,31 ]. So, the variational technique [4] gives better accuracy than conventional methods in the case of non-synchronous coupling [ 1,31 ]. However, when the two fibres are identical, the amplitude coefficients for the isolated-core mode fields representing the supermode fields are already determined by symmetry to be identical for the even supermode, and to have equal magnitude and opposite sign for the odd supermode, i.e. exactly the same way as by the conventional methods. Since any improvement is only achieved by the better way to determine the amplitude coefficients, the variational principle does not generally improve over the conventional meth581

ods in the case of such symmetric refractive-index distributions considered here. Corrections for the supermode propagation constants are then obtained only from a superposition of unperturbed isolatedcore fields, and are hence reliable in general only to first order according to eq. ( l 1 ). The second-order corrections for the propagation constants (31 ) are the same for both supermodes, i.e. 8fl~2)+ =6fl~2)_ =~fl~2), no matter what special kind of appropriate normalization has been chosen. Thus, they do not affect the beat length of the supermodes, and hence do no1: influence the rate of power transfer between the fibres. Now, the supermode propagation constants are fl+ = (fib + ~ ) _ + + 6/3~2)) '/2 + O ( T 3) =/~o + (~,)_+/2/~0) +

[41~2~5~2)- (~]~1)+)2]/81~3 + O ( T 3 ) . The secondorder corrections should be u,;eful for the calculation of the cutoff wavelength 2c_ =2~r/kc_ of the antisymmetric supermode. The latter may be obtained from the eigenvalue equation 8 - ( k c - ) = kc_ nc, or

=kc_nc

(32)

similarly to the method of ref. [32], where, in the case of two circular step-index fibres, the conventional coupling coefficient was used as the first-order correction to the supermode propagation constant.

5. Power loss due to weak absorption

Fibre couplers are now considered, which have constant cladding loss ac and a spatially varying absorption a(x, y)=ac+Aa(x, y) in their cores. For weak absorption, standard perturbation theory [ 17,25 ] gives the power attenuation constants of the supermodes + ~

--oo

(33) -oo

The supermode fields then propagate as F + (x, y) e x p [ - (ifl+ +a+/2)z]. Ilaserting the field corrected up to first order in 7, and taking the symmetry of the coupler into account, one finds 582

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Volume 94, number 6

a + =ao+- 8 a + O ( T 2) ,

(34)

where 1

(35)

0~0= O/c + ~aaa core

a

is the zeroth-order attenuation of the fibre modes in isolation from the other core, and

AaFaffbdXdy-(°~°--°lc)Nl)

Sot= ~aaa 1 ( 2 ff~ core

a

(36) is the contribution of first-order in T. The quantity N1 was defined by eqs. (20d) and (23a). The symmetry of the coupler allows to reduce the loss calculation to equations (35) and (36), or to the analogous expressions with a and b interchanged. Contrary to the loss formula derived previously ref. [ 6, sect. V.A. ] ), the more accurate first-order attenuation coefficient (36) involves first-order field corrections. This may be particularly important if absorption or gain [33 ] (as its negative counterpart), is employed as a design parameter [ 15 ]. Obviously, the even and odd supermodes propagate with different attenuation rates. These are, respectively, increased and decreased by the same amount with respect to the isolated-core modes. Thus, if only one core is illuminated, i.e. both supermodes are nearly equally excited at the input, a steady state is reached after some propagation distance L~= 2/I a + - a _ [ = 1/ [Sal, from where on the supermode with the smaller loss predominates. Hence, for large coupler lengths L >> L,, both cores carry nearly the same power, no matter what distribution was launched at the input. If the fibres have longitudinally constant gain, the inverse behaviour again leads to equal, but now amplified powers in both cores after a length L > L,~, resulting in a beam splitting amplifier, as discussed later in sect. 6. Aa(x, y) and 6n2(x, y) may have different transverse dependence [33]. For two circular step-index fibres with mode fields according to eq. (14), one has

Naa=~p2V2K2( W) /U2 .

(37)

Then, making use of the somewhat more involved expression for Nab obtained elsewhere [8,34], one

derives from eqs. (20d), (23a), ( 1 3 b ) - ( 1 4 ) , (16a), and (16b): N~

(a)

D=2

u3

D

=p2 - ~/K, (WD) - 2 K o ( W D ) X [Ko(

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Volume 94, number 6

O

W)Io(W) +K, ( W)I, (W) ]

> O

2Ko(WD)

L_

-t'- U2 W2 V2Kl (W) { ( W4 -- U4)K' (W) +UZW2V2II(W)I[K2(W)-K2(W)] } .

(38)

(Note that N1 had not to be known in calculating the polarisation beam splitting of two circular fibres in sect. 3 due to the absence of isolated-fibre birefringence Z ~ - Z ~ = 0 . ) The attenuation constants for arbitrarily distributed absorption is then calculated via eqs. (35) and (36) as illustrated in the following. Consider first constant absorption or gain inside very small circles around the fibre axes [ 33]

I

I

i

2

?

V

1.0 .......,.'""

(b)

gl I/) 0

0.5 >

Aa(x,y)=&p2/p 2, O
=0,

else,

q=aandb.

(39)

D=2

Then for po/p<< U, W one finds from eqs. (35), 0

(14), and (37): ao = a¢ + ~ w ~ / [ v~J ~,( u ) ] .

(40)

For V~oo, one has W ~ Vand U~ Uoo=2.405 where Jo(Uoo) = 0 [ 17], so that a 0 ( V ~ o o ) ~ a ¢ + 3 . 7 1 0 & From eqs. ( 3 6 ) - ( 3 8 ) , (14), (16) and (13), one finds & W 2 U2NI

8a=

-

np2V4jZl ( U)K~( W) "

(41)

Note that the integral in eq. (36) has been neglected here, since, for example, Fb vanishes on the axis of core a. For lossless cladding ac = 0, the resulting supermode attenuation constants (34) and (35) are plotted in fig. 5a versus Vfor touching cores (D = 2). For low V ( > odd-mode cutoff Vc_, the latter being obtained as V c _ ( D ) = (0.121+0.5 In D) -~/2 from ref. [32]), the even supermode experiences the stronger attenuation, which is by a factor 1.18 larger than the isolated-core mode loss at V~ V¢_. Defining Vo~ by a + = a o = a _ , i.e. 6a(Vo,~)=0, one has Vo~=1.731 with ao(Vo,~)=l.08 &. For larger V> Vo,~,the odd supermode is more rapidly damped. For D = 3 , Vo,~ shifts to 1.397 with ao=0.71 &.

I

I

I

I

1

2

3

/.

5

v

Fig. 5. Normalized lossesof the supermodesand the isolated-core modes for touching (D=2) circular step-index fibres: (a) absorption distribution concentrated to the fibre axes, (b) constant absorption inside the cores. Dashed lines in fig. 6 show the ratios a+/ao and a+/ a _ versus the core separation D at the corresponding odd-mode cutoffs. Typically, a +/a_ amounts to 1.4 ... 1.5 near Vc_(D) for 2~
W2[

U2K~(W) ]

OL0=ac~-(O~l--ac)~--~ 1+ W 2 K ~ ( W ) A ,

(42)

with ao( V--,oo) .~ a~. To evaluate 6a, the integral in eq. (36) may be calculated using the step-index result

f f Faffb dxdy=~p2W2Ko(WD)/UZV 2 .

(43)

core a

The resulting attenuation constants are plotted in fig. 5b versus V for D = 2 and o~c=0. Again, the even 583

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is the constant core absorption in two step-index fibres. With eq. (10), and with the weak-guidance assumption fl~ knc, one finds

1.5

o~+_=O~o + k d O t-

1.4

-~/°tO

q) o

t 3

1.0

2

-_

I 4

D

5

Fig. 6. Ratios of the losses of the even supermode to that of the odd supermode (upper curves), and to that of the isolated-core

modes (lower curves) at the odd-mode cutoff Vc(D). Dashed lines: absorption concentrated to the fibre axes. Solid lines: Absorption equally distributed inside the cores. mode is more strongly d a m p e d for small V> Vc_, while the odd mode experiences the larger attenuation for V> Vo~=2.402. For example, a + = 1.27ao at Vc_ (D = 2 ). At Vow,one has ao = 0.83 a 1- For D = 3, the losses are equalized with a o = 0 . 5 3 a , at Vo,= 1.480. Figure 6 shows the ratios a+/O~o and ol+/a_ at V~_(D) versus D by solid lines. Within 2 ~
A s ( x , y) =M[n2(x, y) -n2c] ,

k

dk +oo

=

(46)

accurately to first order in T. Then, one only has to calculate the spectral derivative of the coupling coefficients, and no first-order fields are required to obtain first-order corrections to the loss. In couplers, for which eq. (44) holds, both supermodes are damped at the same rate if the spectral derivative of k times the coupling coefficient C vanishes. To achieve this within a certain spectral range, one has to ensure that Coc2~z/k=2. In the cases where the cores have constant absorption oq and refractive index nl, the second terms in eqs. (33) and (45) are identical with, respectively, o q - ac times and n 2 _ n 2 times the power fraction in the cores. The differences of the core power fractions of both supermodes have been numerically calculated for two circular step-index fibres from the spectral derivative of the corresponding coupling coefficients in ref. [ 8 ]. Applying eq. (46) to step-index fibres with constant core-absorption, the same supermode losses as those of the analytical results from eqs. (34), ( 36 ) - ( 38 ), (42) and (43) can be obtained. However, for arbitrarily distributed absorption, one has to use firstorder corrected supermode fields to calculate the differential supermode attenuation from the perturbation formula (33), or equivalently from eqs. ( 3 4 ) (36).

6. Excitation and power evolution

It is now investigated, how an isolated-core mode field Fa with power Naa launched into fibre a at z = 0 would excite the associated set o f the even and odd supermodes. The total field of the latter propagates through the coupler as

p_+ d#_+

vg+k -

(kC),

(44)

where M is a constant, then the second term in eq. (33) can alternatively be expressed by the group velocity vg_+ of the supermodes, i.e.

c/L_

15 December 1992

n~]F+_ dxdy -- oo

+oo

× (~I F~ dxdy)

F(x, y, z) =A÷ ( O)F + (x, y) exp( - i f l + z)

-1

,

(45,

-oo

where c is the free-space speed of light. A special case 584

+ A _ ( 0 ) F - (x, y) exp ( - ifl_ z) .

(47)

On multiplying the input field F(x, y, z = 0 ) = F a ( x , y) by the supermode fields, integrating over the full

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OPTICS COMMUNICATIONS

cross section, and using the normalized power orthogonality relation 4-oo

=2[Na~+(N~b+2QN, a ) I + O ( T 2) ,

(48a)

+oo

,48 , -oo

sition. The Aq'S, too, depend on the choice of the amplitude for the homogeneous solution Fq in the contribution to the first-order field fq. However, consider the full process: The isolated-fibre mode a is launched at the input z = 0 resulting into eq. (50). The propagation along the coupler according to eq. (47) determines the supermode amplitudes at z as a+ =A+ (z) =A+ (0) e x p ( - i f l + z ) . Thus the amplitude excitation of an individual-fibre mode by one of the two supermodes is A~ +) = A ] + ) ( z )

where

= e x p ( - i f l + z ) (l+_Nab/Naa)/2+O(T 2)

4-oo

Q-~- f~FqLdxdy/Sqq ,

15 December 1992

q=aorb

(49)

or, e.g., Q = (Nab-Nab)/Na~, one readily obtains A + ( 0 ) = ( I - T - Q ) / 2 + O ( T 2) .

(50)

Q is of first order in T, and depends on the choice of the amplitude of the isolated-core field contribution to the first-order fields. The power P+ (0) excited in each of the supermodes is then

P+ (0) = (N~a + N ~ b ) / 2 + O ( T 2) ,

(51)

so that the sum of both supermode powers is identical up to first order in T with that of the exciting isolated-fibre mode. Obviously, the even mode takes up an amount of power, which is greater by the firstorder difference Nab than that taken up by the odd mode. Further, the effect of arbitrarily choosing the isolated-core field contribution to the first-order field fq is cancelled out in th power (51 ) due to the appearance of Q both in the orthogonality relation (48) and in the excitation amplitude (50). Consider now the field distributions a+F +(x, y), where a+ and a_ are arbitrary supermode amplitudes. By these, the individual modes of fibre a or b may be excited with amplitudes

A~+)=a+[I+_(Q+N~b/Na~)]+O(T2),

(52a)

A~±)=+-a+[l+-(Q+Nab/Naa)]+O(T2).

(52b)

The superscripts on the 1.h.s. of eqs. (52) indicate excitmion of the individual-fibre mode a or b due to the corresponding even ( + ) or odd ( - ) supermode. Combined terms result from linear superpo-

(53a)

or

Ab~+) =Ab~+) (Z) = _+exp( --ifl+ z)

× ( 1 +Nab/Naa)/2+O(T 2) ,

(53b)

where the dependence on the choice of the first-order fields is cancelled out. Hence, when the coupler is lossless and the fl+'s are real, the individual-fibre mode q is excited with power Pq(z)= ]Aq(z) lZNaa, where Aq(z) =Aq~+ ) (z) +Aa~- ) (z), according to

P~(z) =N~ cos2(Cz) + O ( T 2) ,

(54a)

Pb(Z) = N ~ sinZ(Cz) + O ( T 2) .

(54b)

Leaving any inaccuracies of the increasing phase Cz out of consideration, eqs. (54) are accurate up to first order in T with regard to the power amplitudes. Thus, appart from the otherwise corrected 'speed' of power exchange, the power evolution is not affected with regard to shape and amount of the power swapping by taking first-order field corrections into account, as long as one individual-fibre mode is launched at the input, and the excitation of one individual-fibre mode is considered at the output. Note that, in general, such considerations regarding the excitation up to first order in T are power preserving only, if any mode excited at the coupler is power orthogonal to all other modes under consideration at least up to the same order in T. This necessitates that the subsequent fibre, which supports the excited mode, is isolated from other fibres, or is separated from them by at least twice the distance of the fibres of the exciting coupler, so that fields are decayed to order T 2 towards any neighbouring similar fibres. Alternatively, one may allow 585

Volume 94, number 6

only one individual-fibre mode to be excited at the coupler output, as done above. These conditions arise from the fact, that the orthogonality of the individual-fibre modes is broken in first order of weak coupling [3,8], as obvious from N a b = O ( T 1) 5 0 in eq. (12b). In general, the first-order field corrections will improve results for excitation problems to first order in weak coupling. To account for the polarization properties, one would have to insert Cs from eq. (24) instead of C into the power evolution equations (54) for s-polarized light. The difference Cx-Cy leads to a splitting of the light of the two polarization states after length Lx~.=zt/21Cx-Cyl, when C,Lxy/Zr for s=x or y is an integer [ 131. Then, if fibre q is illuminated, the s-polarized light is carried in the same fibre q at the output of the coupler of length L~y. Launching the individual-fibre mode a with field Fa(x, y) into the lossy coupler at the input z = 0 , the total field propagates as

F(x, y, z) =A+ ( 0 ) F + (x, y) exp[ - (ifl+ + a+/2)z] +A_(O)F-(x,y)exp[-(ifl_ +a_/2)z],

(55)

with the supermode amplitudes according to eq. (50). Then, the exponential function in the expression for the amplitude of the respective individualfibre mode (53) excited at the coupler output after length z has to be replaced by exp [ - (ifl+ + a +/2) z ]. The corresponding power carried in the mode is

P~(z) =N~a e x p ( - a o Z ) [cosZ(Cz) +sinh2(Saz/2 ) - (N~b/N~a) sinh ( 6 a z ) + O ( T 2) ]

(56a)

OF

Pb (z) = N~ exp ( - ao z) [ sin 2 (Cz)

+ sinh 2 ( S a z / 2 )

-(N~b/N~a) sinh(6az)+O(T2)] ,

(56b)

correct to first order in T. Altematively, after having calculated C = 8fl, and 8 a from corrected supermode fields, one may disregard first-order field terms in eq. (55 ) and approximate the total field propagating within the coupler by F(x, y, z)~A~(z)F~(x, y) +Ab(z)Fb(x, y)+O(T~). Then, on letting the supermode input amplitudes A + (0) and A_ (0) be quite arbitrary, and differentiating with respect to z, the corresponding coupled amplitude equations of 586

15 December 1992

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the isolated-core modes propagating on the coupler for arbitrary input excitation are obtained as d----~ +

iflo+

A,(z)

= - (iC+ ~-~)Ab(z) , dz

+

( 2) iflo+

(57a)

Ab(Z)

= - (iC+ ~ ) A . ( z ) ,

(57b)

where the coupling coefficient C follows from eqs. ( 9 ) - ( 1 2 ) . Thus, - 6 a / 2 acts as the imaginary part of a complex coupling coefficient. Due to the imperfect orthogonality of the isolated-core modes, it is physically reasonable only to rely on the zeroth-order accuracy of the mode amplitudes obtained from eqs. (57). When 18alz << 1, the powers in eq. (56) reduce to

P,(z)=N~, e x p ( - a o Z ) cosZ(Cz),

(58a)

Pb(z)=N, aexp(-aoz)sin2(Cz),

(58b)

being then identical with results from those analyses, which rely on isolated-core mode fields only [ 14,15 ]. However, when I~alz is sufficiently large such as after some distance z>L,~, one has, after neglecting power terms of first order in T in eq. (56),

Pa(z) ~ e b ( z ) ~Naa exp[ -- (ao-- I~al ) z ] / 4 ,

(59)

i.e., both coupler arms carry same powers, which are damped with the attenuation rate of the less lossy supermode. If the a ' s in eq. (59) represent gain [ 33 ], a o - [ ~ a l has to be replaced with the larger one of the amplification constants of the two supermodes. If only one individual-fibre mode is excited at the coupler output, the corresponding equation (56) allows to include first-order power terms. The second sinh functions in eqs. (56) change sign as 6a does, and hence indicate predominating contribution from the even ( i f 6 a < 0 ) o r odd ( i f ~ a > 0 ) supermode. When [~al is comparatively large, such as about ao/5 near the odd-mode cutoff from sect. 5, then eqs. (58) are no longer accurate enough to fully include the effects of loss or gain, and one has to employ eqs.

Volume 94, number 6

(56), even for short propagation distances.

7. Conclusions In conclusion, a scalar wave equation for coupled identical weakly-guiding waveguides of first order in weak coupling has been derived, which relates to each other the isolated-core fields and the first-order corrections for both the fields and the propagation constants of the supermodes. It allows us to derive the scalar coupling coefficient in a most general way. General properties such as the polarization behaviour, attenuation, excitation, and power evolution have been discussed. It is essential to include accurate first-order field corrections in calculating polarisation properties and supermode losses in general. For practical loss distributions, the two beating supermodes have been found to propagate with different attenuation constants. Scalar second-order corrections to the propagation constants do not affect the rate of power transfer, but may be useful in cutoff calculations. The present analysis applies to arbitrary sets of associated even and odd supermodes. Its validity also extends both to TM modes on weakly guiding, and to TE modes in arbitrary planar-waveguide couplers. All elements of the analysis can be generalized to more than two fibres, which moreover may be dissimilar and may be surrounded by inhomogeneous cladding materials.

Appendix

Continued perturbation theories It is shown that the choice of the amplitude of the isolated-core field contribution to the first-order field corrections (sect. 2) does not affect the results of continued perturbation methods (sects 3-5 ), if only it is ensured to be of first order in the perturbation parameter (such as T in the present analysis). The scalar Helmholtz equation for a bound mode with field F=F(x, y), eq. (1) can be put into the form [17]

[V2t + k 2 n 2 ( x , y ) - f l 2 ] F ( x , y ) = L F = O ,

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(A.1)

where F is everywhere finite, continuous, and has

continous partial first derivatives. Often, the general problem can be solved by splitting eq. (A.1) into contributions of different orders in a smallness parameter e, which can be regarded as a measure for the deviation from a simpler problem LoFo=0 with known solutions [25 ]. For the lowest-order terms of F=Fo+FI + .... where Fo=O(~ °) and F~ = O ( e 1), one then obtains E°:LoF0 = 0 ,

(A.2)

l:LoF1 =J.

(A.3)

Thus, the first-order contribution Ft has to obey the zeroth-order equation where an additional source term J of first order in E has been added to the right. FI is fully determined except for a contribution TF0 (where z is a constant factor), which is a solution of eq. (A.3) with the r.h.s, set to zero, i.e. of the homogeneous equation (A.2). Hence, FI =F'~ +zFo, where F'l is any particular solution of eq. (A.3) of first order in e. The only restriction on the amplitude r is to be not larger than of first order in ~, so that the contribution zFo to F~ is ensured to be not larger than of first order in e, too. Within this condition, z can be determined by the otherwise arbitrary choice of normalization of the complete field

F=Fo +FI +O(E 2) = ( 1 +v)Fo +/'~ + O ( ~ 2) , (A.4) for example by requiring F to have an unchanged value at some point (x, y) with regard to Fo, as done in eqs. (16), or otherwise. A perturbation about the (scalar) problem (A. 1 ), such as due to the polarization term in the vectorial Helmholtz equation [ 17 ], or due to a distribution of weak absorption, can be taken into account via the usual perturbation theory [17,25], leading to an expression for the perturbation of the propagation constant +oo

+oo

--oo

-oo

1

(A.5) where ~e is a linear differential operator. Inserting the field according to eq. (A.4), and allowing only for terms up to first order in e, one arrives at 587

OPTICS COMMUNICATIONS

Volume 94, number 6 +oo

+o¢

-oo

References

--1

--oo +oo

×

(FoS~{F',}+F'~Se{Fo})dxdy

1+ -oo -I-<3o

--1

-oo +oo

-t-oo

-oo

--o~

-- 1

(A.6) T h i s result does n o t d e p e n d o n r. So it has b e e n s h o w n that the Fo t e r m in F~ does n o t affect results o f c o n t i n u e d p e r t u r b a t i o n m e t h o d s , if only its a m p l i t u d e is c h o s e n such that the whole expression is at least o f first o r d e r in E. I n the case, that the e°-con t r i b u t i o n is zero, i.e. the integral o f FoSe{Fo} v a n ishes, the a b o v e c o n s i d e r a t i o n s do n o t apply. H o w ever, t h e n there is o f course n o c o n t r i b u t i o n f r o m the rFo t e r m u p to first o r d e r i n E either. W i t h view to the p r e s e n t s t u d y o f s u p e r m o d e s o f fibre couplers, e represents the w e a k - c o u p l i n g par a m e t e r T. F u r t h e r , Fo s t a n d s for F,+_Fb, where higher-order effects, such as the o v e r l a p o f Fb with core a a n d that o f F , with core b, have b e e n neglected i n the z e r o t h - o r d e r e q u a t i o n ( A . 2 ) . F~ represents the c o r r e s p o n d i n g l i n e a r c o m b i n a t i o n s + f , +fb, a n d J i n eq. ( A . 3 ) follows f r o m accordingly c o m b i n i n g the r i g h t - h a n d sides o f eqs. ( 8 ) . T h e n 5 f l o f eq. ( A . 5 ) m a y be the a n a l o g u e to ~ff+ o f e q . ( 1 8 ) o r a + o f e q . (33). S i m i l a r c o n s i d e r a t i o n s as to the c o n t i n u e d p e r t u r b a t i o n t h e o r y show that the scalar s e c o n d - o r d e r corrections o f sect. 4 do n o t d e p e n d o n the choice o f the a m p l i t u d e r o f the z e r o t h - o r d e r field in the first-order term, as long as it is of first order in weak coupling.

Acknowledgements I wish to t h a n k a n a n o n y m o u s referee for s t i m u lating the d i s c u s s i o n o f s e c o n d - o r d e r t e r m s o b t a i n e d b y the v a r i a t i o n a l m e t h o d [4 ]. 588

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