Analysis of Scattering from Fiber Waveguides with Irregular Core Surfaces Eric G. Rawson The scattering from clad fiber optical waveguides that have cores whose surfaces are perturbed in various ways is calculated with the induced dipole method. Such scattering is of interest because it can couple the guided modes of a multimode waveguide, thereby reducing the temporal dispersion. The five surface de- viations considered are a rectangular radial step, a sinusoidal perturbation of the radius, a sinusoidal me- ander of the core, a deviation where the core alternates periodically along the core axis between a y-ex- tended elipse and a z-extended elipse (x is the core axis), and a helical meander of the core axis. 1. Introduction Marcuse has calculated the scattering from slab waveguides' and round waveguides2, 3 that have cores whose dimensions are sinusoidally perturbed by solv- ing Maxwell's equations and using a first-order per- turbation theory. The scattering that results from such deformation is of interest because in a multi- mode waveguide, such scattering can couple the guid- ed modes and thereby reduce the temporal disper- sion. 4 ' 5 Recently, 6 I derived the angular distribution of light scattered from aligned, finite dielectric nee- dles, embedded in cylindrical fiber waveguides by calculating the radiation from dipoles induced in the needles by the incident light. In this paper, I use the induced-dipole approach to derive closed-form solu- tions for the angular scattering from cylindrical waveguides that have various deviations of the core- cladding interface, including deviations that might be deliberately impressed on the core during or after fabrication. The calculation of interface deviation scattering in this way has certain advantages and disadvantages when compared to the Maxwell equation-perturba- tion theory technique. Very complicated surface ir- regularities can be analyzed relatively simply, and the cylindrical waveguide (three-dimensional) case can be treated as easily as its slab waveguide (two- dimensional) analog. The induced-dipole analysis, as presented here, is limited to the case where the surface deviations are small compared to the radius and the refractive index change at the core surface is The author was with Bell Laboratories when this work was done; he is now at Xerox Research Center, Palo Alto, California 94304. Received 18 April 1974. small compared to unity. (This limitation is not fun- damental. A more general, and more complicated induced-dipole analysis could be made.) This analy- sis is therefore less general than Marcuse's perturba- tion theory.1 4 For the cases of interest mentioned earlier, however, these small-parameter assumptions are valid. In this paper I calculate the induced-dipole scat- tering for several cases. These include (1) a rectan- gular step perturbation of the radius; (2) a sinusoidal perturbation of the radius; (3) a sinusoidal meander of the core axis; (4) a helical meander of the core axis; and, (5) a deformation in which the core cross section alternates periodically along the fiber between an el- ipse extended along the y axis and an elipse extended along the z axis, where y and z are orthogonal axes in the cross-section plane. The extension of these cal- culations to other interface irregularities is straight- forward and will be evident from these examples. Because the analyses of these cases are very similar to one another and to the analysis of scattering from aligned dielectric needles, 6 the full analysis of the rectangular radial deviation is given in the Appendix, together with additional notes related to the analysis of the other cases. Only the final results of this deri- vation are given in the following sections. 11. A Rectangular Radial Deviation Consider a dielectric fiber core, as shown in Fig. 1. The rectangular deviation can be viewed as a hollow dielectric sleeve about an otherwise ideal (i.e., loss- less) guide, in which the dielectric constant is per- turbed by an amount A = - o. Such a sleeve is shown in Fig. 2(a). The scattering is thus the in- duced-dipole radiation from this sleeve. The total scattered power per steradian in the direction 0 is given by [Eq. (Oa)] ( is the direction inside the core, 2370 APPLIED OPTICS / Vol. 13, No. 10 / October 1974
Transcript
Analysis of Scattering from Fiber Waveguides withIrregular Core Surfaces
Eric G. Rawson
The scattering from clad fiber optical waveguides that have cores whose surfaces are perturbed in variousways is calculated with the induced dipole method. Such scattering is of interest because it can couple theguided modes of a multimode waveguide, thereby reducing the temporal dispersion. The five surface de-viations considered are a rectangular radial step, a sinusoidal perturbation of the radius, a sinusoidal me-ander of the core, a deviation where the core alternates periodically along the core axis between a y-ex-tended elipse and a z-extended elipse (x is the core axis), and a helical meander of the core axis.
1. Introduction
Marcuse has calculated the scattering from slabwaveguides' and round waveguides2,3 that have coreswhose dimensions are sinusoidally perturbed by solv-ing Maxwell's equations and using a first-order per-turbation theory. The scattering that results fromsuch deformation is of interest because in a multi-mode waveguide, such scattering can couple the guid-ed modes and thereby reduce the temporal disper-sion.4'5 Recently,6 I derived the angular distributionof light scattered from aligned, finite dielectric nee-dles, embedded in cylindrical fiber waveguides bycalculating the radiation from dipoles induced in theneedles by the incident light. In this paper, I use theinduced-dipole approach to derive closed-form solu-tions for the angular scattering from cylindricalwaveguides that have various deviations of the core-cladding interface, including deviations that might bedeliberately impressed on the core during or afterfabrication.
The calculation of interface deviation scattering inthis way has certain advantages and disadvantageswhen compared to the Maxwell equation-perturba-tion theory technique. Very complicated surface ir-regularities can be analyzed relatively simply, andthe cylindrical waveguide (three-dimensional) casecan be treated as easily as its slab waveguide (two-dimensional) analog. The induced-dipole analysis,as presented here, is limited to the case where thesurface deviations are small compared to the radiusand the refractive index change at the core surface is
The author was with Bell Laboratories when this work was done;he is now at Xerox Research Center, Palo Alto, California 94304.
Received 18 April 1974.
small compared to unity. (This limitation is not fun-damental. A more general, and more complicatedinduced-dipole analysis could be made.) This analy-sis is therefore less general than Marcuse's perturba-tion theory.14 For the cases of interest mentionedearlier, however, these small-parameter assumptionsare valid.
In this paper I calculate the induced-dipole scat-tering for several cases. These include (1) a rectan-gular step perturbation of the radius; (2) a sinusoidalperturbation of the radius; (3) a sinusoidal meanderof the core axis; (4) a helical meander of the core axis;and, (5) a deformation in which the core cross sectionalternates periodically along the fiber between an el-ipse extended along the y axis and an elipse extendedalong the z axis, where y and z are orthogonal axes inthe cross-section plane. The extension of these cal-culations to other interface irregularities is straight-forward and will be evident from these examples.
Because the analyses of these cases are very similarto one another and to the analysis of scattering fromaligned dielectric needles,6 the full analysis of therectangular radial deviation is given in the Appendix,together with additional notes related to the analysisof the other cases. Only the final results of this deri-vation are given in the following sections.
11. A Rectangular Radial Deviation
Consider a dielectric fiber core, as shown in Fig. 1.The rectangular deviation can be viewed as a hollowdielectric sleeve about an otherwise ideal (i.e., loss-less) guide, in which the dielectric constant is per-turbed by an amount A = - o. Such a sleeve isshown in Fig. 2(a). The scattering is thus the in-duced-dipole radiation from this sleeve. The totalscattered power per steradian in the direction 0 isgiven by [Eq. (Oa)] ( is the direction inside the core,
Fig. 1. A section of a dielectric fiber waveguide with a rectangularstep perturbation of the radius.
-I3 dx <d /
(a)
that is, before refraction at the core-cladding inter-face):
OT(O) = (k 2a2A/2)(I, + I cos2O)(AzE/E0)2 T(0)J02 (kr sine)
x {[1 - coskL(1 - cosO)]/(l - cose)2}, (la)
where k -27rno/X, no = ,/Eo, X is the incident lightwavelength, a, r, and L are dimensions defined in Fig.1, In and III are the incident powers per unit area as-sociated with each polarization in the vicinity of thecore surface, T(0) 1 for Oc < 0 < -r-0, (0, is the criti-cal angle for total internal reflection), otherwise T(O)- 0, and Jo is the Bessel function of order zero. Inthe limit where Ae << so and a << r, the scattering oc-curs mostly at small angles. In the small-0 approxi-mation, Eq. (la) simplifies to [Eq. (lob)]
where sincx =sin( rx)/rx, and Ii I + III.Figure 3 shows the normalized angular distribution
tT(0)/a 2Ii calculated with Eq. (lb) for L = 10 Am, X =0.633 Aim, no = 1.5, and An = 0.005, with the core ra-dius r as a parameter. The zero near 17° is the firstzero of the sinc term; for r < 0.5 ,im, there are no Bes-sel zeroes in the range 00 < 0 < 17°. As r increases,the total scattered intensity generally increases andthe zeroes associated with the r-dependent Besselfunction dominate the distribution.
Figure 4 shows the normalized scattering for a sim-ilar case, but with the core radius r = 0.5 gim, andwith length L of the scattering region as a parameter.For r = 0.5 ,gm, and first Bessel zero occurs around170; this is the only zero seen in the L = 10 gm curve.As L increases, the total scattering increases, the dis-tribution narrows, and the zeroes associated with theL-dependent sinc term dominate the distortion.Thus, lengthening the scattering region and increas-
x
(b)
Fig. 2. (a) The step perturbation can be viewed as a hollow sleeve
about an otherwise ideal (lossless) waveguide, in which the dielec-
tric constant is perturbed by an amount Ac = El - co. (b) The in-
duced-dipole radiation from each element dV at an angle 0 reachesa point P on a distant reference sphere.
Fig. 3. The normalized angular scattering distribution for a rec-
tangular radial perturbation with the core radius r as a parameter.
Fig. 4. The normalized scatter for a rectangular perturbationwith the perturbation length L as a parameter.
ing the core radius yield qualitatively similar results,although the length parameter is operating throughthe sinc function and the radius parameter is operat-ing through the Bessel function.
It is of interest to know the total power that is scat-tered at all angles by a rectangular radial deviation.We define 4i as the integral over all angles of Eq. (la).Thus,
) a, e )=j I (8)-27T sinOd0, (2)
where 2r sinOdO is a conical element of solid angle.Figure 5 shows 4/a 2Ir evaluated numerically with Eq.(la) for T(O) as a function of L, with the indexchange An as a parameter. It can be seen that (D in-creases linearly with L for a while, and then settlesdown to some constant value. As the index changeAn increases, the length L at the first maximum de-creases, and the limiting value of '1 increases.
Physically, these effects are due to the fact that thescattering is associated with the two radial discon-tinuities at each end of the cylinder, and not to thebody of the cylinder itself. The ringing seen in Fig. 5is due to interference effects between the two ends,and the ringing disappears when the ends are suffi-ciently separated. Note that Eq. (la) has no termproportional to L, and the L2 term in Eq. (lb) is can-celled by that in the denominator of the sinc2 term.
Marcuse3 has calculated and measured the totalscattering loss in a Teflon waveguide that has beenexcited at millimeter wave frequencies and that has arectangular step perturbation. His results are shownin Fig. 6. The two dashed curves are the losses cal-culated by Marcuse by two different methods, and
the two x points represent his measured losses. Hisresults are not directly comparable with the theorypresented here because the radial steps used werelarge (Ar/r = 0.13 and 0.27) and the index change waslarge (the cladding was air; thus An = 0.432). Thesevalues violate our assumptions that r/r << 1 andAn/n << 1. When Ar/r is large, most of the scatteredradiation emerges through the ends of the cylindricalsleeve, and not through the cylindrical surface. Thisend loss is not considered in the present theory. Inorder to make an approximate comparison, the in-duced-dipole theory was modified to include as lossthe fraction of the scattered radiation below the criti-cal angle that strikes the end face either directly orafter one internal reflection. The calculated lossesare shown in Fig. 6. The solid line shows the calcu-lated induced-dipole scattering losses, including endlosses and assuming a length L = 5 cm7 and showsorder-of-magnitude agreement with Marcuse'svalues. The dotted line shows the induced-dipolescattering losses, excluding end losses. It is evident,for this case, that most of the loss is end loss and thatany theory that ignores end loss is inadequate to han-dle large Ar and large An cases. However, as was re-marked earlier, most cases of practical interest in-volve both small Ar and small An.
111. A Sinusoidal Radial DeviationWhen the core radius is sinusoidally perturbed, as
shown in Fig. 7, the perturbation functions as a grat-ing and diffracts energy at or near an angle OR, deter-mined by the sinusoid wavelength Xm. The sharp-ness of the radiation lobe about the angular directionOR is determined by the length L of the perturbed re-
14
0_N
L (m)
Fig. 5. The normalized total scatter at all angles as a function oflength with the core-cladding index change An as a parameter.
Fig. 6. Comparison of induced-dipole scatter loss with the calcu-lations and measurements by Marcuse.
gion, and the intensity is determined by the indexdifference An and the amplitude a.
The angular distribution in the small-O approxima-tion is [Eq. (13)]
9T(0) = {87rarXm(zE/E)T()J(krO)[sin(kLO 2/4)
/(4xo- Xm204 )]}2 X Ir. (3)
Intense radiation only occurs near the resonanceangle OR, given by OR2 = 2XO/Xm, and only if Oc < OR <X- O, where 0c cos 1 (1- AE/2EO) is the criticalangle for total internal reflection. This inequalitycondition can be rewritten as [Eq. (14)]
AnXm < X. (4)
When this inequality is reversed, the resonancepeak is retained by total internal reflection withinthe core and it may serve to couple various guidedmodes. As mentioned in the Introduction, such cou-pling may be of value in the reduction of temporaldispersion in multimode waveguides.
Figure 8 shows the normalized scattering distribu-tion 4T()/a 2Ir, calculated with Eq. (3), with r = 1 ,um,X = 0.633 gim, no = 1.5, Xm = 60 gim, and with L as aparameter. The resonance scattering angle is 6.800.
Fig. 8. The normalized scatter for a sinusoidal radial perturba-tion with the perturbation length L as a parameter.
It can be seen that as L increases, more energy isscattered and the scattering direction becomes moresharply defined. When L is only a few cycles long,the scattered peak is broad and the maximum is dis-placed to smaller angles. Only when L is many cy-cles long does the scattered peak approach OR.
A small value for the core radius (r = 1 im) waschosen in order to keep zeroes from the J0
2 term frominterfering with the resonance peak. For r = 1 gim,the first zero of Jo occurs at 0 - 90 (see Fig. 3). As rincreases, the first zero occurs at smaller angles until,at r 1.36 ,gm, it coincides with the resonance angle,6.80. Thus, for radii r 3 1.36 ,um, the angular distri-bution is complicated and may or may not showstrong resonance scattering at the resonance angle,depending upon the proximity of the resonance angleto a Bessel zero.
IV. Other Periodic Surface Deviations
Sinusoidal core meander (Fig. 9) and a deviation inwhich the core oscillates periodically along the coreaxis between an elipse extended along the y axis andan elipse extended along the z axis (Fig. 11) scatteraccording to equations identical to Eq. (3) exceptthat the Bessel function is of order one for the sinus-
CLADDING' CORE
Fig. 7. A sinusoidal perturbation of the radius, like all periodic
perturbations, functions as a diffraction grating.Fig. 9.
SCHEMATIC POLAR DIAGRAMS OF SCATTERED INTENSITIESY Y
LI LI Yi
OSCILLATINGELLIPSE
(c)
HELICALMEANDER
(d)
Fig. 10. Schematic polar diagrams of the scattered intensity for various core surface perturbations.
oidal meander, and of order two for the oscillating el-ipse. Symmetry properties indicate that there is noscatter in certain planes, as indicated schematicallyin Fig. 10. A helically meandering core scatters inthe same manner as a planar meandering core, exceptthat the radiation is the same in all scattering planes;there are no null planes.
Thus, aside from the presence or absence of nullplanes, the differences between the various periodi-cally deviated surfaces are the differences betweenthe various orders of the Bessel function squared.For example, Fig. 12 is a plot of J 2(kr sinO) vs 0 for i= 0, 1, and 2, assuming that = 0.6328 gim, no = 1.5,and r = 2 im. Note that for a given core radius, theoscillating elipse perturbation (which goes as J 2 2) of-fers the largest angular range free of Bessel zeroes.The presence or absence of Bessel zeroes should beconsidered in the design of diffractive couplers orwhen implementing coupling between guided modesof a multimode waveguide, as discussed earlier. Formode coupling applications the sinusoidal radial per-turbation (i = 0) is the most efficient at small angles.However, the ease or difficulty with which differenttypes of perturbations can be impressed upon thecore may provide additional constraints.
V. Summary
The induced-dipole scattering method has beenapplied to the problem of calculating the angular dis-tribution of various surface perturbations. The
method is relatively simple to apply, but the resultsare less general than the perturbation theory ap-proach. The theory of such scattering has applica-tion to the coupling and tapping into fiber wave-guides and to the reduction of pulse dispersion inmultimode waveguides. Different types of surfaceperturbations have scattering nulls in different direc-tions. Equations are presented that allow the scatternull angles of particular cases to be calculated.
I would like to acknowledge several helpful discus-sions with D. Marcuse and E. A. J. Marcatilli of BellLaboratories.
1.0
X 0.6328jum
-j2 ( ) r 2.0Lm
0~~~~~n 1. 5
.~0.5-
/ " ~ ~~~2 ).''/ \\
/: .
0 ... ...0 5 0
8 (DEG)
Fig. 11. A deviation in which the core oscillates periodically alongthe core axis between an elipse extended along the y axis and an el-
ipse extended along the z axis.
Fig. 12. The dependence of the angular distribution on the coreradius goes as the square of the Bessel function of order i, where i
Consider a dielectric fiber of core radius r, core di-electric constant o, with a rectangular radial devia-tion of height a << r and length L, as shown in Fig. 1.Ell and E 1 are the incident field components in thevicinity of the -core-cladding interface, respectivelyparallel to and perpendicular to the x-y plane. Forsimplicity, circular symmetric guided modes are as-sumed; hence, Ell and E 1 are constant around the in-terface circumference. Given cq, Eo, r, and X (theguided light's vacuum, wavelength), and the totalpower propagating along the waveguide, one can cal-culate9 the electric field amplitudes at the interface,Ell and El; the values of Ell and E 1 are here assumedto be known.
The rectangular deviation of Fig. 1 can be viewedas a hollow cylindrical sleeve about an otherwise ideal(i.e., lossless) guide, in which the dielectric constantis perturbed in an amount Ae = El - Eo. Similarly, arectangular decrease in radius (a < 0) corresponds toa sleeve with a dielectric constant change of -Ae.Thus, the scattering is the induced-dipole radiationfrom the hollow dielectric sleeve shown in Fig. 2 (a).Assuming Ae << co, the differential polarizability a ofa volume element dV is given byl
2 = [( 1 - E0)/E0] (dV/47r), (a > 0), (5a)
and
a = [(Eo - E1)/E1] (dV/4ir), (a < 0), (5b)
where dV = lair dx df: is the volume element at (xf3)and where x and fi are cylindrical coordinates definedin Fig. 2. Replacing El by co in the denominator ofEq. (5b) above is consistent with the assumption ofsmall Ae. Thus, noting that a is an algebraic quanti-ty, we can write
a (ar/4r)(AE/Eo)dfSdx. (5c)
As shown in Fig. 2(b), the induced-dipole radiationfrom each element dV at an angle 6 in the scattering(x-y) plane reaches a point P on a distant referencesphere of radius R >> L; for simplicity, refraction atthe cladding's outer surface is ignored by assumingthat the cladding radius is infinite. Since R >> L, thescattered light peak amplitudes at P from all ele-ments dV are equal and given by the well-known ex-pressions for dipole radiation"
dE 1'(0) = k2aE_/R (6a)
anddE,,'(0) = (k 2 aE,/R) coso, (6b)
where k = 27rno/X is the wave constant in the clad-ding and no = +/EO. We define a polarization-depen-dent function HI such that 1 - 1 for perpendicularpolarized light EI and II -cosO for parallel polar-ized light Ell. Thus, the total peak amplitudes at Pare
where sp(x,fl) is the phase at P of the ray from the vol-ume element at (x,o) relative to some reference rayFigure 2 illustrates the relevant geometry. Formathematical convenience we choose the referenceray to be the ray from the element at x = 0, f = r/2.Thus, (p(x,o) is the phase shift of the point A with re-spect to the point B in Fig. 2(b):
(p (x, j3) = k (x - x cosO - r cos3 sin8). (8)
Equation (8) involves two approximations. First, theguided wave actually propagates with an effectivewave constant k*, which obeys the inequality k < k*< kcore.9 Thus, the first x in Eq. (8) should have acoefficient k*/k, which is close to unity. The effectof this approximation is to change the effectivelength L of the deviated region and to slightly shiftthe angular distribution. Secondly, for volume ele-ments on the bottom side of the cylinder, the scat-tered light passes through the core some distance be-fore emerging into the cladding. Light from such el-ements suffers a small additional phase delay equalto the product of (k - k) and the pathlengththrough the core. The effect of this approximation isto cause small shifts in the angular distribution. Inorder to simplify the analysis, and because we areprimarily interested in the broad features of the an-gular distribution, we ignore these effects and use Eq,(8) for the phase shift. Substituting into Eq. (7),
= 2ar E)IIE, ,exp[ik(1 - cosO)xldx
x f exp(-ikr sinO cos3)d3. (9)
The first integral is of a standard form. The secondintegral is a standard integral form8 of the Besselfunction of order zero and real argument 7rJ0 (-krsinO). Performing both integrations yields (since Jois an even function)
ErO f = ( ) r( )nE,,,Jokr sinG) {sink(l - cosO)L
+ i[ - cosk(l - cos0)L]/(1 - cosO).
Thus, the scattered power per cm2 at P on the refer-ence sphere, for Gaussian units, is12
= (n0c/8T)EL,,,'(0) E±l,,'*(0)
= [(noc/87T)E, 2] H2 (k2 a2r2/2R 2) (AE/EO)2
x J02 (kr sino){[l -cosk(l - cosO)L]/(l - cosO)2}
where c is the vacuum light velocity and the asteriskdenotes the complex conjugate. The quantities I =nocE± 2/8ir and III = nocEI12/87r are the incident pow-ers per cm2 that are associated with each polarizationin the region of the core-cladding interface. We de-fine I'T(O) -I' (0) + I'I(O) as the total scatteredpower per cm2 at P. Also, OT() 3 R 21'T(O) is thetotal power per steradian in the direction 0. Thus,
OT(0) - [k2a2r2 (I + cos2 0)/2](AE/EO)2T(0)J02(kr sin0)
x {[1 - coskL( - cos0)]/(l - cos0)2 . (l0a)
The factor T(0) is introduced at this point in orderto account for total internal reflection (TIR). Thus,T(0) 1 for 0Oc < 0 < (7r - Oc) and is zero otherwise,where Oc- cos' (1 - Ae/2co) is the critical angle forTIR. Fresnel reflection when 0c < 0 < 7r- 0c can beignored, since the reflected fraction emerges from thecore after a few reflections, either at +0 or -0, andthus inevitably appears as radiation.
For the special case when 0 is small, Eq. (a)simplifies to
OT(0) = 2k2a22I1(AE/E0)2T(0)j02(kro)
x {[1 - cos(kL02/2)]/0 4},
where Ir = II + Ill. Using the identity 1 - cos2x = 2sin 2x,
Defining dimensionless length and volume parame-ters for the scattering sleeve I L/(AX/no) and v-27rraL/(X/no)3, and defining Pr3 27rralr as the totalpower incident on the end of the scattering sleeve,one obtains
This expression, except for the presence of the Besselfunction and the TIR term, is identical to Eq. (2c) inRef. 6, which gave the scattering from end-illumi-nated, finite dielectric needles. For dielectric nee-dles of radius r << , TIR trapping of scattered lightdoes not occur, because the needle surface areas aretoo small compared to the wavelength to sharply de-fine the reflected direction. Thus, since J(0) = 1,the dielectric needle case considered in Ref. 6 is seento be a limiting case of Eq. (10c) in which the cylin-der radius r - 0.
B. A Sinusoidal Radial Deviation
Consider a fiber with a sinusoidally perturbed ra-dius, as shown in Fig. 7. In analogy to Eq. (5c), thedifferential polarizability at (x,o is
a ar/47r(AE/ E0)sinKxdBdx,
where K 27r/m is the wave constant of the pertur-bation of wavelength m.
Thus, we obtain an equation identical to Eq. (a),except that the first integral contains the additionalfactor sinKx in the integrand;
E1,,,'(0) = (yR (I) HE,Jo (kr sin0)
x fL exp[ik(l - cos0)x]sinKxdx. (11)
A change of variables (Kx - x') yields a standard in-tegral form, and the integral part of Eq. (11) becomes
(1/K) *{exp[ikL( - cos8)][i(k/K)( - cos0)sinKL
- cosKL] + 1}/[1 - (k2/K2)(l - coso)2].
For simplicity, we restrict ourselves to the case wherethe sinusoidal perturbation is an integral number Nof cycles in length. (One result of the assumption ofintegral N is that the induced-dipole radiation at 0 =0 is always zero.) Thus, KL = 27rN, sinKL = 0, andcosKL =1. By adopting the notation Xo =X/no, theintegral becomes
(1/K) {1 - exp[ikL(l - cose)]}/{1 - [(xm/xo)
x (1 - coso)]2}.
By proceeding as before, this expression is substitut-ed into Eq. (11) and E' 1,11 is multiplied by its com-plex conjugate and by nc/87r in order to obtain thescattered intensities I 11 (0). By defining I'T(O),OT(O), and T(0) as before, we obtain
Thus, tT(O) has a resonance when 0 = R, given bycosOR = 1 - (o/Xm). By assuming again that 0 issmall and by applying the identity 1 - cos2x = 2sin 2x, we obtain
OT(0) {87Tarxm.(/Eo)T(0)J(kr)[sin(kLO2/4)
/(4X02 - m20 4)]} I,, (13)
where I = I + Ill as before, and the resonant scat-tering direction simplifies to OR (2Xo/Xm)/2.
It is evident that significant scattering occurs onlywhen 0 < OR < ( -0,)- These inequalities can berewritten as cosO > cosOR, or 1 - (An/no) > 1 - (o/Xm), or, since X =-noo,
Anx m < X. (14)
Thus, the requirement that the resonance scatteringescape appears as an upper limit to the productAnXm. When AnXm > X, the resonance scatter is re-tained in the core due to TIR and may contribute tomode coupling in a multimode waveguide.
C. Sinusoidal Meander of the CoreWhen the core has a constant diameter but the
center axis meanders sinusoidally with wave constantK and amplitude a in the x-y plane, as shown in Fig.9, the differential polarizability at (x,o) becomes
The first integral is the same as in Eq. (7); the secondis a standard integral representations for J, (-krsin0), the Bessel function of order one. Thus, we ob-tainEn, I' (0) =- (i7rarxm/XOR)(AE/Eo)llE, J(kr sin0)
By proceeding as before, we obtain an equation thatis identical to Eq. (12), except that the Bessel func-tion is now of order one. Thus, with the same changeof order, Eq. (13) also gives the angular distributionin the small-0 approximation.
When the core axis meanders in the x-z plane,however, the scattering in the y-z plane is identicallyzero. This is due to the following symmetry proper-ty. Every point P in the x-z plane is illuminated bythe radiation from a volume element dV1 at (x, + )[see Fig. 2(a)]. There is a second volume elementdV2 at (x, - i) with an equal but opposite surfacedisplacement (An of opposite sign) that therefore ra-diates out of phase with element dV,. Since both el-ements are the same distance from the point P1,these scattered fields cancel. Since all scatteredfields are similarly paired, the total scatter orthogon-al to the meander plane must be zero. This is de-picted schematically in Fig. 10, which shows polarplots of scattered intensity for various core surfacedeviations. In Fig. 10(b), the meander plane is as-sumed to be the x-y plane.
D. An Eliptically Oscillating Cross Section
Figure 11 illustrates a core whose cross section al-ternates periodically along the core axis between anelipse that is extended along the y axis and an elipsethat is extended along the z axis. Thus, the differen-tial polarizability is
a = (ar/47r)(Ac/E0 ) sinKx cos2,3dpdx.
A straightforward repetition of the previous analysisyields scatter equations identical to Eqs. (12) and(13), except that the Bessel function is now of ordertwo.
Fig. 10(c) schematically shows the polar distribu-tion of the total scattered intensity. Symmetryproperties analogous to those discussed in Sec. Cshow scattering maxima in the x-y and x-z planes,and nulls in the scattering planes lying at 450 to thex-y and x-z planes.
E. A Helically Meandering Core
Here the differential polarizability is
a = (ar/47r)(AE/E0) sin(G + Kx)digdx,
and Eq (7) becomes, after substituting for (p(x,o) withEq. (8),
Ej.,1,'() = ( r (AC) HEjw', JfJ exp(ik[x(1 - cose)
- r cos3 sin0])sin(3 + Kx)dfBdx.
Expansion of sin( + Kx) as (cosf sinKx + sinfcosKx) allows straightforward integration, whichconfirms the intuitive conclusion that the helical me-andering core scatters according to the same equa-tions as the planar meandering core; that is, Eqs. (12)and (13) with J, replacing Jo. However, the scatter-ing is now identical in all scattering planes, as indi-cated schematically in Fig. 10(d).
References1. D. Marcuse, Bell Syst. Tech. J. 48(10), 3233 (1969).2. D. Marcuse and R. M. Derosier, Bell Syst. Tech. J. 48(10),
3217 (1969).3. D. Marcuse, Bell Syst. Tech. J. 49(8), 1655 (1970).4. S. D. Personick, Bell Syst. Tech. J. 50(3), 843 (1971).5. D. Marcuse, private communication.6. E. G. Rawson, Digest of Technical Papers, Topical Meeting in
Intergrated Optics (February 1972), Optical Society of Ameri-ca, Washington, D.C.
7. The value L = 5 cm was provided by D. Marcuse (private com-munication) and is approximate.
8. M. Abramowitz and I. Stegun, Handbook of MathematicalFunctions (Nat. Bur. Stds., Washington, 1964), Eq. (9.1.21).
9. D. Gloge, Appl. Opt. 10, 2252 (1971).10. H. C. van de Hulst, Light Scattering by Small Particles
(Wiley, New York, 1957), p. 68.11. For example, Ref. 10, p. 64, gives the scattered electric field
amplitude in terms of the induced-dipole moment p aA andsin7, where y is the angle between p and the scattering direc-tion. When E = Ell, siny = cosO, and when E = E_, siny = 1,giving Eqs. (6a) and (6b) for the peak amplitudes.
12. J. D. Jackson, Classical Electrodynamics (Wiley, New York,1962), p 205.
13. See, for example, R. Bracewell, The Fourier Transform andIts Application (McGraw-Hill, New York, 1965), pp. 62 and63.
14. Marcuse (private communication) observes that the method ofanalysis presented here yields the same result for the total ra-diation loss coefficient as he obtained' 5 using a different meth-od and making the assumption that the radiation modes of thefiber can be approximated by the radiation modes of freespace. He has calculated that at very low scattering angles,such an approximation can result in quantitative error of theorder of 30%,15 although at the same time the angular scatter-ing distribution is substantially unchanged in shape. I con-clude that at the intermediate angles in which we are primarilyinterested (at or slightly beyond the critical angle) errors dueto this approximation are of little importance.
15. D. Marcuse, Theory of Dielectric Optical Waveguides (Aca-demic Press, New York, 1974), pp. 155, 156.
