Applicability of vector potentials witha single non-zero component for wavesin uniaxially anisotropic media

Applicability of vector potentials witha single non-zero component for wavesin uniaxially anisotropic media

S Z . G A �AL , E . L �OR I N C Z , M . B A R A B �AS , P . R I C H T E R

Technical University of Budapest, Department of Atomic Physics,H-1111 Budapest, Budafoki uÂt 8, Hungary

Received 11 February; revised and accepted 25 March 1998

In this paper we derive conditions under which the electromagnetic ®eld of a wavepropagating in a uniaxial medium may be expressed with the aid of two vector potentialswhich are parallel to the optic axis. Unlike the coordinates of the electric ®eld vectorwhich are usually coupled owing to anisotropy, such single-component vector potentialsform a pair of independent scalar ®eld quantities. We prove that if the optic axis is in thetransverse plane then the ordinary refractive index must be constant along that plane inorder for the vector potential description to be applicable, but there is no restriction forthe extraordinary refractive index or the dielectric interface. It is also shown that if theoptic axis is in the longitudinal direction then the vector potential description can be usedeither in materials where the ordinary refractive index is constant in each region or incircularly symmetric ®bres where the ordinary and extraordinary refractive indices havecircular symmetry, being constant along the dielectric interface(s), or in slab wave-guides, where the ordinary and extraordinary refractive index must be constant on theboundary of the slab.

1. IntroductionThe general use of integrated optical devices makes it necessary to develop analytical andnumerical methods by means of which the guided modes can be determined in a widefrequency range with great accuracy, in both isotropic and anisotropic materials. Thisrequires us to solve the Maxwell equations or, in the case of a time dependence of exp(jxt)and a longitudinal coordinate dependence of exp(ÿjbz), to ®nd the propagation constant band the electromagnetic ®eld distributions so that the solution satis®es the boundaryconditions. In order to solve the Maxwell equations we will introduce either scalar orvectorial auxiliary functions (the potentials) by means of which all the components of the~E and ~H vectors can be expressed.

If the materials in which the wave propagates are isotropic, the homogenous Maxwellequations lead to vectorial Helmholtz equations. These two vectorial Helmholtz equationssplit into six uncoupled scalar equations with six unknown components of ~E and ~H .Keeping two arbitrary components as unknowns, the remaining four components can beexpressed by them [1, 2].

Optical and Quantum Electronics 30 (1998) 413±425

0306±8919 Ó 1998 Chapman & Hall 413

In the case of general anisotropy of the permittivity and permeability tensors with non-zero o�-diagonal elements the ®nite element method may use either two vectorialpotentials [3] or one scalar and one vectorial potential [4].

This paper deals with a situation which is in between the two above discussed cases, i.e.we have to ®nd potentials by means of which the ®elds in uniaxially anisotropic materialscan be described. The following two cases must be considered: (i) when the optic axis isparallel with one of the transverse axes (in our notation with the x axis) and (ii) when theoptic axis is parallel with the longitudinal axis of the waveguide. The potentials used by®nite element methods [4] for generally anisotropic cases can also be used in these cases toobtain an exact solution of the Maxwell equations, but the computation time wouldincrease.

In case (ii) we get two, uncoupled Helmholtz equations for Ez and Hz and the transverse®elds Et and Ht can be expressed by means of them [5].

Case (i) is more complicated. In this case homogenous Helmholtz equations can bederived for the ®eld components which are parallel to the optic axis (in our notation Ex

and Hx), but the other ®eld components cannot be expressed with them. For the ®eldcomponents other than Ex and Hx one can get coupled wave equations, and it can beshown (see e.g. [6]) that the coupling is caused by the anisotropy. In wave guidingstructures not only the anisotropy but also the dielectric interfaces couple the ®eld com-ponents.

There are several methods in use to decouple the di�erential equations. If one of the®eld components is supposed to be much smaller than all the others then the ®eld com-ponents can be described with one potential satisfying the Helmholtz equations. Thisapproximation can be made, for example, to calculate the eigenmodes of a channelwaveguide far from cuto� [7]. Then the ®eld components can be expressed with one of thetransverse components of ~E, while the other transverse component is set to zero.

If anisotropy is small then the coupling term can be regarded as a perturbation of thehomogenous Helmholtz equations and the problem can be solved by a perturbationmethod [6].

The disadvantage of the methods of [6] and [7] is that they can be used only at relativelyhigh frequencies or in the case of small anisotropy. To avoid these limitations [8] presentsthe vector potential description of the electromagnetic ®eld in a bi-anisotropic, homoge-nous medium to calculate the Green's function. The advantage of this description is thatthe ®eld structure can be described with two vector potentials, each of them having onlyone component instead of the vector variables used in [3, 4]. As calculations based onsingle-component vector potentials need less computation time and provide the sameaccuracy as the equations using three-component vector variables, it is important to knowthe ordinary and extraordinary refractive index pro®les and waveguide geometries forwhich the vector potential description of [8] can be used. In this paper we give a su�cientcondition for case (i) and a necessary and su�cient condition for case (ii) which must holdfor the applicability of this description. A similar analysis can be found in [9] where therefractive indices are derived for an isotropic slab waveguide in which TE and TM modescan exist.

If the ®elds can be described with uncoupled single-component vector potentials then incase (i) each eigenmode can be considered as a superposition of a longitudinal sectionelectric (LSE) and a longitudinal section magnetic (LSM) wave. It can be shown analo-gously that in case (ii) the eigenmodes can be regarded as superpositions of TE and TM

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waves. As the eigenmodes form a complete set of functions, an arbitrary transverse ®eldstructure can be regarded as a sum of LSE and LSM modes in case (i) and TE and TMmodes in case (ii).

2. Introduction of vector potentialsAs mentioned in the introduction, in a uniaxially anisotropic (and possibly inhomoge-neous) medium we cannot choose two components of ~E and ~H by means of which theremaining four components can be expressed. To overcome this di�culty we decomposethe ®eld into a component called the electric wave and into another one called the mag-netic wave. For this purpose we introduce the electric vector potential~A e and the magneticvector potential ~A m. The displacement vector ~D e of the electric wave is de®ned by

~D e � rot ~A e �1�and induction vector ~B m of the magnetic wave, by

~B m � rot ~A m �2�The material equations connecting ~D with ~E and ~B with ~H are

~D � e$~E �3�

and

~B � l0~H �4�

Because of linearity they are valid also for the electric or magnetic waves alone. If theelectromagnetic ®eld has a time dependence of exp(jxt) then the rotational Maxwellequations are

rot ~H � jx~D �5�and

rot ~E � ÿjx~B �6�Substituting ~D e and ~B m into Equations 3 and 4 and making use of Equations 5 and 6 onecan ®nd the displacement ~D m of the magnetic wave and the induction ~B e of the electricwave as follows:

~B e � ÿ�jx�ÿ1rot e$ÿ1

rot~A e �7�

~D m � �jx�ÿ1rot l0ÿ1rot~A m �8�

The resultant displacement and induction vectors are the sums of electric components ~D e,~B e and magnetic components ~D m, ~B m:

~D � ~D m � ~D e �9�~B � ~B m �~B e �10�

Using Equations 5±10 we can express the resultant ~D and ~B by ~A e and ~A m:

~D � rot~A e � �jx�ÿ1rot l0ÿ1rot~A m �11�

~B � ÿ�jx�ÿ1rot e$ÿ1

rot~A e � rot~A m �12�415

Applicability of vector potentials with a single non-zero component

Now we derive the wave equation for the electric vector potential ~A e. We substituteEquation 4 into Equation 5 to get

rot �l0ÿ1~B e� � jx~D e �13�

Substituting the ~B e given by Equation 7 and the ~D e given by Equation 1 into Equation 13we obtain

rot �l0ÿ1rot e

$ÿ1rot~A e ÿ x2~A e� � 0 �14�

As the rotation of l0ÿ1rot e

$ÿ1rot~A e ÿ x2~A e equals zero, there exists a scalar Ue (called an

electric gauge function) for which

l0ÿ1rot e

$ÿ1rot~A e ÿ x2~A e � gradUe �15�

A similar equation is obtained for ~A m and for the corresponding scalar Um (i.e. themagnetic gauge function):

l0ÿ1 e$ÿ1

rot rot~A m ÿ x2~A m � gradUm �16�At this point Ue and Um are arbitrary. Later we show (see Equations 24±25 and 31±32below) they can be expressed by means of the vector potentials~A e and~A m. Therefore theyare determined by the shape of boundary as well as the permittivities of the wave guidingstructure.

Introducing the notation W�a� � ÿ�1=x2�U�a�, where a =(e,m), Equations 15 and 16 canbe transformed in a more convenient form:

l0ÿ1rot e

$ÿ1rot~A e ÿ x2�~A e ÿ gradWe� � 0 �17�

l0ÿ1 e$ÿ1

rot rot~A m ÿ x2�~A m ÿ gradWm� � 0 �18�The vector potentials ~A �a� in the ®rst terms of the above equations could obviously be

replaced by ~A �a� ÿ gradW�a� without changing the ®elds derived from them (see Equations19 and 20 below). Therefore we can attribute no physical meaning to the scalar functionsW�a�a =(e,m). They represent merely a gauge transformation and will be used later (seeEquations 24 and 25 and Equations 43 and 44) to eliminate the undesirable components ofthe rotations of ~A e and ~A m in the further sections.

Using Equations 11, 12, 3 and 4, the electric and magnetic ®elds can be expressed as:

~E � e$ÿ1�rot~A e � �jxl0�ÿ1 rot rot~A m� �19�

~H � l0ÿ1�ÿeor�jx�ÿ1 rot rot~A e � rot~A m� �20�

and Equation 20 represents the electromagnetic ®eld distribution for an arbitrary per-mittivity tensor where~A e and~A m have three components. In the following sections we ®ndthe permittivity distributions and the possible waveguide geometries where ~A e and ~A m areparallel to the optic axis having only one non-zero component.

3. The vector potentials in a translation invariant uniaxial mediumwith the optic axis lying in the transverse plane

In this section we assume that optic axis is perpendicular to the direction of wave prop-agation and that the permittivity tensor is of the form

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e$ �

ee�~rt� 0 00 eor�~rt� 00 0 eor�~rt�

0@ 1A �21�

where the indices e and or refer to the extraordinary and ordinary permittivity, respec-tively, and~rt=[x; y]. The x axis has been chosen to coincide with the optic axis.

Our goal is to ®nd a permittivity tensor of the form Equation 21 where the electro-magnetic ®eld distribution can be expressed with the single-component vector potentials

~A e � �Aex; 0; 0� �22�

~A m � �Amx ; 0; 0� �23�

Here ~A e and ~A m are two independent variables. If both of them di�er from zero then thesolution is the superposition of LSE and LSM waves.Let us consider the y and z coordinates of Equation 17 with ~A e given by Equation 22

describing the LSE polarized ®eld:

1

l0

@x@yAe

x

eor

� �� x2@yW

e � 0 �24�

1

l0

@xÿjbAe

x

eor

� �ÿ jbx2We � 0 �25�

If assumption 22 for ~A e is correct then the y and z coordinates of Equation 17 (i.e.Equations 24 and 25) must hold simultaneously. This condition is necessary and su�cient.To prove this let us ®rst express @xW

e from the ®rst coordinate of Equation 17, and, afterdi�erentiating both sides of the second and third component of (17) with respect to x,insert it into them. Keeping in mind that the electric vector potential ~A e must disappear inin®nity, we get Equation 26. If we express We from the third coordinate of Equation 17 or@yW

e from the second coordinate form of Equation 17 and substitute to the other twocoordinates, we also get Equation 26.

If we di�erentiate Equation 25 with respect to y and then subtract it from Equation 24,we get

@x@yAe

x

eorÿ @y

�Ae

x

eor

�� 0 �26�

which holds if

Aex@yeor�~rt� � e2or�~rt�C�y� �27�

where C�y� is arbitrary.An obvious solution of Equation 26 for eor is

eor�x; y� � eor�x� �28�This means that in materials having a permittivity of the form 27 there are purely LSE

polarized modes. There may be other structures too where LSE modes exist, since fromEquation 27 one can express ~A e with eor and solve the boundary value problem for theelectromagnetic ®eld trying to ®nd the appropriate functions C�y�, but this problem is toocomplicated to be solved for the general case. Thus Equation 27 is only a su�cient but nota necessary condition for ~A e to have only a single component.

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Now let us ®nd the governing equation for the x coordinate of ~A e. To do this ®rst onehas to express @xW

e from Equation 25,

ÿ 1

x2l0

@2xxAe

x

eor

� �� @xW

e �29�

and then substitute this expression into the x coordinate of Equation 17. If the ordinarypermittivity is constant then we get the following Helmholtz equation:

�@2xx � @2yy�Aex � �eorl0x

2 ÿ b2�Aex � 0 �30�

Now we consider the magnetic vector potential ~A m describing the LSM wave. Thenecessary and su�cient condition for ~A m to have a single component can be obtained inthe same manner as the corresponding condition for the LSE case. We begin with the yand z coordinates of Equation 18:

1

l0eor@2xyAm

x � x2@yWm � 0 �31�

1

l0eor@2xyAm

x � x2Wm � 0 �32�

We di�erentiate both sides of Equation 32 with respect to y and then subtract it fromEquation 31 to get

@y1

l0eor

� �@xAm

x � 0 �33�

from which it follows that either the permittivity distribution must be of the form

eor�x; y� � eor�x� �34�or @xAm

x must be zero. This can occur only if the wave guiding structure is a slab where thedielectric interface is parallel to the optic axis (that is now the x axis).

There are, however, methods using the electromagnetic vector potentials in a form ofEquation 23 even if the ordinary refractive index varies in the y direction [10]. Thesemethods divide the space in subregions in which the refractive index can be consideredconstant, and at the dielectric interfaces they apply the boundary conditions.

We can determine the governing equation for the x component of~A m similarly as for theLSE case. As a result we get the Sturm±Liouville equation:

ee�~rt�@x1

eor@x

� �� @2yy

� �Am

x � �ee�~rt�l0x2 ÿ b2�Am

x � 0 �35�

where ee�~rt� is arbitrary.In view of Equations 28 and 34 we can say that if a medium is characterized by the

diagonal permittivity tensor Equation 21, then the vector potentials have a single non-zerocomponent only if the ordinary refractive index is dependent only on the x coordinate. Ifthe wave guiding system consists of two or more uniaxially anisotropic regions andcondition 34 holds for all the ordinary permittivities, then the shapes of the dielectricinterfaces between the regions and the extraordinary permittivity pro®le are arbitrary, andall the electromagnetic ®elds that can exist in the structure can be expressed by means ofthe single-component vector potentials of Equations 22 and 23.

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SZ. GaaÂl et al.

It is an important practical use of this theory to determine the ®eld distributions in ahomogenous, uniaxially anisotropic medium. Using the vector potentials of Equations 22and 23 one can get the following expressions with the help of Equations 11 and 12:

~H � ÿ1jxl0eor

ÿ@2yy � b2

@2xyÿjb@x

24 35Aex �

1

l0

0ÿjbÿ@y

24 35Amx �36�

~E � 1

jxl0

eÿ1ÿ@2yy � b2

@2xyÿjb@x

24 35Amx � e

$ÿ1 0ÿjbÿ@y

24 35Aex �37�

yielding all the possible solutions.It can be seen that in Equations 36 and 37 the electric and magnetic ®elds have been

obtained as a sum of LSE and LSM waves.If the waveguide consists of several homogenous, anisotropic regions then the tangential

components of ~E and ~H and, equivalently, the normal components of ~D and ~B are con-tinuous on the dielectric interface. By means of these boundary conditions we candetermine the electric and magnetic vector potentials and the propagation constant b[10,11].

4. The vector potentials in a medium having the optic axis parallelwith the longitudinal coordinate

Let the permittivity tensor be the following

e$ �

eor�~rt� 0 00 eor�~rt� 00 0 ee�~rt�

0@ 1A �38�

and let us search for the electric and magnetic vector potentials in the following form:

~A e � � 0 0 Aez � �39�

~A m � � 0 0 Amz � �40�

Our goal is to ®nd e$�~rt� of form 38 that the electromagnetic ®eld distribution, satisfying

Equations 5 and 6 Maxwell equations and the boundary conditions, can be described withelectric and magnetic vector potentials in a form of Equations 39 and 40, respectively, sothat all the ®elds that can exist can be described by Equations 39 and 40. If the systemconsists of two or more regions, the shape of the dielectric interfaces is to be determined aswell so that Equations 39 and 40 hold.We are searching for the ordinary and extraordinary permittivity pro®les in the form

eor � �l0fo�ho�x; y��ÿ1 �41�

ee � �l0fe�he�x; y��ÿ1 �42�The functions ho and he are speci®ed later on in this section. Although this notation seemsvery cumbersome, it will facilitate the further derivations. Having introduced notations 41and 42 our task is reduced to ®nding functions ho�x; y� and he�x; y� so that Equations 39and 40 hold.

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Let us consider Equations 17 and 18. Because of the exp(ÿjbz) dependence of theelectromagnetic ®eld the ®rst and second coordinates of Equations 17 and 18 are re-spectively the same, so we can write:

1

l0eor@2xz A�a�z ÿ x2@xW

�a� � 0 �43�

1

l0eor@2yz A�a�z ÿ x2@yW

�a� � 0 �44�where (a) = (e, m) and

ÿ@x1

eor�x; y� @xAez

� �ÿ @y

1

eor�x; y� @yAez

� �ÿ x2l0�@zW

e � Aez� � 0 �45�

ÿ 1

ee�x; y� @x�@xAmz � ÿ

1

ee�x; y� @y�@yAmz � ÿ x2l0�@zW

m � Amz � � 0 �46�

The necessary and su�cient condition for Equations 39 and 40 is that Equations 43 and 44must be valid simultaneously which can be proven in the same way as for Equations 24and 25. From this condition we will derive relations for the ordinary permittivity pro®lesgiving us the solution. In this derivation ®rst we determine all the possible solutions fromEquations 43 and 44 and then we substitute the possible solutions into Equations 45 and46 to ®nd those refractive index distributions for which Equations 39 and 40 hold.

We di�erentiate Equation 43 with respect to y, Equation 44 with respect to x, thensubtract the two equations from each other to obtain

@y1

l0eor

� �@xA�a�z ÿ @x

1

l0eor

� �@yA�a�z � 0 �47�

Equation 47 is a ®rst order di�erential equation in two variables for A�a�z . As a consequenceof Equations 46 and 47 ho must be equal to he as we will see later.

Let us consider ®rst the case when eor is piecewise constant. In this case Equation 47 willbe an identity, and from Equations 43±46 we can derive the Helmholtz equations forAz�a��a � e,m�:

�@2xx � @2yy�Aex � �eorl0x

2 ÿ b2�Aez � 0 �48�

�@2xx � @2yy�Amz � eel0x

2 ÿ eeeor

b2

� �Am

z � 0 �49�

We can derive the boundary conditions for the electric and magnetic vector potentials. If weexpress Ez and Hz from Equations 19 and 20, then apply the continuity relation to them,then making use of Equations 48 and 49 we get the following conditions for Ae

x and Amx :

l0x2 ÿ b2

e�1�or

!Am;�1�

x � l0x2 ÿ b2

e�2�or

!Am;�2�

x �50a�

l0x2 ÿ b2

e�1�or

!Ae;�1�

x � l0x2 ÿ b2

e�2�or

!Ae;�2�

x �50b�

where the upper indices (1) and (2) refer to the ®rst and second medium, respectively.

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SZ. GaaÂl et al.

The corresponding equations for Ez and Hz can be found for example in [5]. In the furtherderivation we assume that eor is not constant.

According to Equation 47

~ez grad?1

eor� grad?A�a�z

� �� 0 �51�

where ~ez is the unit vector in the z direction and grad? denotes the transverse gradientoperator whose components are �@x; @y�. This means that grad�1=eor� and grad?A�a�z areparallel to each other, and therefore their contour lines are the same. Therefore the zcomponents of the vector potentials as well as the `gauge functions' W�a� are in the fol-lowing form:

1

eorl0

� fo�ho�x; y��

A�a�z � A�a�z �fo�ho�x; y��W�a� � W�a��fo�ho�x; y��

�52�

where the fo dependence of W�a��a � e,m� follows directly from Equations 43 and 44. (Thesame result can be obtained using the method of characteristics [12] for the ®rst orderpartial di�erential Equation 47).

Let us introduce the following notations:

pe B fo�ho� dA�e�z

dho� fo

dA�e�z

dfo

dfodho

�53a�

qe Bdpedho� dfo

dho

� �2dA�e�zdfo� fo

d2A�e�zdf 2

o

!� fo

dA�e�z

dfo

d2fodh2

o

�53b�

pm BdA�m�

dho� dA�m�z

dfe

dfedh

�53c�

qm Bdpmdhe� d2A�m�z

df 2e

dfedho

� �2

� dA�m�z

dfe

d2fedh2

o

�53d�

If one substitutes expressions 52 into Equation 45 setting (a)=(e) and expressingW�a��a� � �e� from Equation 43 one gets with the aid of Equation 53a±53d:

pe @2xx � @2yy

� �ho�x; y� � qe �@xho�2 � �@yho�2

h i� lx2�jbWe ÿ Ae

z� � 0

We � ÿjbx2l0

Zfo

dAez

dhodho

�54�

The corresponding equations for A�m�z can be derived similarly:

pm @2xx � @2yy

� �ho � qm �@xho�2 � �@yho�2

h i� lx2ee�jbWm ÿ Am

z � � 0

Wm � ÿjbx2l0

Zfo

dAmz

dhodho

�55�

The results for the gauge functions W�a� show how the W�a� must be chosen to get rid ofthe undesired transverse components of the vector potentials A�a�.

In order that the solution is in the form of Equation, 52 i.e. that the solution dependsonly on f �ho�x; y��, it is necessary that the x and y dependence must be eliminated from

421


Equations 54 and 55. Thus the problem of ®nding the appropriate pro®le ho�x; y� can bereduced to solving the following set of simultaneous equations:

@xho�x; y�� 2� @yho�x; y�� 2� F1�ho�x; y�� 56�

@2xx � @2yy

h iho�x; y� � F2�ho�x; y�� 57�

where F1�ho� and F2�ho� are also unknown functions. In the detailed derivation, that canbe found in the Appendix we show that Equations 56 and 57 can be ful®lled only if thesecond partial derivative of ho�x; y� with respect to x equals its second partial derivativewith respect to y. Hence one can determine ho�x; y� as well as the unknown functions,F1�ho�x; y�� and F2�ho�x; y��. As a result we get that the pro®les satisfying Equations 56 and57 are of the following form:

ho�x; y� � x2 � y2 � a1 � a2 �58�ho�x; y� � �d1 � d2�x� �d1 ÿ d2�y �59�

If

d2f

d�ho�2� 0 �60�

which holds even if f �ho� is a slowly varying function of ho then solutions 58 and 59 givethe only possible solutions.

If Equation 60 does not hold then the right hand sides of Equations 56 and 57 must beconstant to eliminate the ho dependence from Equations 54 and 55 in order that thesolution is in the form of Equation 52. It can be shown in a similar manner as Equations58 and 59 were derived, that the solution for the electric and magnetic vector potentials arein the form of 52 only if ho is in the form of Equation 59.

From Equations 58 and 59 we conclude that the contour lines of the permittivity pro®lesde®ned by Equations 41 and 42 are either circles or straight lines of arbitrary inclination.On these contour lines the A�a�z �a � e;m� are constant. If our structure consists of severalregions then we have to match the ®elds at the boundary to ful®l the boundary conditions.Since the components of ~E and ~H are also constant on these contour lines, the dielectricinterfaces must coincide with one of the contour lines in order that the boundary condi-tions can be imposed. This means that if the contour lines are circles then the structuremust also be circularly symmetric, for example it must be an optical ®bre. If the contourlines are straight lines then the structure must be a dielectric slab waveguide with adielectric interface parallel with the contour lines. It follows directly from our derivationthat there are no more possible pro®les or arrangements for which the vector potentials arein the form of Equations 39 and 40.

5. ConclusionWe have given conditions that must hold if the electric and magnetic vector potentialshave a single non-zero component.

We distinguished two main cases: when the optic axis coincides with one of the trans-verse axes and when the optic axis is parallel with the longitudinal coordinate. If the opticaxis coincides with one of the transverse axes, we can assume that both vector potentialsare parallel with the optic axis if the ordinary refractive index varies only along that axis.

422

SZ. GaaÂl et al.

In this case one can obtain homogenous second order partial di�erential equations for Aex

and Amx . We showed that in this case the extraordinary permittivity (or refractive index) as

well as the geometry of the waveguiding structure are arbitrary.If the optic axis is parallel with the longitudinal coordinate then the vector potentials

may be parallel with that axis either in structures, where the ordinary refractive index isconstant in each region, or in dielectric slabs or in optical ®bres with circular symmetry. Ifthe ordinary refractive index is constant in each region, then the shape of the dielectricinterface is arbitrary. If the index is graded, then both the ordinary and the extraordinaryrefractive indices must have either circular symmetry (with the refractive index beingconstant on the boundary) or they must de®ne a dielectric slab.

AcknowledgementsParts of this work were supported by the Hungarian National Scienti®c Research Fund(OTKA), grant No. T 016318. We thank Hugo Hoekstra (Universiteit Twente, TheNetherlands) for the useful discussions.

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8. W. S. WEIGELHOFERW. S. WEIGELHOFER andand I. LINDELLI. LINDELL,, Int. J. Appl. Electromagn. Mat.Int. J. Appl. Electromagn. Mat. 44 (1994) 211.(1994) 211.

9. A. W. SNYDERA. W. SNYDER andand J. D. LOVEJ. D. LOVE,, Optical Waveguide TheoryOptical Waveguide Theory (Chapman and Hall, London, 1983).(Chapman and Hall, London, 1983).

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AppendixThe appendix is dedicated to the derivation of ho. To determine ho we ®rst di�erentiateEquation 57 with respect to x then with respect to y and eliminate @2xyho to get

�@xho�2�F 01 ÿ 2@2xxho� � �@yho�2�F 01 ÿ 2@2yyho� �A1�where F 01 � �dF1�=�dho�. Expressing �@ii�2ho�x; y� i � �x; y� from Equation 57 and�@�i�ho�x; y��2 i � �x; y� from Equation 56 and substituting them into Equation A1 we get

2@2xxho � F 01 ÿ 2F2 � �@yho�2G �A2�2@2yyho � F 01 ÿ 2F2 � �@xho�2G �A3�

where G � G�ho� � 2�F 01 ÿ F2�=F1.Here we can distinguish two cases again. In case (1) we suppose that G 6� 0 and in case

(2) we deal with G � 0. Now we show that the supposition that G 6� 0 is not correct, soG � 0, thus we can focus attention on case (2).

To prove that G � 0 we ®rst express �@yho�2 from Equation A1 and substitute it intoEquation 56 to obtain

G�@xho�2 � 2@2xxho � F1G� F 01 �A4�

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A similar expression can be obtained for the partial derivatives with respect to y. If we ®rstsubstitute @xh � p then introduce notation c for p2 � c we get a ®rst order linear di�er-ential equation for c. If we introduce the notation

Sp � exp

ZG�ho� dho

� � Z�F1Gÿ F 0� exp

ZG�ho� dho

� �� dho �A5�

then we get the following:

dho � dx C1x�y� exp ÿZ

G dh� �

� C2x�y� exp ÿZ

Gdh� �

� Sp

� �1=2�A6�

dho � dy C1y�x� exp ÿZ

G dh� �

� C2y�x� exp ÿZ

G dh� �

� Sp

� �1=2�A7�

Let us denote the square root terms on the right hand sides of Equations A6 and A7with rx and ry respectively. If we divide Equation A6 by rx, Equation A7 by ry andintegrate both parts we get Z

dho

rx� x� D1�y� �A8�Z

dho

ry� y � D2�x� �A9�

Equations A8 and A9 must hold even if D1 � 0 and D2 � 0. Therefore C1x, C2x, C1y , C2y

must be constants. Hence it follows that the left hand sides of Equations A8 and A9depend only on ho. This can only be possible if

ho � const1x� const2y �A10�yielding F1 � �const1�2 � �const2�2 from Equation 56 and G � 0 from Equations A1 andA2. It follows directly from G � 0 that F2 � F 01 and therefore

@2xxho � @2yyho � F 01 �A11�holds. It also follows from Equations A2 and A3 that ho must satisfy the followingrelation:

@2xxho ÿ @2yyho � 0 �A12�giving the general solution [12]

ho � �C1�x� y� � C2�xÿ y��=2 �A13�for ho where C1 and C2 functions are to be determined as follows. Let us introduce thefollowing notations:

n � x� y f � xÿ y �A14�and substitute Equation A13 into Equation 56 to get

0:5 �@nC1�2 � �@fC2�2h i

� F1 �A15�

Di�erentiating Equation A15 with respect to ho using the chain rule of di�erentiation

424

SZ. GaaÂl et al.

@C1=2

@ho� @xC1=2

1

@xho� @yC1=2

1

@yho�A16�

we obtain

@2nnC1 � @2ffC2 �@nC1@

2nnC1 � @fC2@

2ffC2

@xho� @nC1@

2nnC1 ÿ @fC2@

2ffC2

@yho�A17�

Multiplying both sides of Equation A17 with @xho@yho yields

�@nC1�2�@2nnC1 ÿ @2ffC2� � �@fC2�2�@2ffC2 ÿ @2nnC1� �A18�The solutions of Equation A18 are given by

C1 � a1 � b1n� n2 �A19a�

C2 � a2 � b2f� f2 �A19b�

C1 � d1 �A20a�

C2 � d2 �A20b�where the constants ai and bi i � �1; 2� must be de®ned that Equation 56 holds resulting inbi=0 for i � �1; 2�, d1 and d2 are arbitrary. This means that the required pro®les ho�x; y�have one of the following forms:

ho�x; y� � x2 � y2 � a1 � a2 �A21�

ho�x; y� � �d1 � d2�x� �d1 ÿ d2�y �A22�We note that pro®le A22 has already been obtained in Equation A10.

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