A Heuristic Approach to the Computation of 3B-ray Trajectories in Step

A Heuristic Approach to the Computation of 3B-ray Trajectories in Step Index Optical Fibers Athanasios Nassopoulos and Ron Pieper Electrical and Computer Engineering Naval Postgraduate School Monterey, CA 93943 correspondence between waves and rays has been proposed as a substitute for thle first order WKB analysis in producing the Eikonail-equivalent of the ray propagation equations 161. This correspondence principle has been used t o qualitatively demonstrate that there is a direct association between the exact wave modes and the allowed ray trajectories in an optical fiber.

Abstract 30-ray trajectories in an optical fiber are derived through use of a simple correspondence principle. The ray trajectories are linked to the mode number of the exact waveguide solutions. The analysis and simulations presented will be in terms of dimensionless parameters needed to characterize the optical fiber. Specifically this includes the V-parameter, the core index of refraction, and the numerical aperture. The curves prepared are presented for allowed radial and azimuthal mode numbers and are presented in terms of dimensions which are normalized b y the core radius. Although this approach is extendible to graded index fibers this development willnot be presented in this report.

Background Starting from the Hemholtz equation which is the phasor form of the wave equation in linear, isotropic source-free homogeneous media, w e have

wlhere k(7) is the wave number aind E is the electric field. It is noted that this equation is only aplproximately valid in a graded index fiber. Consistent with the form given in (1) w e assumed that

Introduction Exact waveguide solutions have been derived for a select number of index of refraction profiles. Among these exact solutions, the derivation for the step-index case commonly appears in most introductory and intermediate level optical fiber texts e.g. [I]. Despite the mathematical importance of having an exact solution for the propagation in an optical fiber, the wave solutions can be difficult t o interpret physically. It is not surprising that an alternate method t o treating optical fibers, the approximate Eikonal ray approach [21, which in turn is based on Fermat’s extremum principle, has been repeatedly applied in order t o visualize effects which occur in optical fibers [31. For example, bending losses [41 and dispersion effects I51 in optical fibers have been treated using this approach. However, in the Eikonal approach the derived ray trajectories are not automatically related t o the radial and azimuthal mode numbers. Recently an approach based on a simple

k 2 ( r )= n 2( r )k2=k:+k$+kz, ( 2 ) where k is the vacuum wave number, n(r) is the index of refraction, and r, Qj and z are cylindrical coordinates needed t o describe points in the fiber. Following a standard procedure [I1 in waveguide analysis w e assume that

E,

( 31

where v , the azimuthal mode number, is forced t o be an integer due t o periodic boundary conditions. p is the phase constant of the wave guide. F(r) is the radial solution t o (1 1. In a stlep index fiber the exact waveguide solution for F(r) can be expressed on terms of Bessel Functions [1,31. Using a simple corresponrjence rule, V+j.jk,

179

0094-2898193$03.000 1993 IEEE

( r )= F ( r )ejvae-jPz

k , = k ( r )sine c o s t .

(9)

Letting d l represent the line variation in the ray location defined by * A A A d l = d z e , + d r e,+r dc$ e4

and that the incremental components d@, dr, dz are tangential t o (7),(8) and (9) respectively, it follows

Figure 1: G e o m e t r y o f t h e p r o b l e m

d z = d l cos0

(11)

d r = d l sine cos[

(12)

r d+=dl s i n e sin{.

(13)

it is easily shown [71 that

k4z-y V

(4)

Through combinations of (71413)the initial form of the trajectory equations becomes

A condition of phase synchronization [11 on the radial part of the wave vector results in an approximate integral expansion

I-=-d4 k 4 -

where the radial mode number m takes on positive integer values. Equations (41, ( 5 ) and ( 6 ) become the basis for the three dimensional ray modeling that will be the focus of the discussion that follows. I t is noted that the components (kr, k,, k,) will describe the direction of the ray and that (r, @, z) locate the ray. For purposes of analysis the geometrical relationships are more carefully defined in the next section.

r

=tan[

(15) The evolution equations (14) and (151, that describe the ray trajectories, can be shown t o be equivalent t o the Eikonal solution [71. Because of the 3D emphasis in this report w e have chosen t o concentrate on skew rays ( v f 0 ) instead of the less interesting meridional rays ( v = 0 ) .

Geometry of the problem

Modeling Analysis. Step Index case

With reference t o Fig. 1 the orientation of the wave vector is defined in terms of 0 and f . It follows that

k + = k (sine ~ ) sint

- _V

We follow the usual step index notation n(r)=n, for r i a and n(r)=n, for r>a.On Fig. 2 the sum of the t w o terms in (61, k'n,'-(n/r)', is shown plotted versus r. The condition that the integration (6) must remain positive would indicate that the hatched region in Fig. 2 defines the allowed extremities of the possible ray trajectories. It therefore follows that

(71

180

Specification of a radial number m and the azimuthal mode numbers will dictate a specific phase constant p,, which are ordered E11 by convention

where the upper and lower bounds on p,, are predictable from a combination of wave and ray analysis [ I 1. The turning point equation for rmin is obtained by setting the integrand in (16) to zero

/

“h

a

7

L

Figure 2 : Step index case

its maximum value. As seen from (5) and (8)this occurs for the minimum p. It follolws from (1 7 ) that the maximum value of C is the V parameter. After direct substitution into (221, mma,:(v ) satisfies

The maximum allowed v is determined by setting fl,,=n2k2 and noting from Fig. 2 that rmin+a. It follows that

where for where the quantity in the brackets is the well known V parameter. The lntE 1 operator takes the integer part to produce a valid mode number.The general expression for rmincan be obtained from (18) as

v =O

in agreement with a mode space analysis 181. From the set of equations (20) and (22) it cain easily be shown that

where the second equation follows from geometric considerations. From (5) and (81, we find that (16 ) can be expressed as

-Given that m and v are known, the exact value of rmirlcan be found by numerically solving (25).

Normalized ray trajectories where C =akn,sinO, Integration leads to

-

r = r/a

m ( v ) =1nt[2 (d--vsec-l(:)) x

and

-

rmin = rmin/a.

In this section the normalized trajectories for rays defined by a combination of standard optical parameters (V,NA, n,) and mode indices m and v . Consistent with the purpose OF this report the res,ults for meridional rays ( v = 0) will be presented without significant discussion. Starting from (4) and ( 7 ) it easy to demonstrate that

I (22)

To find m m a x ( vit) is necessary to allow C to take

181

v

d x = 0.2

= I D 0

d y = 0.2

NA = 0.5

“i

=

dz

1.5

=

1

zia

v

=

m

= 10

-t

i

Figure 3 : 3D representation showing 4 turning points v = - r k4=-r k n, sine sine.

and (27) @(TI= O for v =O. z(7) can be found by solving the integral form of(14) which is

(26)

After substitution of (20) into (26) w e have that

sin{=--,r m i n

(27)

I

With a second application of(27) w e find which is needed in the analysis t o follow. The integral form of (15) which is

where z=z/a. The angle 6 can be easily calculated from (20) as

(28) can be solved by substituting (27). Finally w e have

-

4 (3=sec-l(=-- r

)

and where Twincan be obtained from (25). For v = O it can be shown that f(r)=7ctnf3 and sin6 = (mNA)/(Vn,) are applicable t o trajectory calculations of meridional rays. Along with the input parameters V, NA, n,,

(29)

rmin

which is not valid for Y =O. However from (1 5)

182

decrease.

Atbrradl

A2

m

V

0.1 2

17.25 12.57 09.84 12.37 07.38 06.10

10 10

02 10

10

18 20 20 20

0.44 0.64 0.98 0.53 0.44

05 15 20

'Table 1:Representative variation i:n ray parameters with wave mode numbers.

Projection in polar plane

Figure 4 :

Conclusions The main point of this report on optical fiber ray calculations is t o demonstrate the existence of a pedagogically attractive alternative t o more formal methods such as WI