We present a model of a graded-index taper for which the field solutions can be obtained directly byseparation of variables. Those fields tightly concentrated about the axis of the taper are given very accuratelyby remarkably simple expressions which clearly illustrate the influence of the tapering. Frequent compari-sons with a geometrical optics analysis demonstrate the link between the ray and field approaches and alsoassist in the physical interpretation of several results. Examples of the application of these field solutions arealso described.
1. Introduction
It has become clear that tapered optical waveguides(of both slab and circular geometry) will become inte-gral components of various optoelectronic systems.They will be used, for example, as coupling andbranching devices in fiber-optic communications sys-tems1 and to perform specialized functions, such asmode filtering, in integrated optics systems.2 In addi-tion, the focusing properties of tapered graded-indexmedia may find applications as focusing elements,3image reducers and enlargers,4 and light concentra-tors.5 Consequently, detailed understanding of thecharacteristics of light propagation in such media ishighly desirable.
Unfortunately, due to the loss of translational in-variance, we cannot speak of modes in the usual sensewhen discussing the fields in such media, and even thesimplest tapered waveguides present immense mathe-matical difficulties. This has led to the developmentof various approximation methods, the most widelyused being coupled-mode theory and the adiabaticapproximation.
Any exact field solutions are clearly very valuable,not only for elucidating the nature of wave propagationin tapered media but also by providing a test casefor directly checking the accuracy of the commonlyused approximation methods. This is particularly im-
C. Pask is with Australian Defence Force Academy, University,College Mathematics Department, Campbell, ACT 2600, Australia;the other authors are with Australian National University, Depart-ment of Applied Mathematics, Research School of Physical Sci-ences, Canberra, ACT 2600, Australia.
portant in view of the rarity of exact solutions (seeMarcatili6 for another discussion on exact field solu-tions).
In this paper, we show how the fields inside a partic-ular graded-index taper can be analyzed exactly. Themodel that we use is the standard parabolic-indexmedium but with a characteristic radius that variesquadratically with distance along the axis. We showthat the field solutions which are tightly concentratedabout the taper axis are given to an excellent approxi-mation by remarkably simple expressions which veryclearly illustrate how the tapering affects the fields.This work complements a previous study of the exactray analysis of this taper that was recently made by twoof the authors.7 We make frequent comparisons be-tween the ray and field pictures.
We begin in Sec. II with a description of the tapermodel (with slab geometry). In Sec. III we show howthe fields can be obtained directly by separation ofvariables. We then calculate the small-angle form ofthe fields; this leads to simple accurate expressions forthe fields which are tightly concentrated about the axisof the taper. The properties of these fields are dis-cussed in Sec. IV, and several examples of their appli-cation are given in Sec. V. In Sec. VI we describebriefly the fields in a graded-index taper of circulargeometry. Because the physical interpretation ofthese fields is completly analogous to that of the planarcase (apart from the added complication of a furtherdimension) we go no further than to derive the explicitform of the fields. Section VII contains some conclud-ing remarks.
11. Model of 2-D Taper
First, consider the class of graded-index media mod-eled by the following refractive-index distribution:
Fig. 1. Coordinate systems. The origin of the R - 0 plane-polarcoordinate system is located at the point x = 0, z = -D of the
Cartesian coordinate system.
Here (R,O) are plane-polar coordinates (see Fig. 1),G(0) is an arbitrary 27r-periodic function of 0, and 2 isa constant with the dimensions length squared.
A large number of fundamentally different graded-index media can be modeled by Eq. (1) depending onthe choice of G(O). The particular choice that is ofmost interest to us is
G(O) = sin2 0/cos4 0. (2)
This can clearly be seen to describe a graded-indextaper when we change to a Cartesian coordinate system(x,z) displaced relative to the origin of the plane-polarsystem (see Fig. 1):
x = R sinO, z = R cos0-D. (3)
D is a constant which for convenience we choose to bepositive. The index distribution can now be written[Eqs. (1)-(3)]
- n 1- 2Ax 2 I(D + Z)4]. (4)
We can put Eq. (4) into a more familiar form:
n2 =n[1 - 26x2/p2 (z)], (5)
where
p(z) = po(I + z/D) 2
Fig. 2. Constant refractive-index contours for the graded-indextaper defined by Eq. (4). Index values are ordered n0 > no > n22> n 3.
dence exp(-icwt), where X = ck is the monochromaticsource frequency. A well-known simplification re-sults when the situation is two-dimensional, as thisallows the electromagnetic field to be constructed fromthe solution of a scalar problem: the electromagneticfield is expressed as the sum of a TE field [E = (0,EyO)]and a TM field [H = (0,Hy,0)]. The TE-field compo-nent Ey is a solution of the scalar wave equation, whilethe TM-field component Hy is a solution of a morecomplicated scalar equation that simplifies to the sca-lar wave equation under the weak guidance approxi-mation.8 Thus the complete electromagnetic field canbe constructed from solutions of the scalar wave equa-tion:
(V2 + k2n 2)q/ = 0. (8)
Ill. Solutions
A. Exact Solution of Scalar Wave Equation
Exact solutions for the scalar wave equation [Eq. (8)]can be found for all members belonging to the generalclass of graded-index media [Eq. (1)] by the method ofseparation of variables. Writing
(6) t(R,O) = A(R)B(0),
is the characteristic radius of the grading in the trans-verse direction (and varies quadratically with distancealong the taper axis), and
26 = 2Apo/D4(7)
is a dimensionless parameter. From Eqs. (5) and (6)we see that the medium can be described as an infiniteparabolic-index medium with parabolic tapering.Some constant refractive-index contours are plotted inFig. 2.
The reason for introducing the more general class ofmedia [Eq. (1)] becomes clear in Sec. II when it isshown that exact field solutions can be found for allmembers of this class.
It should be pointed out at this stage that the indexdistribution described by Eq. (5) becomes unphysicalfor large values of /I2Ixj/p(z) [since n2 -X as2IxI/xjp(z) - o]. However, this will be of no concern
provided we focus attention on those fields which arenegligible when VIxj >> p(z).
The electromagnetic field is found by solving Max-well's equations. We assume an implicit time depen-
(9)
we find that the R-dependence is a linear combinationof Bessel functions9:
A(R) = Yf(knR),
and the 0-dependence is given by the solution ofd 2B/dO2 + [M2
- a2G(0)JB(0) = 0.
Here
a = kno+2E,
(10)
(11)
(12)
and JU2 is the separation constant.Regardless of the specific form of G(O), the R-depen-
dence of the field is always given by Eq. (10). For ourgraded-index taper, G(O) is given by Eq. (2), and wefind the 0 dependence of the field solutions by solving
d2B/d0 2+ (2 - a
2 sin20/cos
40)B(O) = 0. (13)
The solution of Eq. (13) also determines the admissiblevalues of the separation constant , 2.
Although the solution of Eq. (13) cannot be readilyexpressed in terms of commonly used functions, it has
a particularly simple form for small 0. This is dis-cussed in the next section.
B. Small-Angle Approximation
To look at the form of the solution of Eq. (13) forsmall 0, we make the approximation
G(O) _ 02
to second order in 0, and so Eq. (13) becomesd2B/d02 + ( 2
- a20
2)B(0) = 0.
(14)
n2 = n'(1 - 25x2/p2) (19)
[where p0 is the constant characteristic radius of themedium, compare with Eqs. (5) and (6)]. The propa-gating Hermite-Gaussian modes of such a medium arewell known1":
Clearly, if the actual solution [of Eq. (13)] leads to afield which is tightly concentrated about the axis of thetaper and is negligible at larger values of 0, the solutionof Eq. (15) will be an excellent approximation to theactual solution. (This point is discussed more quanti-tatively later.) Thus we are interested in the solutionsof Eq. (15) which tend to zero for large 101.
This restriction on the physically acceptable solu-tions leads to the condition10
J2= a(2n + 1), n = 0,1,2,.... (16)
and to the corresponding solutions
Bn(0) = exp(-a0 2/2)Hn(4a0), (17)
where the Hn functions are standard Hermite polyno-mials. 9
We have thus determined a set of basis fields whichcan be used to describe the propagation of fields con-centrated about the axis of the taper. Writing themout explicitly,
f(l)(knoR) and ](2)(knoR) [where = a(2n + 1)]could be used in place of the J and Y Bessel functions,since both sets obviously provide two linearly indepen-dent solutions of the R equation.)
Clearly, the basis fields have an unchanging fieldshape over any radial arc (i.e., at any fixed R) and inthis sense are an extension of the mode concept oftranslationally invariant media. (This is discussedagain in Sec. IV.C when we consider the basis fieldsfrom the point of view of transverse resonance.)
The actual combination of the Bessel functions to beused depends on the boundary conditions (ie., theregion of the taper in which propagation occurs). Thisfollows from the requirement that the basis fields mustbe able to exist in isolation of each other (so that theycan, at least in principle, be excited individually).This point becomes clearer when several applicationsof the basis fields are discussed in Sec. V.
C. Comparison with Untapered Medium
Because of their simplicity, the basis fields demon-strate the effect that the tapering has on wave propaga-tion in the medium with remarkable transparency.We compare the basis fields [Eq. (18)] with the modesof a translationally invariant infinite parabolic-indexmedium:
(20)
where
On = [k2n2 (q/po)(2n + 1)]1/2 (21)
are the propagation constants, and the parameter q isdefined by
q = kn 0 J_. (22)
We make the comparison with the basis fields [Eq.(18)] by converting to Cartesian coordinates. Notingthat within the small-angle approximation we have
Comparing Eqs. (20) and (24) reveals that in anycross section of the taper the field distribution is iden-tical to that within a translationally invariant mediumwhich matches the taper at that particular value of z.(This incidentally illustrates the power of local modeapproaches to wave propagation in tapered media.)Only the propagation characteristics of the two sets offields differ: in the untapered case propagation isdetermined by complex exponentials, whereas for ta-pered media it is determined by some combination ofBessel functions.
We can also check the limiting behavior of the basisfields by looking at the form they take when the degreeof tapering is made negligible. This is formally carriedout by finding the asymptotic form of Eq. (24) for largeD (provided we constrain z to lie within fixed limits;i.e., we consider a finite section of the taper). Fromthe definition of the taper profile [Eqs. (5) and (6)] it isclear that taking this limit reduces the medium to thetranslationally invariant profile Eq. (19).
In the Appendix we show that if we choose to de-scribe the propagation characteristics of the basisfields by the WI') and (2) Hankel functions (ratherthan the J., and Y, Bessel functions), these basis fieldsbecome the propagating modes of the untapered medi-um [Eq. (20)] in the large D limit. Thus the basis fieldsdo display the correct limiting behavior.
IV. Field PropertiesA. Properties of the Basis Fields
As we mentioned previously, the basis fields aretightly concentrated about the axis of the taper. Toillustrate this, in Fig. 3 we plotted B,(0) [Eq. (17)] for n
Fig. 3. Plots of Bn(O) for n = 0 (solid line) and n = 5 (broken line)and with a = 100. (Note that the curves have been nomalized to givea maximum value of unity.) Also shown are ~0 and 0
5 where thesecond derivative disappears.
= 0 and n = 5 and with typical parameter values. Ingeneral, Bn(0) has an oscillatory part centered aboutthe origin (with n zeros) and has an exponential drop-off at larger values of 101 (the wings of the distribution).Thus we see that the field can be considered confinedto the region
0, < < in, (25)
where An is some value of 0 in the wings of the distribu-tion where the decaying exponential behavior has be-come dominant. We can quantify this by choosing Anto be the value of 0 where Bn changes from concavedownward to concave upward (i.e., d2Bn/d0 2
= 0).From Eq. (15) we see that
in = [(2n + 1)/a]1/2.
c/•-_00
Fig. 4. Schematic drawing of a typical ray trajectory inside thegraded-index taper. Ray caustics are shown at R = Rmin and 0 =dmax, confining the ray to the region inside a truncated wedge. Farfrom the taper apex the ray tends asymptotically to a straight line,here denoted 0_ (as the ray comes in) and 0+. (as the ray leaves).
ray trajectory. Then ray trajectories are given para-metrically by [R(t),0(t)],-O <t < a. An invariant forthe ray paths is
K = R2 (t)[n2 - P2
(t)] = R2(0)[n2 - A2 (0)] (28)
(where we introduced the dot convention to denote a tderivative), and this equation can be easily integratedto obtain the R component of the ray path. The 0component is found to satisfy the equation
R402 = K - 2Arn2G(O). (29)
For the graded-index taper, a typical ray trajectoryis illustrated in' Fig. 4. Ray paths are found to beconfined to the region
-°max - < max'(26)
(30)
Thus the field is confined to the region inside awedge of half-angle an, and from Eq. (26) it is clear thatbasis fields with larger values of n are spread out morethan those with smaller values. This leads us to asimple condition for the validity of the basis fieldexpressions:
[(2n + 1)/]'/ 2 << 1. (27)
When Eq. (27) is not satisfied, serious errors may beincurred by solving Eq. (15) in place of Eq. (13).
The characteristics of the R-dependence of the fieldare discussed later when specific forms are used inseveral samples illustrating the application of the basisfields.
B. Comparison with Geometrical Optics
In this section we summarize some of the relevantfacts concerning ray propagation in the graded-indextaper. The full analysis can be found in the paper byBertilone and Pask.7 By comparing the ray picture tothe field analysis we obtain a valuable physical inter-pretation of many results.
Due to the separability of the ray-path equation, anexact ray analysis can be carried out for all graded-index media of the general class described by Eq. (1).We introduce a t parameter, defined by the relation dt= ds/n, where ds is an element of path length along the
R ..i < R (31)
as shown in the figure. Here max and Rmin are deter-mined by the initial launching conditions of the ray. Itis found that Rmin is exactly given by
Rmin = VK/1no, (32)
while 0 max is given in the small-angle approximation[Eq. (14)] by
Omax = /nO- (33)
Hence the family of rays launched with initial condi-tions giving the same value of the ray-invariant K isconfined by the same caustics; i.e., they are confined tothe same region inside a wedge of half-angle Omax that istruncated at the apex by an arc of radius Rmin-
As shown in Fig. 4, far away from the origin (R = 0)the ray trajectories are essentially straight lines.(They asymptotically approach straight lines as R a.) As a typical ray approaches the apex it begins tooscillate [always within the limits set by Eq. (30)] untilit reaches its point of nearest approach, R = Rmin Itthen changes direction and leaves the taper, oscillatinga finite number of times on its way out.
Straightaway we notice the agreement between theray and field analyses concerning the confinement ofthe light to the inside of a wedge. Comparing Owax ofthe ray analysis [Eq. (33)] with 0n of the field analysis
reflection at the caustics (at So and S1) must be equalto an integer multiple of 2ir. This is the transverseresonance condition. Noting that reflection at a caus-tic12 introduces a phase change of -7r/2, we have
A4a + 2(-2r/2) = 27rn,(n an integer)
where
A = fA [kn(R,0)Rb/(P2 + R2O2)112 ]RdO.so-s2
(36)
(37)
Fig. 5. Single period of a typical ray path is shown from SO to S2.Also illustrated is the local plane-wave vector k1,0 (R,0) at a point(R,0) on the trajectory and its radial component kR and transverse-
radial component ko.
[Eq. (26)] allows us to associate the basis field Aln withthe family of rays with ray-invariant K. Setting Omax =OnZ we find that
K = a(2n + 1)/k2; (34)
i.e., the above relation gives the criterion for a family ofrays to correspond to a basis field. This result isdeveloped more formally in the next section.
Finally we point out that there are many other strik-ing agreements between the ray and field pictureswhich will become evident when we look at applica-tions of the basis fields in Sec. V.
C. Basis Field as a Transverse ResonanceIn this section we provide a simple and intuitive
explanation of the ray-field relationship expressed inEq. (34). This will be a straightforward extension ofthe transverse resonance concept that has been suc-cessfully applied to translationally invariant wave-guides."1
We begin by noting that when we attempt to asso-ciate the ray description of light propagation with thewave description, we must incorporate phase with theray trajectories. Rays become associated with localplane waves with propagation vectors directed alongthe ray paths.
If (R,0) are the plane-polar coordinates of a point onthe trajectory, the local plane-wave propagation vectoris
(where the dot convention is used as in Sec. IV.B, andeR,eo are plane-polar unit vectors in the radial andtransverse-radial directions).
Now the basis fields [Eq. (18)] are characterized byan unchanging field distribution over any radial arc.This indicates that the 60 component of the localplane-wave vector must satisfy a self-consistency con-dition to describe a basis field. In Fig. 5 we show atypical ray path over a single period (s = S = S2,where s denotes path length along the ray trajectory).Clearly, to describe a basis field, the phase change ofthe eo component due to the optical path length tra-versed, added to the phase change suffered due to
Using Eqs. (28) and (29) we can reduce Eq. (37) toI Ad = 2 f |+OmaX
_ Om..[G(Omax) - G(0)]11
/2d. (38)
Within the small-angle approximation we can re-place G(0) and G(Omax) with 02 and max) respectively.A simple integration then yields
A = rk2K/a. (39)
Substitution of Eq. (39) into the transverse reso-nance condition [Eq. (36)] leads to the ray-basis fieldrelationship, Eq. (34).
V. Examples of Applications of the Basis Fields
In this section we present three simple examplesillustrating the application of the basis fields. Fre-quent comparisons are made with geometrical opticsto obtain a more intuitive appreciation of the propaga-tion process.
A. Fields Launched down the Complete Taper
As a first example, suppose we look at propagation inthe complete graded-index taper (i.e., we assume thatthe taper is not truncated anywhere so that it narrowsto a point). With all sources located in the z > zoregion, we are interested in determining the field in thez < z region. This example is illustrated schematical-ly in Fig. 6(a). (The reader is warned that there aresome physical difficulties associated with the mathe-matical model described below. They are discussedat the conclusion of this section.)
In this case, because the taper extends to R = 0, theY, part of the radial dependence [Eq. (10)] must berejected (since these solutions become unbounded forsmall R). Hence the basis fields take the explicit form
These basis fields are clearly standing-wave-type so-lutions; the net time-averaged power passing throughany given surface with fixed R is zero. Furthermore,from the general behavior of Bessel functions we caninfer a further important characteristic of these fields.First, we note the well known fact that the Besselfunction J,(x) (, >> 1) has rather different asymptoticbehavior in the regions x'<< , and x >> /i. In the regionx << A it behaves like a decaying exponential, whereas ithas oscillatory behavior when x >> y. This leads us toconclude that the basis fields must be considered asbeing confined to some region
ward- and forward-propagating fields interfere witheach other to create the standing wavefield. The loca-tion of the caustic [Eq. (42)] corresponds physically tothe cutoff of the associated mode of the untaperedparabolic-index medium which is made to match thetapered medium by allowing the characteristic radius[p0 in Eq. (19)] to depend parametrically on z. FromEq. (21) we see that an untapered parabolic-indexmedium with characteristic radius p(z) (i.e., matchingthe tapered medium at z) supports modes with propa-gation constants
On = k2 n - [q/p(z)](2n + 1)11/2
ZZ- ~ ~ ~ ~ ~ ~ -.
,Z. t ''
SOURCES
Fig. 6. Schematic illustrations of the examples discussed in Sec.V.A (a), V.B (b), and V.C (c). Also shown are some typical ray
trajectories encountered in these cases.
R > Rn, (41)
where Rn is some value of R below which the decayingexponential behavior becomes dominant. Takingequality between order and argument to denote thisvalue,13 we have
Rn = [a(2n + 1)J'12 /kn,. (42)
We shall refer to the surfaces R = Rn and 0 = 0n ascaustics because they can be associated with the raycaustics R = Rmin and 0 = Omax, respectively (see later inthis section).
There is a very simple physical explanation of thisresult. Writing
J,(knoR) = (1/2) [W(1)(knR) + Y1P)(kn0R)] (43)
allows us to interpret the basis field as a backward-propagating field [with radial dependene W(2)(knoR)]traveling into the taper and reflecting off the caustic atRn, thereby converting it into a forward-propagatingfield [with radial-dependence ](1)(knoR)]. The back-
(44)
When the characteristic radius is too small, the propa-gating mode is cut off and becomes evanescent (i.e., Onbecomes imaginary). From Eqs. (44) and (23) and thedefinitions of the various parameters, we find that thisoccurs at Rn given by Eq. (42). Thus the caustic occurswhere the taper is too narrow to support the nth propa-gating mode.
Thus, for this first example, we have shown that thebasis fields are confined to the region inside a truncat-ed wedge (-an 0 0n and R 2 Rn). This is inexcellent agreement with the ray picture7: all rays'launched into the z < z region are also confined to atruncated wedge, as illustrated in Fig. 4. All such raystravel down the taper (executing oscillations on theway), reflect off the caustic, andthen travel out again.Comparing the field caustic R1n [Eq. (42)] with the raycaustic Rmin [Eq. (32)] brings us back to the ray-basisfield relationship of Eq. (34).
A general field launched down the taper is express-ible as a superposition of the basis fields (provided thatthis incident field is sufficiently concentrated aboutthe taper axis). Because of the finite number of physi-cally meaningful basis fields [see Eq. (27)] this is clear-ly not an exact representation but will be a very goodapproximation for many situations of practical inter-est. Remembering that on any particular plane eachbasis field has both forward- and backward-propagat-ing components [Eq. (43)] we must match the incidentfield to the backward-propagating (-z-direction) com-ponent and use the orthogonality of Hermite polyno-mials9 to obtain the amplitude of each basis field ap-pearing in the superposition. The field reflected outof the taper can then be found.
Unfortunately, the mathematical model we have de-scribed is inadequate at two small regions within thecaustic surface: the index distribution is unphysical(less than unity) in a small region near each cusp of thecaustic surface [i.e., at the points (Rn, ±:n)]. There-fore, some errors will be incurred when we use thismodel to represent a real physical graded-index taper.Nevertheless, the model is still useful in illustratingthe physics behind the reflection of an incident fieldout of a taper.
B. Fields Launched up the Complete Taper
For the second example, we once again considerpropagation in the complete graded-index taper, butthis time all sources are located in the z < zo region, and
we are interested in the field in the region z > z0. Weillustrate this schematically in Fig. 6(b).
It is more convenient, in this case, to consider the Rdependence of the basis fields to be a linear combina-tion of the K(i)(knoR) and ](1)(knoB) Hankel func-tions (as discussed in Sec. III.). This is because, from aconsideration of the asymptotic form of these func-tions9 for large R,
[where i refers to K(1) and (2), respectively] it is clearthat we must reject the Y2) solution because there canbe no backward-propagating (i.e., in the -z direction)field at infinity. [As noted in Sec. II we are assuming atime dependence exp(-iwt).] Hence, in this examplethe basis fields take the form
V,(RO) = exp(-a0'/2)Hfi(a/0)f/)(,+,)(k fOR). (46)
These basis fields represent progressive waves andin the limit of negligible tapering become the forward-propagating modes of the untapered infinite parabol-ic-index medium (see Sec. III.C).
These results are consistent with the ray optics anal-ysis, which predicts that rays launched into the z > zoregion will continue to travel away from the origin (R =0), executing a finite number of oscillations, and willthen tend asymptotically toward straight lines. Thusa forward-propagating wave solution is to be expected.
Once again, it should be mentioned that a generalfield launched up the taper can be represented (ap-proximately) by a superposition of the basis fields, Eq.(46). However, in this case the problem is simplifiedby the absence of any backscattered field componentsso that the incident field (on the plane z = zo) isexpanded directly in terms of the functions in Eq. (46)evaluated on this plane.
C. Fields launched in a Section of the Taper
The problem of most practical interest concernswave propagation inside a section of the taper [saybetween the planes z = zo and z = z > zo, as illustratedschematically in Fig. 6(c)] and assuming that allsources lie outside this region.
Within the restrictions discussed in the previousexamples, a general field incident on either endface (z= z or z = z) will create a field inside the taper sectionwhich is expressible as a superposition of the basisfields:
E = a,, exp(-ao2/2)Hn(JaO)n
x [J (,n+,)(knoR) + bY ,(,n-a )(k2nOR)I (47)
where the constants an and bn are complex amplitudesdetermined from boundary conditions.
Some of the added complications in dealing with afinite section of the taper are made clearer by lookingat it from the point of view of geometrical optics.Considering the situation shown in Fig. 6(c) with thesources in the region z > z1 , we see that some of thelaunched rays pass right through the taper (when the
' - I
-7 I /-/ I
R,eiP)PXYZ)
Fig. 7. Coordinate systems. The origin of the R - - 0 spherical-polar coordinate system is located at the point x = O, y = 0, z = -D ofthe Cartesian system. We follow the usual convention: O S R < ; O
< 0 < r; 0 < < 27r.
ray caustic lies outside of the taper section), whileothers will be turned around and exit the same waythey entered (ray caustic inside the taper). Thus, evenignoring reflections from the interfaces, the ray propa-gation problem is clearly more complicated than in theprevious two examples with the complete taper.
VI. Propagation Inside a Taper of Circular Geometry
We now briefly analyze the situation occurring in-side a graded-index taper of circular geometry. Be-cause the general characteristics of the fields here areclosely analogous to those found in the taper of planargeometry, the analysis is presented in much less detail.
A. Model of 3-D Taper
As in the planar case, we start by defining a generalclass of graded-index medium:
n2 = n2 [1 - 2AG(O)/R2], '
but this time (R,O,0) are spherical-polar coordinates(see Fig. 7). The function G(O) is an arbitrary 2r-periodic function of 0.
In the next section we show how to obtain exact fieldsolutions for all members of the class of medium de-scribed by Eq. (48). For the moment, however, wenote that the particular member of this class whichmodels a graded-index taper is obtained by choosing
G(O) = sin20/cos4 0. (49)
This can be seen by changing to a Cartesian coordinatesystem (xy,z) displaced relative to the origin of thespherical-polar system (Fig. 7):
We can write Eq. (51) in a form more familiar to thosein the optical waveguide field
n2 = n' [1- 2r 2/p2(z)], (53)
where 26 is the dimensionless parameter defined in Eq.(7), and p(z) is the (z-dependent) characteristic radiusdefined in Eq. (6). Thus the model describes an infi-nite parabolic-index medium of circular geometry withparabolic tapering.
The actual electromagnetic field inside the taper willbe a solution of Maxwell's equations. However, withthe well-known weak-guidance approximations we canconstruct the fields from the solutions of the scalarwave equation.
B. Exact Solution of Scalar Wave Equation
Using the method of separation of variables, we canfind exact solutions for all media having an index dis-tribution in the form of Eq. (48). Writing
+(R,0,0) = A(R)B(0)Co), (54)
we find that
A(R) = (1/FR)J/V+(,, 4)(knoR).WNlW) YF-(noR),
CM = { cosm m = 0,1,2,..., (56)
and B(0) is found by solving
sin20d2 B/dO2 + sinG cos~dB/dO+ [v sin20 - a2 sin 2OG(0) - m2]B(O) = 0. (57)
The solution of Eq. (57) also determines the admissiblevalues of the separation constant v.
For the graded-index taper G(0) is given by Eq. (49).However, as in the planar case, the solution of Eq. (57)cannot be readily expressed in terms of commonly usedfunctions. Fortunately, however, within the small-angle approximation the solutions take on a particu-larly simple form.
C. Small-Angle Approximation
Under the small-angle approximation we have G(0)02 [see Eq. (14)], and we likewise expand the coeffi-
cients appearing in Eq. (57) to second order in 0. Thisleads to the simplified equation
The solution of this equation provides a good de-scription of those field solutions which are concentrat-ed about the axis of the taper. We are thus led to seeksolutions of Eq. (58) which tend to zero for large 0. It isfound that the only solutions which give fields bound-ed on the taper axis while satisfying the above restric-tion are of the form
where L(m) are associated Laguerre polynomials as de-fined in Abramowitz and Stegun.9 The separationconstant takes on the discrete values
v = 2a(m + 2n + 1). (60)
Thus we found a set of basis fields describing wavepropagation close to the axis of the graded-index taperof circular geometry. Writing them out explicitly,
o = R-1/2 0- exp(-a02/2)L(m)(a02) Sin(mO) fJ{(k°nR)n~~m n Cos I~~~Y~(kn0R)
m,n = 0,1,2,..., (61)
where
= (/2)[1 + 8a(m + 2n + 1)11/2. (62)
Note that the superscripts o and e which appear in thesymbol A, denote the basis fields which are odd/evenwith respect to 0. Thus, for ^6°,m the 0 dependence isgiven by sinmok and for Ane, by cosmo. (For m F- 0these are degenerate, of course, 44Am and en,,, haveidentical field distributions, the only difference beingin a rotation by 90° about the axis of the taper.)
The characteristics and application of these basisfields can be obtained in the same manner as describedin the earlier sections of this paper for tapers of planargeometry.
VII. Conclusion
The basis fields presented in this paper are veryaccurate descriptions of those field solutions which aretightly concentrated about the axis of the taper. Theirusefulness lies in their remarkable simplicity and theresultant clarity with which effects associated with thetapering are illustrated. As well as elucidating thephysics involved in wave propagation in tapered me-dia, they can be applied in a variety of studies. Forexample, they can be used to test the accuracy of theadiabatic approximation and also of higher-order iter-ative solutions of the coupled-mode equations. Suchinformation would clearly be of much assistance todesigners of optoelectronic devices having taperedcomponents. They can also be used in accurate stud-ies of focusing and imaging in tapered graded-indexrods.
Appendix: Limiting Behavior of the Basis Fields
Here we outline the procedure for examining thebehavior of the basis fields in the limit of negligibletapering.
From the definition of the index distribution [Eqs.(5) and (6)] we keep 26 and po fixed and allow p(z) todepend parametrically on D. Clearly, the degree oftapering is determined by D. If we consider a givensection of the taper (so that z is constrained to liewithin fixed limits) and let D - a, the tapering disap-pears, and we are left with a section of translationallyinvariant infinite parabolic-index medium [Eq. (19)].We now show that the basis fields [Eq. (24)] exhibit thecorrect limiting behavior.
First, we note that we can write the basis fields [Eq.(24)] in the form
(and hence sec >> 1).The asymptotic form of Eq. (Al) for large D follows
once we know the asymptotic form of J,(,u secb) andY,(, secb) for large 1u [since from Eqs. (A2) and (A3) itis clear that taking the asysmptotic limit for large D isequivalent to taking the asymptotic limit for large uwith sec fixed]. The standard result is9
(Here A1 and 01 are amplitude and phase terms whichwe take to be independent of z.)
With the further observation that p(z) p Po for largeD, we obtain the required limiting expression for thebasis fields [Eq. (Al)]:
4,6n - exp(-qx 2 /2p 0 )Hn[(qp 0 )'12 x]
Xcosl[kno - V2(2n + 1)/2polz + 0,1
lsinl[kno - V2(2n + 1)/2po]z + (fil. (A9)
We compare expression (A9) with the modes of anuntapered infinite parabolic-index medium [Eqs. (20)and (21)]. Under the simplifying assumption [Eq.(A4)] we find that the modal propagation constants[Eq. (21)] reduce to
On _ kn0 - j2(2n + 1)/2 po, (A10)
and we conclude that the basis fields do indeed havethe appropriate limiting behavior. Using the Hankelfunctions W(') and J(2) in place of the J, and Y, Besselfunctions (see (Sec. iLI.B) leads to the complex expo-nential propagating mode solutions of Eq. (20).
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