### Q

**Fi.g 2.7 ** Eigenvalues for the first two modes of the doubly-clad step-
profile (W— ) fiber. (a) U vs. V for a fixed Q = 1.5 and
different values of P, and (b) U vs. Q for a fixed P = 0.5 and
different values of V.

4. Polarization Corrections to the Step-Profile Fiber

In section 2-2(b) we saw that by using the "weakly-guiding", or A -*■ 0 , approximation the vector wave equation could be replaced by the scalar wave equation, and the fields can be described by plane waves which are called LP (linearly-polarized) modes.

An (a,m) mode will be designated as LP throughout this work, unless otherwise specified. We stress that LP modes are not the true modes of the fiber, they are only an approximation in which all polarization properties are ignored. However, we will show in later chapters that in fact the LP modes can be an accurate description of mode propagation if the fiber is perturbed in some manner.

In our study of birefringence in Chapter 6 we will have cause to use the vector properties of the modal field, so here we review the scheme in which vector fields are constructed from scalar fields, and consider a specific example. There are four solutions to the circularly-symmetric form of the scalar wave equation (2.19), which all have the same S's :

e = F (R) c o s H x
~xe **l*** e* = F (R) cosiU y
-ye

*~ i = F„(R) sinib x ~xo*

**l***(2.37) i = F .(R) sinJU y -yo*

**l***i*

**l**where F^(R) is the solution to eqn. (2.21).

Using symmetry methods these scalar solutions can be linearly combined to form the true modes [7];

**®1** e ~xe + e ~yo
e 2
**S3** = 9 + e _{~ye}
(

### 2

.### 38

) e - e ~xo -yeMote that for the two fundamental * ( l* = 0) modes the exact: modes

are the same as the scalar modes. The correction to the propagation constant is now given by eqn. (2.14), with eu given by any one of eqn.

**(2.38), **and e, given by either of eqns. **(2.37) **that make **u d** e u .

As an example for determining the proper modes of a fiber, we consider the singly-clad step-profile fiber, whose scalar fields are given by eqn. (2.23). The scalar propagation constants are given by eqn.

(2.18a) :

3 = kn ( 1 - 2 4 — )1/2 (2.39)

**CO ** **V2**

where use has been made of the definition of V, eqn. (2.15), and U is the eigenvalue of the scalar wave eqn. (2.21).

In order to evaluate eqn.(2.14) we use eqn.(2.17) and write Vtin n 2 » -2AVtf. Tor a step-profile f is a step-function at the boundary

R=1, and we can write

-2A V f = -2A 5(R-1) n (2.40)

A

where 5 is the Dirac delta function. The unit outward vector n is on the boundary interface in the waveguide cross-section. Furthermore

32 - 3 2 = 2s(ß - 3) = 2kn **(3** - §) = --- -

C0 (2A) o

where we have used the approximation eqn. (2.12). Eqn. (2.14) then reduces to

56 = **o(2A)**
3/2

2V **\**

### (VSJ **5t’2 ds **

**'****1"**

**a**

**it-Sfc dA**

(2.42)
**K e r e ^ 5 = ** **r^ a n ** **<?f p r t L e/\< jln or> - f t a ** **oour\cU <N j i r \ { £ ^ ^ c c -**

In Table 2.1 we present the corrections to 3 for the four exact modes, where the modes are labelled in the standard EH and HE notation

(see (1] for a discussion). In figure 2.8 we plot the exact eigenvalue, Uov against V for a fixed A = 0.01 . Uov is defined as

U 2 = o 2(k2n 2 - 3 2) = U 2 - **0**2(s2 - S 2) (2.43)

where U is the scalar eigenvalue given by eqn. (2.18a). Figure 2.8 thus represents the true modes propagating in a singly-clad step-index fiber, and should be compared to the approximate LP modes of figure 2.3. We note that the fundamental mode does not have a degeneracy split.

### 0

### 2

### 4

### 6

### 8

### V

Fig. 2.8 The true modes in a singly-clad step-profile fiber. Uex is given by eqn. (2.43), and the modes are labelled by the HE and EH notation (see text).

Po la rization c o rre ct ion s t o t h e step-pr of ile f i b e r

4. Gaussian Approximation to Modal Fields

The fundamental mode field of an optical fiber is approximately Gaussian in shape, and this fact can be exploited by giving very simple forms

for the modal parameters discussed in Section 2-3. The basis of the

approximation is that the fundamental mode of an arbitrary profile circularly symmetric fiber can be approximated by some Gaussian function:

where R 0 = rQ/p is called the spot-size. The problem then is to find an R 0 such that eqn. (2.44) gives a good approximation to the exact field. In this work we will only be concerned with approximations to step-profile fibers.

Marcuse [12] has determined the spot-size of the fundamental mode by minimizing the difference between the exact and Gaussian fields:

- -R2/2R2 F 0(R)

*l^ ) - S*

**0 [ling--- '2 R d R **

**0 [ling--- '2 R d R**

**(2-45)**

where M (R 0) and x 0 are normalization factors, with x 0 given by eqn. (2.25). The spot-size is then determined by

F 0(R) = exp(- R2/2R2) (2.44)

dl

### 0

(2.46)**o**

For the step-profile fiber we have [12]

A second, variational, method for finding R Q is due to Snyder and Sammut [13].

This method has two advantages: first of all it results in a simple

expression for R 0 ; and secondly it can be generalized to describe higher- order modes [14].

The basis for the approximation is that the exact higher order modes of the infinite parabolic profile (i.e. f = R 2 in eqn. (2.21)) are given by

F (R) = ( f - ) 1 ( ^ ) e x p ( - R2/2R2)
**1 0 ** *** ** **R2**
(2.48)
R 2 = 1 / V . r U )
m-1
0
* ( l )* r R 21
m - 1 ^
m - 1
=

*n*

**Z****( - 1 )**

**( m - 1 + O**n! (n+l)! (ra-1-n)!

**( - ) n**(2.49)

It is assumed that the modes of an arbitrary profile fiber are of the same form as eqn. (2.48). The spot-size is then chosen to minimize the variation in 3 when eqn (2.48) is substituted into a stationary expression for 3 . This expression can be obtained by multiplying the scalar wave equation (2.21) by

RF^(R) and integrating over the profile:

co d 2F . dF

### U

2**= J **

{(— + V 2 f)F - ### --

- -### ^ q-^lF2

R dR/J F 2 R dR (2.50)**o ** **R 2 ** * 1* d R 2

**ri**an

**X/**

**o**

*****

where U 2 is related to 3 2 by eqn. (2.18a). profile, the value of R Q is found by solving value of U [13]

Given f(R) for a particular

the equation for the extrem/um

(2.51)

2m + * l* V 2(m - 1)!

[ z+m- 1)! J 0 dR f — F 2 R2 dR * I* (2.52)

Given the solution of this equation for R 0 , the corresponding value of U, and

hence S , is found by substituting the extremum value of R 0 back into

eqn. (2.50).

For the step-profile fiber f(R) is given by eqn. (2.22), and the

derivative of f(R) is the Dirac delta function: df/dR = S(R-1) . For the

fundamental (z = 0, m = 1) mode eqn. (2.52) reduces to [13,14]

R 0 = (l/ZnV2)1/2 (2.53)

In figure 2.9 we compare eqns. (2.47) and (2.53). The two spot-

sizes behave similarly over a large range of V, although eqn. (2.53) will

break down as V + 1 . We discuss this later.

In Table 2.2 we list F (R), R 0 and U for the first two modes of the

step-profile fiber. The relative error of U with its exact value (i.e. the

eigenvalue equation (2.24)) is at most only a few percent [14].

**Table 2.2: Gaussian Approximation for the Step-profile Fiber**

Mode (z,m)
**F * ( R )** **Ro** **U** 2
(0,1) 8 -R2/2R2 u u **= ** e - 1/Ro **(1 + J-)**
- **V**2
(1,1) **—**R e-R2/2R2
2 1 -1/R2
— **= ** — e 0 2 ( — **+ ** 1 *** ** R ? )
**°o** _{V}_{2} _{R 2 }
□

### eqn.(2.53)

### 1

### 2

### 3

### 4

### 5

### 6

### V

Fig. 2.9 Comparison of spot-size definitions for the Gaussian approximation to the fundamental mode.

### (1/R02).exp(-1/R02) - 2/V2

■ 1 *i* T ■ 3 ' ' t ' T r

### 0.0

### 0.5

### 1.0

### 1.5

### 2.0

### R

### 2

### no

Fig. 2.10 The spot-size equation for the LP,, modes vs. . The curve that has no roots is below cut-off.

In order to find R0 for the 2, > 1 modes one must use a zero finding method as outlined in the Appendix. In general the equation for RQ for these modes has more than one zero.

**I**

As an example for determining the spot-size for some mode we
consider the LP, , mode (refer to Table 2.2), and in figure 2.10 we plot the
function
**h ( f ? o >**
1 -1/R2
= — e 0
R 2
**uo**
2_
V 2

against R Q for several values of V. The curves that have no zero's are for V values below cut-off (for the step-profile fiber the L P l1 modes does not propagate for V < 2.405). The other curves may have multiple zero s, and in this case the correct value of R 0 (that value which gives the most accurate value of U) is given by the first zero.

In Table 2.2 R 0 is not the same for different modes, and we may write

**l**** , m**

(V) = (2.54)

In figure. 2.11 we plot the function M against V for the first four modes. As mentioned previously, eqn.(2.53) will break down as V 1 . Henceforth, when using the Gaussian approximation to describe mode properties on single-mode fibers (i.e., those fibers with V < 2.4 ) we will use the spot-size given by eqn.(2.47). However, when we deal with the more general case of few-mode fibers we will use the spotsize defined by eqn.(2.52) for the

U , m ) mode, and represent this spot-size by ^ .

Later we will use the results of this section in an effort to simplify the description of mode propagation on bent fibers (Chapter 3).

### 1.2

**0.8**

### 0.4

**0.0**

Fig. 2.11 Spot-size M for different modes vs. V. The spot-sizes are numerically evaluated from the zero's of eqn. (2.52).

**max**

**m m**

6. Aüpendix

(i) Zero finding routine for functions

To find the zero, x Q , of the function f(x) we write g = f(x) and

choose m i n i m u m and maximum values of x such that

"min < x, Xmax

as shown in fig. 2.12. We then determine x Q by the following routine: (with

reference to fig. 2.12)
ROUTINE ZERO ff, x . , x ; x j
^ ' min' m a x ’ *0J*
start: x = 1/2 (x_._ + x
min ‘max
g = f(x)
is |g| < e , g < 0 , g > 0 ?
if g < 0 then x = x
max
return to start
if g > 0 then x . = x

**° **

**m m**

return to start
if !g| < s then x Q = x
(2A.1)
e << 1 determines the accuracy of x. . x . and x can be found

*J* 0 m m max

automatically, and the routine can be generalized to find any zero of a mulci-

**(ii) ** **Eigenvalues and fields for arbitrary pofiles**
The differential equation for ehe core region is

### y" + tl y

1 - — y +### (u

2 -### v

2 f]y =### o

R R 2

l2

(2A.2)

where f is the profile shape, eqn.(2.17). y must be continuous and have a continuous derivative y' at the fiber boundary R = 1, giving an eigenvalue equation for U:

Z_