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Copyright © 1974 American Telephone and Telegraph Company 
The Bell System Technical Journal 
Vol. 53, No. 8, October 1974 
Printed in U.S.A. 



Pulse Spreading in Multimode, Planar, 
Optical Fibers 

By J. A. ARNAUD 

(Manuscript received March 8, 1974) 

A dielectric slab can keep optical beams confined transversely in its 
plane if it is tapered, with the slab thickness having a maximum along some 
straight line. When the square of the local wave number of the slab (k 2 ) is 
a quadratic function of the transverse coordinate (y), the rays in the plane 
of the slab are sinusoids whose optical length is almost independent of the 
amplitude. For thin slabs (2d « X) as well as for thick slabs (2d >>> X), 
pulse spreading is large because the ratio of the local phase to group velocity 
is strongly dependent on the distance (y) from axis. We show that pulse 
spreading is almost negligible, however, if the thickness of the slab is 
properly chosen. For example, if the slab thickness on axis is 2.5 microm- 
eters and the refractive index of the slab is 1 percent higher than that of the 
surrounding medium, pulse spreading is only 0.05 nanosecond per kilome- 
ter at a wavelength of 1 micrometer. Pulses in clad fibers having the same 
width (0.2 millimeter) and carrying the same number of modes (15) 
spread 50 times faster. Splicing and matching to injection lasers may be 
easier with planar fibers than with conventional fibers. Low-dispersion 
planar fibers are therefore attractive when used in conjunction with sources 
that are multimoded in one dimension. Closed-form expressions are given 
for square-law and linear-law profiles. 

I. INTRODUCTION 

This introduction gives first a brief review of the general concepts 
of pulse transmission in multimode waveguides, 12 and subsequently 
considers the case of planar structures that ensure transverse confine- 
ment of the optical beams. 

The most important parameters of optical fibers for communication 
are loss (perhaps a few decibels per kilometer) and pulse spreading 
(perhaps a few tens of nanoseconds per kilometer). Given these two 

1599 



parameters, the maximum repeater spacing and the transmission 
capacity of the fiber are pretty much determined, considering the 
limitations that presently exist in source power (L.E.D. or injection 
lasers) and detector sensitivity (avalanche photodiodes). If the loss 
is the limiting factor, a reduction in bandwidth allows an increase in 
repeater spacing because of the increased receiver sensitivity, but only 
by a modest distance. Inversely, baseband equalization allows the 
transmission capacity to be increased at the expense of optical power, 
but not by a very large factor. In this paper, we consider only the 
problem of pulse spreading. 

Consider first a single-mode waveguide; for instance, a rectan- 
gular waveguide whose width is less than a wavelength. The wave 
number may be a rapidly varying function of <*>, particularly near 
cut-off. The transit time of a pulse of radiation is equal to the ratio 
Ldp/dw of the path length L and the group velocity dco/d/3. Because a 
pulse of small duration has a broad frequency spectrum, some com- 
ponents arrive ahead of the others if dp/ da varies with a> ; that is, if 
d 2 p/dw 2 ?± 0. The pulse duration, r, is of the order of (LcPP/du*)*. If 
the waveguide is filled with a material having dispersion, the phe- 
nomenon remains essentially the same. Single-mode pulse spreading 
is small at optical frequencies when the carrier is almost monochro- 
matic (e.g., injection lasers) because, for a given kind of waveguide, 
single-mode pulse spreading is inversely proportional to the square 
root of the frequency ; that is, it is 100 times smaller at optical fre- 
quencies than at microwave frequencies. This effect can therefore be 
neglected. 

A quite different mechanism for pulse spreading is found in multi- 
mode waveguides (with modes of the order m = 0, 1, 2, • • •) excited 
by multimode sources. In most waveguides, different modes have 
different group velocities. Thus, a pulse decomposes into a train of 
pulses, one for each mode, having times of arrival Ldp m /du, m = 0, 
1, 2, • • •. This effect has similarities with the multipath effects ob- 
served in open space. Multimode pulse spreading is observed even 
when a single mode is excited because, soon after, the power is trans- 
ferred to other modes and back to the first mode, as a result of the 
irregularities of the fiber or of the bends (see Ref. 2 and references 
therein). In this paper, we assume that the fiber is perfectly straight 
and uniform, and investigate ways of minimizing the dependence of 
d& m /do) on m. 

To appreciate the magnitude of the problem, let us consider first a 
nondispersive homogeneous dielectric slab with refractive index n close 

1600 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 



to unity. By comparing the length of rays at the critical angle (0 O ) 
to the length of axial rays, we find that the pulse spreading is AT 
= (L/c)[(cos 9 )~ l — 1] « (L/c)(n — 1). This pulse spreading can be 
written as a function of M, the number of modes that we want to trans- 
mit (a characteristic of the source used) and of the slab width Y: AT 
= 400 M 2 (X/F) 2 ns/km. For example, if we want to transmit 20 modes 
and Y = 70 X, pulse spreading is 33 ns/km, a value that seriously limits 
the transmission capacity for long-distance applications. The guide 
width Y cannot be increased very much because the bending losses 
would rapidly increase and because it is difficult to fabricate clad fibers 
with very small differences in refractive index. 

The difficulty is solved in principle if the permittivity e of the 
medium varies as the square of the transverse coordinate y: t(y) 
= 1 — y 2 . In a square-law medium, the optical length of the rays is 
almost independent of their amplitude. If the permittivity has the 
form e(y) = (cosh y)~ 2 « 1 - y 2 + f y* + ■ • ■, rays have in fact all 
exactly the same optical length. 1 - 3 ~ 7 Because most glasses have negligible 
dispersion, such media exhibit very small pulse spreading. * Multimode 
square-law fibers are certainly attractive. However, it may prove 
difficult to obtain with sufficient accuracy the desired variation of 
permittivity. Furthermore, the losses (impurities and scattering) are 
usually higher for heterogeneous material than for homogeneous mate- 
rial such as fused quartz. It is therefore interesting to investigate 
whether a dimensional change can replace the continuous change in 
the refractive index considered above. 

A proposal to that effect was first made by Kawakami and Nishi- 
zawa. 1 They have shown that optical beams can be confined trans- 
versely in the plane of a slab if the slab thickness has a maximum along 
some straight line (z-axis) . This can be understood from a geometrical 
optics point of view. The slab thickness can be considered a constant 
over a small interval of the transverse coordinate y. Various modes can 
propagate in this uniform slab. Let k denote the wave vector of one of 
them, e.g., the Hi mode. Because of isotropy, the magnitude k of k is 
the same in all directions. Once the local properties of the waveguide 
characterized by the wave number k(y) have been obtained, the prop- 
agation of optical beams can be found, in the semiclassical approxima- 
tion. We need deal only with k(y). For instance, if k 2 {y) is a quadratic 
function of y, e.g., k 2 (y) = k 2 — Q 2 y 2 , the rays are sinusoids and they 
have almost all the same optical length. Diffraction effects in the 



* The properties of graded-index fibers that depart somewhat from a quadratic 
law have also been investigated (Refs. 8 to 10). 

PULSE SPREADING IN OPTICAL FIBERS 1601 



yz plane can be taken into account, to some extent, as the Hamil- 
tonian theory of beam modes shows. 11 For the quadratic variation 
considered above, for example, the modes of propagation are Hermite- 
gauss, 111 regardless of the physical origin of the variation of k with 
y (that is, whether the variation of k with y results from a genuine 
variation in refractive index or from a change in slab thickness). Be- 
cause we are interested in highly multimoded fibers, we consider only 
the geometrical optics field. In that approximation, a mode is repre- 
sented by a manifold of rays y(z + f), < J" < Z, where Z denotes the 
ray period. The main result of this representation is that the axial 
propagation constant (fc z ) of the guide is the value assumed by A; at 
the turning point y = £ of the trajectory. Therefore, we need only 
solve a ray equation. 

The preceding discussion is applicable to the propagation of waves 
at one angular frequency, oj . To obtain information concerning the 
propagation of optical pulses, we need to know, not only k(y), but also 
the variation of the local group velocity u with y. If the ratio (<o /fc) 
• (dk/du) of the local phase velocity (v = w /k) and group velocity 
(u = du/dk) happens to be independent of the y coordinate, the time 
of flight of a pulse along a ray trajectory is proportional to the optical 
length of that ray. In that case (but only in that case) , equal optical 
lengths imply equal times of flight. The above condition (v/u inde- 
pendent of y) is rather well satisfied for most materials with low dis- 
persion, such as fused quartz, whose refractive index is changed slightly 
by such processes as ion implantation. (For normal quartz n = 1.4564, 
dn/d\ = -0.27 X 10 -5 at X = 0.6563 /mi.) In cases where there is a 
physical change in the refractive index, it is sufficient to consider the 
optical lengths of rays with different amplitudes to obtain with good 
approximation the value of the pulse spreading. For a homogeneous 
dielectric slab, however, the ratio of the local phase to group velocities 
is strongly dependent on the slab thickness (2d), and therefore on y, 
when either 2d » X* or when 2d «; X. (The latter approximation is 
made in Ref. 1 ; pulse spreading for tapered slabs is not discussed in 
Ref. 1). We will show that small pulse spreading is obtained only for a 
precise value of the slab thickness on axis. For simplicity, we have con- 
sidered only quadratic and linear dependences of k 2 on y. The optimum 
profile may be different, however. In Section II we give the essential 
formulas for the ray trajectories and times of flight in structures with 



* We are indebted to E. A. J. Marcatili for pointing out that pulse spreading in 
thick, quadratically tapered slabs is almost as large as in clad slabs. This observation, 
at first surprising, stimulated our interest in the problem. 

1602 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 



known local phase and group velocities. In Section III we consider in 
detail the case of tapered slabs and given design values for low pulse 
spreading. General results are given in Appendix A, and analytic 
solutions for square-law and linear-law tapers are given in Appendix B. 

II. GENERAL RESULTS 

The local value k of the wave number of a slab mode is given in 
Section III. In the present section we assume that the local wave num- 
ber k = o) v and the inverse dk/dw of the local group velocity u are 
known functions of y at the operating angular frequency (a> ). We give 
the general form of the ray equations and the time of flight of a pulse 
in a mode m, in the geometrical optics (J.W.K.B.) approximation. 
The derivations are given in Appendix A. 

In a medium that is isotropic, time-invariant, and independent of 
the axial coordinate (z), that is, in a uniform fiber, the ray equations 
y(z) are most convenient in the form 

k\ = *(«, y) - k 2 v , (la) 

dy/dz = -dkjdky = k y /k z , (lb) 

dk v /dz = dk z /dy = %(dk 2 /dy)/k z , (lc) 

dt/dz = dk z /du = \{dk 2 /dw)/k z . (Id) 

Because of the t and z invariance of the medium, « and k z are constant 
along any given ray (constants of motion). The x coordinate is ignored. 
The first equation, (la), says that, because of local isotropy, k\ + kl 
is equal to k 2 . In (lb) to (Id), k z is considered a function of k u , co, and 
y. Equations 1(b) and (lc) are the ray equations. They give the in- 
crements in ray position (dy) and momentum* (dk u ) for an increment 
dz of z. As indicated before, k z characterizes a ray trajectory, that is, 
it is different from one ray to another, but it remains the same along 
any given ray. We can eliminate k v from (lb) and (lc) by differentia- 
tion. We obtain 

ffiy/d* = h(dW/dy)/K (2) 

We first select, as an initial condition, the angle 6 that the ray makes 
with the z axis at the origin of the coordinate system (y = z = 0). We 
then evaluate the constant of motion k z from 

k z = fc(0) cos do. (3) 



* The ray momentum is the transverse component of the wave vector. Ray 
momenta and photon momenta ( Ak) are essentially equivalent concepts. 

PULSE SPREADING IN OPTICAL FIBERS 1603 



The ray trajectory y{z) is obtained step by step from (la) and (lb), 

y i+ i = Vi + Lk 2 (Vi)/kl - l]*Ai, (4) 

Az being the increment in z, and y = 0. Note that, because of sym- 
metry, it is sufficient to evaluate y(z) from y = to the turning point 
y = £, with £ given by fc(£) = k z . 

To any given value of 6 (or fc z ) we can associate a mode number m. 
The mode number is the area enclosed in phase space (k y , y) by a ray 
trajectory, divided by 2x minus \ (see Appendix A). Thus, if the 
integration is stopped at the turning point y = £ (one-fourth of the 
ray trajectory), we have 

m= (2/x) [*k y dy-l (5) 

Jo 

Strictly speaking, only those values d om of O should be considered that 
make m an integer in (5) . However, because we are interested in modes 
of high order, m can be considered a continuous parameter. An approxi- 
mate value for m is 7r0„£/\, where A denotes the wavelength on axis 
[fc(0) = 2tt/A]. 

The time of flight T of a pulse is, for a unit length, the inverse l/v g 
of the axial group velocity. We show in Appendix A that T is obtained 
most easily by integrating along a ray ds/u, where ds = (k/k y )dy 
= (k/k z )dz denotes the elementary ray arc length, and 1/u = dk/dco 
the inverse of the local group velocity. Thus, 



= Z- 1 £ds/u = (2/Z) I (dk 2 /dw)(k 2 - k^-Uy. 



(6) 



Near the turning point (k = fc z ), the integrand in (6) is singular. It is 

therefore preferable from a computational point of view to set ds 

= {k/k z )dz and integrate over z rather than over y. We have [also 

directly from (Id)] 

rzn 
T = (2/Zfc.) / (dk 2 /du)dz. (7) 

Jo 

The purpose of this paper is to find ways to minimize the variation 
A T of T for < m < M, where m is given in (5) and M is the number 
of modes that we want to transmit. It is interesting to compare this 
variation to the variation LT C for a clad fiber having the same width 
Y = 2£ and the same number of modes M. The latter is, as we have 
seen in the introduction, 

AT C = (l/32)M 2 (X/£) 2 c. (8) 

1604 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 



Thus, we want to maximize a quality factor Q defined as 

Q = ATJAT = (1/32) M 2 (\/$ 2 /c&T. (9) 

Note that, since AT and AT C are times of flight for unit lengths, they 
have the dimensions of inverses of velocities. For given k(y) and 
(dk/du)(y), integration of (4), (5), and (7) gives Q(6„) in (9). As O is 
increased, Q increases and reaches a maximum Q ma x, which charac- 
terizes the pulse spreading properties of an optical waveguide for a 
given profile. The best profile is the one that maximizes Q mB . x , provided 
other specifications (number of modes, channel width, ■ • • ) are met. 

III. TAPERED DIELECTRIC SLABS 

Let us now consider the tapered dielectric slab shown in Fig. lb. We 
consider only the Hi mode of the slab. A similar discussion would be 
applicable to the E\ mode (and to higher-order modes if the slab is 
thick enough to support them). Of course, a profile that is optimum for 
the Hi mode need not be optimum for the E\ mode, for example, unless 
e = n 2 is very close to unity. Let us first give expressions applicable to 
slabs with constant thicknesses. We assume that the medium is the 
same on both sides of the slab. (For dissymmetrical media, the formu- 
las in Ref. 13 would be helpful.) 

The dispersion equation k(co) for Hi modes in a slab with relative 
permittivity e and thickness 2d is, as is well known, 

(kd) 2 - (- c d\ = 2 tan 2 0, (10a) 

02 = t ( !? dX - (kd) 2 . (10b) 

From (10) we obtain at u/c = 2ir (that is, X = 1 /im, using the ^m 
as the unit of length), by straightforward substitutions and differ- 
entiations, 

d = (1/2tt)(€ - l)-40/cos0, (Ha) 

k 2 = (2tt) 2 [1 + (e - 1) sin 2 0], (lib) 



n = £ ^i — 27r(e0 tan + e sin 2 + cos 2 0) 
" 2 do) (0 tan 0+1) 



(He) 



Thus, the quantities k 2 and \dk 2 /Bu that enter in our previous expres- 
sions are explicit functions of the parameter 0, related to d by (11a). 
The parameter varies from t/2 f or d — <* to for d = 0. The varia- 



PULSE SPREADING IN OPTICAL FIBERS 1605 





Fig. 1 — Planar fibers, (a) Fiber with constant thickness and variation of the 
permittivity of the form 1/cosh 2 (y). (b) Tapered dielectric slab. The field is shown 
for the Hi slab mode, (c) Coupling between the various slab modes (Hi, Hi, • • •, 
Ei, E 2 , •••) cannot be neglected when the thickness 2d(y) varies abruptly. This 
coupling can eliminate the higher-order modes (Hi, E 2 , • • •) for suitable dimensions 
(see Ref. 12). 



tion of 



e0 tan + e sin 2 <t> + cos 2 <f> 



(12) 



u ~ \ k 2 ) 2 da ~ [1 + (e - 1) sin 2 tf>](tf> tan + 1) 
is plotted in Fig. 2 as a function of for various values of e. For quad- 



1606 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1 974 



ralic fc a (y), the optimum value <f> on axis if close to the maximum of 
the curves, shown by a dotted line, because, near this maximum, times 
of flight are proportional to optical lengths (see Section II). Thus, we 
have for that case a rule for the selection of the slab thickness on axis, 
2d = 2d(0). The optimum value of d may be slightly different, how- 
ever, than the one given by the maximum of the curves in Fig. 2, be- 
cause we want to minimize the variations of T over a finite range of m. 
Instead of specifying the slab profile d(y) or the square of the wave 
number law fc 2 (y), we find it convenient, for the ease of computa- 
tions, to specify <t>(y). If <t> is quadratic in y, both k 2 (y) and d(y) are 
quadratic in y to first order. Thus, we set 



<t> = <Po- Ky\ 



(13) 



where K denotes a constant, in (11), and substitute in the ray equa- 
tions, (lb) and (lc), eq. (5) for m and eq. (7) for T. 

The variation of the time of flight as a function of the angle 6 that 
the ray makes with the z axis at the origin is shown in Fig. 3 for <f> = 1.5 
to 0.2 and n = 1.45, X = 1 /xm. Large pulse spreading is observed 



5 o.oi -i 



0.001 




Fig. 2 — Variation of the ratio of phase to group velocity in a dielectric slab for 
different relative permittivities, as a function of the characteristic angle <f>. The 
optimum points of operation for low pulse spreading in square-law tapered slabs are 
shown by a dashed line (X = 1 /*m). 



PULSE SPREADING IN OPTICAL FIBERS 1607 



0.05 



n=1.45 



T 0.02 



5 001 - 




Fig. 3 — Ratio of vacuum to axial group velocities (c/v a ) as a function of the ray 
angle (0 O ) at the origin, for a tapered dielectric slab with n = 1.45 and a quadratic 
variation of the characteristic angle «:^=*,-4X 10 -6 j/ 2 , for various values of </>„. 
Group delay is related to c/v a by T = (10 4 /3)c/y„ ns/km. The characteristic angle 
on axis <p = 0.65 is seen to give a small variation of c/v„ over a large range of values 
of 6 (X = 1 /mi). 

when the slab is very thick on axis (<f> = 1.5) or very thin (<£„ = 0.2). 
Optimum values are between 0.6 and 0.7. Detailed results will be given 
for the case n = 1.01 (refractive index of the slab is 1 percent higher 
than that of the surrounding medium), which seems of greater practical 
importance. 

For n = 1.01, X = 1 nm, and 4> = <t> — 10" V» we see in Fig. 4 
that the tapered slab can be 50 times superior to the equivalent clad 
fiber (factor Q). The profile of this fiber is shown in Fig. 6 (curve a), 
the thickness on axis being equal to 2.5 pm. The results for the case of 
a linear law <f> = 4>„ — 5 X 10~ 3 1 y \ are shown in Fig. 5 and the cor- 
responding profile in Fig. 6 (curve b). For both quadratic and linear 
laws, we note that a trade-off has to be made between the quality 
factor Q and the mode number M. (Note that the results are meaning- 
ful only when M is large compared with unity.) 

In conclusion, tapered dielectric slabs can exhibit very low pulse 
spreading if properly dimensioned. If the slab material has a refractive 
index 1 percent higher than that of the surrounding medium, the thick- 
ness should be of the order of 2.5 ± 0.2 fim at a wavelength of 1 /xm. 



1608 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 



The waveguide width would be in that case of the order of 0.2 mm. 
Pulse spreading does not exceed 0.05 ns/km for 15 modes. These opti- 
cal waveguides are attractive because they can be stacked for multi- 
channel operation (a possible arrangement is shown in Fig. 7) and 
splicing would perhaps be easier than with conventional fibers (a good 
angular alignment, however, is required for planar fibers). Further 
technological researches are needed to settle this point. 

IV. ACKNOWLEDGMENT 

The author expresses his thanks to E. A. J. Marcatili for stimulating 
discussions. 

APPENDIX A 

Times of Flight in the J.W.K.B. Approximation 

The purpose of this appendix is to derive the ray equations and the 
time-of-flight equations from general principles. We start from the 
Hamilton equations in space-time both for conceptual clarity and to 




25- -10 



0.775 



0.780 0.785 

<fi IN RADIANS 



0.790 



Fig. 4 — Variation with the characteristic angle on axis <f> (or slab thickness on 
axis 2d ) of the quality factor Q ( denned as the ratio of pulse spreading for an equiva- 
lent clad fiber AT C to the actual pulse spreading AT) for c = 1.02 {n = 1.01) and 
<£ = tp — 10~ 6 y 2 . ij denotes the maximum ray excursion, M the total number of 
modes, and AT the pulse spreading. The ray period is 14 mm and 0„ is equal to 2.6° 
for O = 0.785 (X = 1 Mm). 



PULSE SPREADING IN OPTICAL FIBERS 1609 



20- - 20 



10- -10 



AT 
(nsec/km) 




(im 



-100 



0.5- - 50 



0.99 



1.00 



1.01 



<b n IN RADIANS 



Fig. 5 — Variation with the characteristic angle on axis (j> a (or slab thickness on 
axis 2d ) of the quality factor Q for e = 1.02 (n = 1.01) and ;£=</»„- 5 X 1©"*|»|. 
The variations of £, M, and AT are also given. The ray period is 5.6 mm and O is 
equal to 4° for <f> — 1. 

facilitate generalizations to anisotropic or time varying media (which 
are not discussed in detail in the main text, but are of potential 
interest). 

A general medium is described by a function of co, k, t, x 

H(co, k, t, x) = 0. (14a) 

The space- time trajectories (world lines) of particles or wave packets, 
[_t (a), x(<r)] or x(0, are obtained by integrating the Hamilton equations 



(14b) 



dt/dff = -dH/du, 
dx/da = dH/dk, 
da/ da = dH/dt, 
dk/da = -dH/dx, 

where a denotes an arbitrary parameter. * These equations are in a suit- 



* If we define X = {x, id}, K = {k, ioi/c}, the Hamilton equations (14b) are: 
dX/da = dH/dK and dK/da =-dH/dX. The latter follows from the first (see 
Ref. 11) because H = and K = VS. The dynamical significance of the Hamilton 
equations follows from the expression of the canonical stress-energy tensor (Ref. 14) : 



1610 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 




Fig. 6— Slab profiles for n = 1.01. (a) Quadratic case = 0.785 - I0~ b y 2 . (b) 
Linear case * = 1-5X 10~ 3 \y\. 

able form for numerical integration. The initial conditions must, of 
course, be consistent with (14a). Then (14a) remains satisfied at all 
a because, from (14b), dH/do = 0. 
For time-invariant media, the form 



= co(k, x) 



(15) 



is more useful. The motion x(t) of a wave packet is a solution of the 
Hamilton equations 

dx/dt = du/dk, 
dk/dt = -du/dx. 



(16) 



If we are interested only in ray trajectories at some fixed u, we can 
rewrite (15) 

h(k,x) = 0, (17a) 

T = K3£/dK, where £ denotes the averaged Lagrangian density. 3J2/3K is the 
(conserved) wave action, and T is conserved in time-invariant homogeneous media. 
The equality of group and energy velocities readily follows from this expression for 
T. Note that these results are applicable to any linear wave (e.g., matter waves, 
acoustical waves, or optical waves). 



PULSE SPREADING IN OPTICAL FIBERS 1611 




3)im 



n* 1.416 



S^^^^^^^^^M 






& 



iSSSSSSSSSS 



BEAM 



^X\\\\X\^\N\^^V^\^\NN\N\\SXV^ 




0.3mm 

(10 CHANNELS) 



-J 



Fig. 7 — Stacked tapered dielectric slabs. Adjacent slabs are separated by slabs 
with inverted slope and high index to minimize crosstalk caused by scattering. 



and obtain the rays from 



dx/da = dh/dk, 
dk/da = —dh/dx, 



(17b) 



where a is again an arbitrary parameter. Equation (17) is the reduction 
of (14) to three dimensions. Note that the Fermat principle (in three 
dimensions) is applicable to rays x(o-) at a constant frequency co. It is 
unrelated to the time of flight of wave packets, except for nondispersive 
media. It is important for our study that the time of flight of a pulse 
be carefully distinguished from the transit time of the crest of a time- 
harmonic wave (optical length). The latter is the integral of the ray 
index along the ray path, evaluated at a fixed frequency w. 



1612 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 



We need the Hamilton equations in one more form, in which the z 
axis is singled out. For media that are invariant in the z direction, it is 
convenient to solve (14a) for k z . Ignoring the x coordinate, we have 

H = k z - fc,(«, k v , y) = 0, (18a) 

and the ray equations are, from (14b), 

dy/dz = — dk z / dk y , 

dt/dz = dk z /du, (18b) 

dky/dz = dk z /dy, 

where co and k z are constants of motion. If the surface y, z is isotropic, 
k enters only through its magnitude k. Thus, 



and (18) becomes 



k\ = k*(u, y) - kl (19) 

dy/dz = k y /k z , (20a) 

dky/dz = \{dk?/dy)/k z , (20b) 

dt/dz = f (dfc 2 /d6>)/fc,. (20c) 

These are the expressions used in the main text. Equations (20a) and 
(20b) give the rate of change of the ray position and momentum as a 
function of z. Equation (20c) gives the time of flight of a pulse by direct 
integration. We now show that this result can be obtained from the 
J.W.K.B. approximation of the wave optics solution. 

The scalar Helmholtz equation is obtained from the substitution 

ky-^-id/dy (21) 

in (19). We obtain 

Wdy 2 + fc 2 (u, y)> m = k 2 m + m , (22) 

where m = 0, I, 2, • • ■ , for trapped modes. Given fc(w, y), we look for 
solutions of (22) that are square-integrable and obtain the time of 
flight of a pulse in a mode m over a unit length by differentiating k zm 
with respect to o>, 

T = l/v„ = dk zm /du. (23) 

Instead of solving (22) for k z and differentiating with respect to to, 
we may use the Hellmann-Feynman (H.F.) theorem. 15 Let 3C be a 
self-adjoint operator depending on a parameter co, 

3C.(o,)rP m = E m yJ/ m . (24) 

PULSE SPREADING IN OPTICAL FIBERS 1613 



Premultiplying both sides of (24) by yp m we obtain 

E m - <*JC(ct)t.)/<td6.>, (25) 

where 

/+00 
a*(y)b(y)dy. (26) 

-oo 

It is not difficult to show that E m is stationary with respect to a small 
change in \f/ m . Thus, when we differentiate (25) with respect to a> (or 
to 2 ), we can ignore the dependence of \f/ m on co (or a> 2 ). We have for \f/ 
a real 

dE m (t m (dK/da> 2 )* m ) (27) 

do. 2 (+ m + m ) K ' 

In our case, (22), 

JC(w) = d 2 /dy* + fc 2 (co, y). (28) 

Thus, by application of the H.F. theorem we obtain 

(C/V a ) m = (k /k zm )(dkln/dl$) = (k /k zm ) 

X \y (dk*/dk 2 )+ 2 m dy/ J*" + 2 m dy m {k /k tm ) (dk*/dk 2 ) m . (29) 

The J.W.K.B. method shows that, for large m, a mode can be repre- 
sented by a manifold of rays satisfying the Bohr-Sommerfeld condition 



/ 



k v dy = (m + |)2ir, (30) 



where the integral on the lefthand side in (30) is the area enclosed in 
phase space (fc„, y) by a ray trajectory. Equation (30) expresses the 
uniqueness of the phase of the field. At the turning point, k y = 0, 
y = £ m , we have from (19) 

k zm = fc(«, U). (3D 

An alternative way of obtaining the time of flight of a pulse in a 
mode m is to integrate ds/u from z = to 1 along a ray of the mani- 
fold. The arc length is denoted by ds = (k/k t )dz and w _1 = dk/du is 
the inverse of the local group velocity. Thus, 

This expression, (32), in which ( ) denotes an average taken along a 
ray period, is the semiclassical analog of the Hellmann-Feynman theo- 
rem eqs. (27) and (29), and is used in the main text. It can be obtained 

1614 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 



alternatively by noting that the group velocity in a waveguide is the 
ratio of the total energy flow to the energy stored per unit length. 
(This is a special case of the theorem derived in Ref. 16 for periodic 
bi-anisotropic media. To obtain the result applicable to open wave- 
guides, we only have to let the periods go to infinity.) The result, (32), 
follows by integrating the energy density along a ray pencil bounded 
by the rays y(z) and y(z + dz). Let us sketch the proof. If Pdz denotes 
the energy flow in this ray pencil, the total energy flow in the wave- 
guide is PZ. The energy density, on the other hand, is P/u sin 0, 
where is the angle that the ray makes with the z axis. Thus, the energy 
per unit length is obtained by integrating Pds/u along the ray, in 
agreement with (32). 

APPENDIX B 

Square-Law and Linear-Law Media 

In this appendix we work out the case of square-law and linear-law 
media because they lend themselves to exact analytical expressions 
that are useful for comparison with computed solutions. The case in 
which the wave number k varies quadratically with y is also useful to 
obtain first-order solutions. Let us consider this case first. 

fc 2 (a>, y) = *£(») - fl 2 (co)2/ 2 , (33) 

where the functions fc (a>) and ft (to) are arbitrary. The wave equation, 
(22), is 

(d*/dy* + kl - OV) = *& (34) 

where ^ represents, for instance, the y component of the electric field 
for H modes in a dielectric slab. This equation has the well-known 
eigenvalues 

k\ = k\- (2m + 1)12. (35) 

Thus, 

T = 1/ Vg = dk z /do> = [k k - (m + *)n][A£ - (2m + l)ft] _ * 

= h + (n/k )(k /k - ti/n)(m + \) + WHY 

X l(i)k /k - fl/0](m + I) 2 + • • •, (36) 

where upper dots denote differentiation with respect to o>. The condi- 
tion for the removal of the first-order terms in (36), k/k = Q/Q, is the 
same as the condition of stationarity of v/u = a>fc _2 ^(dfc 2 /dw) given in 
the main text. (Note that m is proportional to 6 2 . Thus, first-order 
terms in m correspond to 6 2 terms.) 

PULSE SPREADING IN OPTICAL FIBERS 1615 



Let us now show that this result can be derived from the ray equa- 
tions. Equations (19) and (20) are 

dy/dz = ky/kz, (37a) 

dk y /dz = -Wy/k z , (37b) 

ffly/d* + {Sl/kzYv = 0. (37c) 

The solution of these equations is straightforward. We obtain 

y = (k vo /$) sin [(fl/A; 2 )z], (38a) 

k y = k vo cos [_(Sl/k z )z~\, (38b) 

where 

k vo = tf - kl, (38c) 

if we specify, for simplicity, that 7/(0) = 0, and use (33). The quantum 
condition, (30), is therefore 

kl = (2m + 1)S2. (39) 

Thus, setting fc„ /£2 = £, the axial wave number is given by 

kl = k*(u>, £) = k 2 - fi 2 £ 2 = kl - (2m + 1)12, (40a) 

in exact agreement with (35) (the agreement needs to be exact only 
for square-law media) . 

The ratio of the optical length of a ray period (period Z) to the cor- 
responding length on axis is 



Z kHz 





R = (KZ)- 1 l Z kds = (KKZ)- 1 f 
Jo Jo 

= (k k z Z)~ l f Z [k 2 - fc^sin 2 l(Q./k z )z~\)dz 

Jo 

= (1 - I sin 2 o )/cos d = 1 + dt/8 + • • •, (40b) 

where O denotes the angle between the ray and the z axis at the origin. 
By comparison, we have for a clad slab 

R c = 1/cos 6 « 1 + e 2 /2 + • ■ • . (40c) 

Thus, for small 6 , R — 1 is much smaller than R c — 1, as discussed in 
the introduction. The above results, (40b) and (40c), are significant in 
the problem of pulse spreading in graded-index fibers if the material 
has low dispersion, but they are not relevant to tapered dielectric 
slabs. They are given here only for comparison. 

1616 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974 



Let us now evaluate the group velocity by integrating ds/u along a 
ray of the manifold, following (32). We have 

where Z = 27rfc z /fi denotes the spatial period, and y(z) is given in 
(38a). The integration is straightforward. Using (39), a result identical 
to (36) is obtained. Note that the above results are exact ; the paraxial 
approximation was not made. We have shown in Appendix A that it 
is legitimate to evaluate pulse spreading by integrating the inverse of 
the local group velocity along rays representing the modes of propaga- 
tion, in the limit of large mode numbers. The agreement is now found 
to be exact for square-law media. 
For a linear-law medium with 

Wfay) = /c 2 (co) -2o(«)|y|, (42) 

we shall only give the results. The rays are, from (20), 

y(z) = tan 6 z =F (a/2k 2 cos 2 WJ^ <^ < 7 | (43) 

with a period 

Z = 4k 2 sin O cos d /a. (44) 

The ratio of the optical length of a ray to the length on axis is 



f> 



kds / fk dz = (1 - | sin 2 o )/cos 6 = 1 - 6 2 /G + • • •. (45) 



The situation is opposite to that of a clad fiber: The optical length 
decreases as O increases. Therefore, we may in that case have a small 
increase of v/u when the slab thickness is reduced, that is, work on the 
right side of the dotted line in Fig. 2. This leads to a thicker slab than 
in the case of square-law profiles. These theoretical results are con- 
firmed by the curves in Figs. 5 and 6. We note that the optimum <t> 
is about 1, the maximum of the v/u curve being at only 0.8. The time 
of flight is, using (32), 

T = l/ Vg = (cosd )- l dko/du - (23/6) (k /a) (sin 2 o /cos O ) (da/da). 

Thus, T is independent of O for small O (no terms in 0?) if k (u) and 
a(w) in (42) are related by 

(dk„/du)/k = (23/3) (da/du)/a. (46) 

It can be shown that this condition corresponds to an increase of v/u 
with | y | , in agreement with the previous discussion. 

PULSE SPREADING IN OPTICAL FIBERS 1617 



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1618 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1974