1 1

United States Department of Commerce

| I Technology Administration

1 National Institute of Standards and Technology

N/ST/R 3997

TRANSFER FUNCTIONS FOR CHARACTERIZING

MULTIMODE OPTICAL FIBER COMPONENTS

Shao Yang

NISTIR 3997

TRANSFER FUNCTIONS FOR CHARACTERIZING

MULTIMODE OPTICAL FIBER COMPONENTS

Shao Yang*

Electromagnetic Technology Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

* Prepared at the University of Colorado under contract

with the Optical Electronic Metrology Group

January 1 993

U.S. DEPARTMENT OF COMMERCE, Barbara Hackman Franklin, Secretary

TECHNOLOGY ADMINISTRATION, Robert M. White, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director
ABSTRACT

A mode transfer function approach is proposed to characterize optical

fiber devices. The transfer function is used to analyze the accuracy of the

mode transfer matrix, which is currently being used to characterize optical

fiber devices. The analysis shows that the mode transfer matrix depends on

launch condition. Based on the study of the physical process of two basic

mode coupling mechanisms, that is, the scattering coupling and the overlap

coupling, two basic transfer functions are derived. Mode transfer functions for

fibers/cables, connectors/splices, and power splitters are formed using these

two basic transfer functions. Results of a round-robin test and a concatenation

experiment show that the transfer function is better than the transfer matrix

in that it is independent of launch conditions, and thus can improve both the

repeatability of measurements made by different laboratories and the prediction

of concatenated results. The transfer function can also be used to analyze the

structure of a device.

keywords: fiber optics; mode coupling; multimode fibers; optical fiber

devices; transfer function; transfer matrix
CONTENTS

CHAPTER

1 INTRODUCTION 1

2 THEORETICAL BACKGROUND 5

2.1 Introduction 5

2.2 Modes in an Optical Fiber 6

2.3 Mode Parameter 14

2.4 Mode Continuum 17

2.5 Phase Space 22

3 MODE TRANSFER MATRIX (MTM) AND MODE TRANSFER

FUNCTION (MTF) 28

3.1 Introduction 28

3.2 MTM Technique 28

3.3 Definition of MTF 31

3.4 MTM as a Truncation of MTF 32

3.5 Dependence on Launch Conditions of the Measured Matrix . . 35

3.6 Experimental Setup 41

3.7 Round-Robin Test and Experimental Results 51

4 COUPLING MECHANISMS AND TRANSFER FUNCTIONS ... 63

4.1 Introduction 63

4.2 Transfer Function for Scattering Coupling 65

4.3 Transfer Function for Overlap Coupling 70

5 FIBER-OPTIC DEVICES AND THEIR TRANSFER FUNCTIONS 78
IV

5.1 Introduction 78

5.2 Fibers and Cables 80

5.2.1 Transfer function 80

5.2.2 Experimental results 82

5.3 Connectors and Splices 88

5.3.1 Experimental results of connectors 89

5.3.2 Experimental results of splices and their concatenation 96

5.4 Power Splitters 104

5.4.1 A model of 3-port fused biconical taper power splitters 107

5.4.2 Experimental results Ill

5.5 Conclusions 121

BIBLIOGRAPHY 126
CHAPTER 1

INTRODUCTION

The invention of the laser in 1960 created great interest in communication at optical frequencies, and the search for a suitable transmission medium

was promptly launched. Dielectric waveguides including optical fibers were

studied theoretically but were not useful because of their high loss - of the

order of 100 dB/km. In 1966. however, Kao and Hockman [1] pointed out that

the high loss was a result of impurities in the fiber material and that the losses

could be reduced to less than 20 dB/km, a point where optical waveguides

would be a viable transmission medium. They also asserted that it would be

further reduced below several decibels per kilometer if the purity of the material was further improved, making scattering loss the limiting factor. This was

realized in 1970 when Kapron, Keck, and Maurer [2] fabricated a silica fiber

that had a 20 dB/km attenuation. This success triggered explosive research

in the following years toward the development of practical optical fibers. In

1974, the fiber loss was reduced to less than 2 dB/km [3]. Since scattering

loss is lower at higher wavelengths, optical fibers for longer wavelengths were

also developed and the loss has already reached to 0.16 dB/km at a 1550 nm

wavelength, which is close to its theoretical value of 0.14 dB/km. Some of the

initial field trials of optical fiber telephone systems were carried out in 1977 in

Los Angeles [4] and in Chicago [5].

The first generation of optical fiber links used multimode fibers. Due
2

to the elimination of modal dispersion, single-mode fibers have a much larger

bandwidth than multimode fibers, and since 1984, single-mode fibers have

been used exclusively for long-haul telephone trunks. However, multimode

fibers continued to be used in short-distance communications systems such

as the local area networks (LANs) [6] and premise wiring. Applications of

optical fibers in these areas is increasing and the multimode fiber market is

still growing.

Optical fiber characterization and measurement developed along with

the development of optical fibers and their applications. A single loss value was

originally used for characterizing a multimode fiber. Because different modes

have different loss [7, 8], the measured loss for a multimode fiber is not unique.

It depends on how much power is launched into different modes. After propagating a certain distance in the fiber, however, the modal power will reach the

steady-state or equilibrium distribution and the loss per unit length of the fiber

will then remain constant [9]. This steady-state loss represents the loss of a

multimode fiber for long-distance application and was defined as the loss value

for a multimode fiber. Techniques for the steady-state modal power distribution launch were devised in order to measure the steady-state loss [10, 11, 12).

Good results in the prediction of concatenation for long lengths of fiber were

obtained using the steady-state loss [13, 14]. An interlaboratory comparison

of attenuation measurement using two launch techniques, the limited phase

space launch and the mode filter launch [15], were carried out by the National

Bureau of Standards (now the National Institute of Standards and Technology,

NIST) in cooperation with the Electronic Industries Association (EIA) in 1981

[16]-
3

Multimode optical fibers are not used in long-haul communications

systems any more but are being used more and more in short-haul applications

where the modal power distribution will never reach the steady state. In addition. components other than the fiber such as connectors and power splitters

are intensively used and can alter modal power distribution. Consquently, the

steady-state loss for multimode fiber characterization has lost its usefulness. A

new method, the mode transfer matrix (MTM) method [17], has been adopted

by the industries to characterize multimode optical fibers and other fiber optic

devices. Mode coupling has been included in the mode transfer matrix and

good experimental results have been obtained. The Telecommunications Industry Association (TIA) is considering using MTM as the standard method

for the characterization of multimode optical fibers and devices and an interlaboratory comparison round-robin test similar to that for the loss measurement

mentioned above is being conducted by TIA, NIST and the guided-wave optics

group of the University of Colorado.

Although MTM is an improvement over simple loss measurement, it

has long been realized that MTM is still launch dependent [18, 19]. But no

theoretical analysis and experimental tests have been carried out regarding

this problem. One of the purposes of this technical note is to give a theoretical

analysis of the dependence on input modal power distribution of MTM and to

use the round-robin test results to verify the conclusions of this analysis. The

second purpose of this technical note is to introduce the mode transfer function

(MTF) method as an alternative method for the characterization of multimode

optical fibers and devices. The MTF is, in principle, independent of launch

conditions. MTFs for various multimode fiber devices have been derived from
4

the basic mode coupling mechanisms in optical fibers and devices. The roundrobin data were also treated by the MTF method and the results compared

with those by the MTM method. Significant improvements over the MTM

method has been obtained by the MTF method.

The technical note is organized as follows. Chapter 2 describes the

basic theoretical concepts which are used in the rest of the technical note.

Chapter 3 gives a theoretical analysis of the MTM by means of the MTF and

presents round-robin results of the MTM, which verifies the theoretical analysis.

Chapter 4 derives the transfer functions for the two basic coupling mechanisms

in optical fiber devices: the scattering coupling and the overlap coupling. In

Chapter 5, the two basic transfer functions are applied to different devices,

and the round-robin results treated by the MTF are presented and compared

to those in Chapter 3. The result of a concatenation experiment of four splices

is also presented. Conclusions are given in the last section of Chapter 5.
CHAPTER 2

THEORETICAL BACKGROUND

2.1 Introduction

In this chapter, we will discuss the basic theory of wave propagation

in optical fibers. This is the theoretical basis for the approach presented in this

technical note. The theory of wave propagation in an optical fiber has long been

established and detailed discussions can be found in many books [20, 21, 22].

We will present only those results that are relevant to this technical note.

The most important concept in optical fiber theory is the "mode,” and

the major part of this technical note is about power coupling between different

modes. Section 2.2 is about the basic theory of modes. A mode is a wave theory

concept. However, light propagation in a fiber can also be described by ray

theory. Section 2.3 introduces a parameter R which represents different modes

and bridges the wave theory and ray theory. This mode parameter is used

throughout this technical note. Mode is originally a discrete concept. However,

in practical cases, modes in a fiber could form a continuum, which makes

it possible to use a function to describe the power coupling between modes.

Section 2.4 discusses the mode continuum problem. Due to the introduction

of mode parameter R, it is possible to represent modes in terms of rays. A ray

that represents a mode moves along a certain trajectory which can be described

by its radial position and its propagation direction. These two parameters

form the coordinates of the modes in phase space. The phase space concept is
6

introduced in Section 2.5.

2.2 Modes in an Optical Fiber

An optical fiber is a dielectric cylindrical waveguide,and a mode is a

basic concept in the wave theory of optical fibers. The electromagnetic wave

theory always starts from Maxwell's equations:

VxE(r,i) = -«

VxH(p,l) = J(r,() +

«

v* D(r,t) = p{r,t),

V-B = 0.

The optical fiber is a dielectric medium, so

J(r, t) = 0. (2.2)

We can also reasonably assume that

p{r,t) = 0. (2.3)

The constitutive relations for a nonmagnetic material are

eE, (2.4)

MqH. (2.5)

The refractive index is n — Cy/ep0 . The index profile of a fiber takes the form

(2.6)

, 1 - 2Af( L

), r < a,

2 ( \ 2 / \ 2

J ' a "

c e(r)/z0 = n (r) = nx 1 — 2A, r > a,

where f(r/a) is bounded between 0 and 1 with value 1 at the fiber cladding

and 0 at the center of the core, c is the speed of light and A is the maximum

index difference

A = (nl - nl)/2nl ~ (ni - n2 )/ni, (2.7)
7

Figure 2.1: Sketches of the a profile function for several a values.

where iii = n(0) is the refractive index at the center of the core and n2 = n(r >

a) is the refractive index of the cladding. For a practical fiber, A ~ 0.01 —0.02,

which is small. The fiber's numerical aperture NA is determined by the index

difference A:

NA = yjn\ — n\ = riiV2A. (2.8)

Fibers with a small index difference are called weakly guiding fibers

[23]. The weakly guiding condition also dictates that Ve/e is small. Practical

fibers display an a profile, f(^) = (^)

a

. The value of a ranges from 1 to oo.

The parabolic index profile, for which a = 2, is the most commonly used in

telecommunications because it has the least modal dispersion [21, 24]. a = oo

is the step index profile, which is the most easily fabricated.

Since practical multimode fibers are almost always nearly parabolic,

we will in the following always assume that the fiber has parabolic index profile.

The q index profile is sketched in Figure 2.1 for several values of a. A mode is an electromagnetic disturbance which retains a certain

shape when propagated forward along the fiber; that is, the transverse dependence of its field does not change. It is thus convenient to express the electric
8

and magnetic fields in the fiber in terms of their axial and transverse components separately. Since the fiber is a circular waveguide, cylindrical coordinates

(r, #, z) are used:

E(r,i) = [Et (r,0) + ez E2 (r,9)}e^t -<3 *\ (2.9)

H(r,f) = [Ht (r,6) + e ! Hz (r,8)]ej l“ t -l3 *\ (2.10)

where E< and H* are the transverse components in the r, 0 plane, Ez and Hz

are scalar quantities of the axial components, u is the angular frequency of

light, and /3 is the propagation constant in the axial direction. Substituting

Equations (2.6) and (2.7) into Maxwell's equations and using the weakly guiding assumption, we can obtain the following vectorial wave equation for the

transverse components:

V2 E, + [u> 2 e(r )/2 0 - /?

2 ]E( = 0,

V2 H( + tw

2 £(r)/xo - /?2 ]H( = 0.

Because E and H are linked by Maxwell's equations, we will consider only the

electric field in the following. The weakly guiding assumption dictates that the

fiber modes are linearly polarized (LP) [23] and that the transverse field can

be expressed in the form

E*(r, 0) - ax Ex (r , 6)ex + ay Ey (r, 9)e y

. (2.12)

The wave equation for the electric field in Equation (2.11) can thus be separated

into wave equations of the x and y components:

V

2 Ea + [u)2

e(r)/j.o - p

2 ]Ea = 0 (Ea = Ex , Ey ). (2.13)

These equations are called scalar wave equations.
9

To solve the scalar wave equation, we can try separation of variables

Ea = F(r)e-]ue. (2.14)

Inserting this expression into the scalar wave equation (2.13), we obtain for

F(r)

1 d , dF . r o / \ -o z/ 2

, „ - —(r— ) + [uj

2 t(r)no - P2 -]F = 0. r dr dr

With k

2 = u2 tn, Equation (2.15) becomes

d

2 F l dF , o/ \

• a

r TT +

--r + (/: 2 (r)-/?'

drz r dr

is

4 )F = 0,

(2.15)

(2.16)

where k

2 (r) is given, for the parabolic index profile, by

k

2 (r) = k\[1 - 2A(-) 2

] = k

20 n\{ 1 - 2A(-)2

]. (2.17)

a a

Confinement of the light field in the fiber characterizes a propagating

mode. Because of this property, the core-cladding boundary of the fiber has

little effect on many lower-order modes. This makes it possible to use the

infinite parabolic index approximation [22]. A steady and confined transverse

field is the condition for the existence of a mode, which leads to the solution

of Equation (2.16) and the determination of the propagation constant /3 of a

mode. The solution of F(R) for a propagating mode is then

F^r) = N^e-^LlC-F), (2.18)

a

z

and the corresponding propagation constant is

Puis — &i[l — + v + l)]

1 / 2 — ki[l ——(2/z + if + 1)], (2.19)

k\ a kia

where if is an integer called the azimuthal mode number and is defined in

Equation (2.14); // is an integer called the radial mode number and determines
10

the number of peaks of the radial distribution of the field; V is called the

normalized frequency [21] of the fiber and is defined as

V = V2Akia. (2.20)

L

v (x) is the Laguerre polynomial which is given by [25]

K{x)= £(-!)’

(/* + "V-

;X

( 2 . 21 ) m—o (n - m)\(u + m)\m\

Equation (2.18) shows that the lowest-order mode has a Gaussian radial distribution:

Fqo — Nooe 2

2

. ( 2 . 22 )

N^v in Equation (2.18) is a normalization factor. Figure 2.2 shows the shape

of some of the lowest-order fiu modes.

The principal mode number m is defined as

m — 2/i -T v + 1. (2.23)

Since all the fiv modes with the same principal mode number have the same

propagation constant, they are called degenerate modes [26] and form a group

designated by mode number m. Because weakly guiding modes are linearly

polarized, they can have two orthogonal polarization directions; that is, they

can be x-polarized or y-polarized. Also, each mode can have two azimuthal

orientations, except for modes with v — 0, which have no azimuthal variation.

Thus each fiis mode with v ^ 0 is four fold degenerate, and it is two fold degenerate if v = 0. By counting all the possible [iv combinations for a principal

mode number m, we can show that there are 2m degenerate modes in mode

group m. It is more common to ignore the common factor of two and say that

there are m degenerate modes in mode group m.
11

Foo F01

Figure 2.2: Sketches of the radial variations of the first few LP modes.
12

Power coupling among degenerate modes of the same mode group is

usually much stronger than power coupling between different mode groups.

Even if they are not equally excited, the strong coupling will distribute power

evenly among these degenerate modes [27] after they propagate a short distance. The radial distribution of a mode group will then be the superposition

of those of all these modes. Figure 2.3 shows the radial distribution of some of

the mode groups.

The cutoff condition for a propagating mode is

(3 = n2 k0 . (2.24)

Modes whose propagation constants satisfy

k0 ni > > k0 n2 (2.25)

are propagating modes. When (3 < k0 n2: the field is not confined. Analysis by

the WKB method [28] shows that there are two kinds of modes for (3 < kon2:

leaky modes and radiation modes. When k\n\ —

~

21//4 < (3

2 < k^n^ the

electromagnetic field decays for a short radial distance near the core-cladding

boundary and becomes oscillatary again in the cladding region; thus some of

its energy propagates outward. These kinds of modes are called leaky modes.

They behave like very lossy propagation modes. When (3 becomes even smaller

and satisfies (3

2 < k^n\ —

v the modes become radiation modes, which

have oscillatory field distribution in both the core and cladding regions and

whose transmission loss is much higher than the leaky modes.

The condition for propagation modes of Equation (2.25) can also be

expressed in terns of the principal mode number m as

0 < m < —

“ 2

(2.26)
13

m=15 m=20

Figure 2.3: Radial distributions of mode groups m=5, 10, 15, and 20.
14

2.3 Mode Parameter

A very useful parameter representing degenerate modes will be used

frequently throughout this technical note. This parameter is designated as R.

It is defined in terms of both wave theory and ray theory [29]. It is thus a

bridge between the two theories.

Consider the geometrical optics of an optical fiber. The coordinates

r of a ray path represent the evolution in space of the normal to the phase

front of a local plane wave travelling through a medium whose index varies

slowly (compared to a wavelength). Figure 2.4 shows a sketch of a ray path.

The distance measured along the ray path is s and is defined, in cylindrical

coordinates, as

J

ds = J

\J(dr ) 2 -f (rdcf))

2 -f ( dz ) 2

. (2.27)

The unit vector s along the ray path is

dr dr A d<j> A dz A s — ~r — ~r er + r ~r e<p + ~r ez- ds ds ds ds

(2.28)

The ray equation is [30]

d , , .dr, __ j . ^[n(r)-] = Vn(r).

It can be expressed in the cylindrical coordinates as

(2.29)

(2.30)

where the relation that s = dr/ds has been used. Since fibers are weakly

guiding, the modes are plane wave-like. The direction of the ray at any position

in the fiber is the effective normalized k vector of a local plane wave. Thus

k

k

s = (2.31)
15

n(x,y,z) §

Figure 2.4. A sketch of a ray path in a spatially varying medium of index

n(x,y,z), where s is the distance measured along the ray path, 5(3) is the unit

direction of the ray path at s and r(s) is the vector describing the ray path.

Vector k can be represented by its components in the three coordinates (see

Figure 2.5):

k = kr&T + k^ -f kz ez . (2.32)

The z component of the k vector is just /3, the propagation constant. From

Equations (2.28), (2.31) and (2.32), we obtain the following expression for the

z direction

P = n(r)k(yj-. (2.33)

as

The z component of the ray equation in cylindrical coordinates (Equation

Figure 2.5: A sketch which defines the components of the k vector.
16

The 2 component of the ray equation (Equation (2.30)) in cylindrical coordinates is

d .

. ^dz . Ts

{n{r)

Ts ) = °’

which can be rewritten in the form

(2.34)

dz

n{r)ko — = constant = (3.

as

(2.35)

where the last equal sign comes from Equation (2.33). Note that dz / ds is the

cosine of the angle representing the angular direction of a ray with respect to

the 0 axis. Using this fact, together with Equation (2.6) for parabolic index

profile, Equation (2.35) can be expressed in the form

T = [1 - 2A(^)2

][l - sin^(z)]. (2.36)

The weakly guiding assumption implies that A is small and that 0

2 is of the

same order. We can then expand Equation (2.36) to obtain

R

2 =

1 3

2 —[1 - —

2A

1 k

2

r r (z)i2 , Sin2g ( 2 )

1 a1 2A

fKf)i2 , sin2 ^( z )

1 a

1 (NA/rii)2

' (2.37)

where R is the mode parameter which is the bridge between ray and wave

optics. NA/ni is the numerical aperture inside the fiber. Since (3 is a constant

for a mode, R also is a constant for a mode. This also means that the right

side of the last equal sign in Equation (2.37) defines an invariant along a ray

path. As is sketched in Figure 2.6, a constant R parameter defines rays whose

angles must decrease with increasing radius from the axis. This is by definition

a bound mode. Further, the value of R is between 0 (lowest-order mode) and
17

Figure 2.6: Path of a ray with mode parameter R.

1 (cutoff). Since a group of degenerate modes with the same principal mode

number m have the same (3 (Section 2.2), mode parameter R actually represents

a group of degenerate modes. Thus we will often say mode group R as well as mode R. Using Equations (2.19), (2.20) and (2.37), we obtain

R

2 = y (2.38)

2.4 Mode Continuum

As was stated in Section 2.2, the principal mode number m as well as

the radial and azimuthal mode numbers fi and v can only take integral values.

This means that modes are discrete and the propagation constant (3 of modes

in a fiber also forms a discrete set. As a result, R can take on only discrete

values. This discretization of modes can be depicted in the [3 spectrum by

delta spikes at certain values of /?, as is shown in Figure 2.7(a). The spacing

between the values of (3 of two adjacent modes is

A(3 = 0m+ 1 - (3m

riik0 ^ V2A m + 1 nik0a N

Ul °\

\/2A m

riik0 a N

\/2A

a

(2.39)
18

However, the values of /3 are directly affected by the wavelength of the source

through the factor ko. In fact, a source of linewidth 6 A at a central frequency

A would induce a spread of 8(3 in [3 space, the spread is given by

6/3 = /3(A) — /?(A + SX) = n]fc0 -y, (2.40)

where hi = A ^ — ni is the group index. Clearly, if 8(3 >> A/3, then the (3

spectrum will become a continuum [31] between the cutoff points as is depicted

in Figure 2.7(b).

Using Equation (2.39) and (2.40), we obtain the condition for a continuum of (3:

SX V2A 2A 1 (NAY ,, , n

A hik0 a V n\ V

With the typical numbers for a 50 [im diameter, 0.2 Nfiber, of V=35, the

limit on 8\ is —

8X 4 > 5 x 10~ 4 A

(2.42)

Equation (2.42) is satisfied in most practical cases where multimode

sources are used to excite multimode fibers. The typical value of A is 0.01,

of is 1.46, of a is 50 [im. For wavelength A = 1.33 if AA > IQ

-3

,

Equation (2.42) is satisfied. A typical multimode laser diode has a linewidth of

^ > 5x 10 -3

. Single-mode sources are not used for multimode fiber excitation

because of the severe modal noise [32, 33, 34, 35] caused by the interference

between different propagation modes.

However, the continuum of (3 does not necessarily lead to a continuum

of i?, and it is R that dictates whether a continuum of ray paths exist. The

reason is that the variation of A will cause a variation of k as well as of (3. Thus,

due to Equation (2.37), the (3 continuum caused by the source linewidth does
19

(a)

Figure 2.7. A sketch of a ft spectrum for a multimode waveguide excited with

(a) a monochromatic source and (b) a source whose spectral width induces a

spread which greatly exceeds Ap.
20

not mean a corresponding continuum of R. The condition for the continuum

of R must be sought in a different way. We will use the relation R

2 = 2m/V

(Equation (2.38)). Adjacent modes are separated by

(R + AR)2 2 (m -f 1)

V

(2.43)

Equations (2.38) and (2.43) yield the relation

(ARf + 2RAR = y. (2.44)

Solving Equation (2.44), we obtain

AR- -R + \IR2 + y

- R(J1 +

^

- !) (2.45)

d_R

R

Since V = v

/2Akx a = yjn\ — n\~a depends on A, the variation of A will cause

a variation of V, which will in turn result in the variation of R. First we have

the relation

rlR dV

f- (2-46)

From the expression of V we obtain

dV A dn . dA

V n\ + ri2 d\ A

/A

~ 2^

(n - n) T

dX

A

' (2.47)

From Equations (2.46) and (2.47), we obtain

dR = (2.48)

^ A

For a mode continuum, we must have dR > Ai?. The condition for mode

continuum is then

f

(2 - 491
21

This mode continuum condition depends on the principal mode number m and

can be further approximated to —

d\ 1 > . A 2m (2.50)

A much broader spectrum must be used for a continuum of R than for a

continuum of /3. Most practical light sources can still satisfy the condition of

mode continuum for higher-order modes.

We will use very often a concept of mode density when modes constitute a continuum. With discrete modes, the total power in the guided modes

is given by

M

Pt = ^2rnp(m) 1 (2.51)

i where M is the maximum principal mode number. When the modes become

continuous, Equation (2.51) can be replaced by

rM

Pt = I mp(m)dm,

Jo

(2.52)

where m is now continuous. Changing variables from m to R by using Equation

(2.38), we obtain

V

2 r

1 Pt = — f R

3p(R)dR.

A Jo

Equation (2.53) implies that mode density in R space is

(2.53)

m(R) = —i?

3

, (2.54)

Lj

where m(R) designates the mode density of mode group R. The letter m here

has a different meaning than the principal mode number m. In the following,

we shall often omit the constant factor V

2 / 2 and simply say that the mode

density is R

3

. The total power in mode group R is given by

P(R) = R

3p(R). (2.55)
22

We can infer from Equation (2.55) that in most cases lower-order

modes carry much less power than the higher-order modes. Thus the higherorder modes play a more significant role in the power propagation behavior

of the fiber than the lower-order modes. Therefore, we will use the mode

continuum approximation for all the modes even though the condition is not

satisfied for the lower-order modes.

2.5 Phase Space

Phase space is a two-dimensional coordinate system illustrated in Figure 2.8, in which the horizontal axis is the index profile function f(s). where

s = r/a is the normalized radial position of a ray, and the vertical axis is the

square of the normalized numerical aperture t = smO/NA of the ray in the

fiber. For a parabolic index fiber, f(s) = s

2

. From the definition of mode

parameter R of Equation (2.37), a particular mode with the parameter R is

represented in phase space by a straight line which lies at an angle of 45° to

the axes and intersects the axes where s

2 = R2 and t2 — R2

. All the guided

modes lie in the area under the line R

2 = 1. Above this line, are leaky modes

and radiation modes.

For a step index fiber, modes with parameter R are represented by

a line parallel to the horizontal axis and intercepts with the vertical axis at

t

2 = i?2

, as shown in Figure 2.8(b).

The power in all the degenerate modes with the same parameter R is

uniformly distributed along the line representing mode R.

The relation between the nearfield pattern and the modal power distribution can be derived straightforwardly in the phase space representation.

As is illustrated in Figure 2.9(a), the total power at radial position s is the
23

(a) (b)

Figure 2.8. Phase space presentation of modes for (a) parabolic index fiber,

and (b) step index fiber.

integral of all the power in the modes that are present at this position. Only

those modes with a parameter 1 > R > s will be present at 5. Since the mode

spacing in the phase space is R

2

, the total power at 5 in the area element ds2 dt 2

in phase space is

p(s)dA = I\ p{R)ds 2 dt

2

, (2.56)

where K is mode density in the phase space. I\ can be determined in the

following way. Since mode density in R space is V

2 R3

/ 2, we have

v2 V2 —R

3 dR = —

R

2 dR2

. (2.57)

2 4

v '

This equation means that mode density is V

2 R2 / 4 in R2 space. Phase space

represents the R

2 space. The length of the straight line representing mode

group R in phase space is R

2

. Thus the mode density in phase space is V

2 /4,
24

which is constant throughout phase space and is the K of Equation (2.56).

In phase space we also have the relation R

2 = s2 + t 2

. Thus Equation (2.56)

becomes

V

2 p(s)dA = —p(R)ds2 d(R2 — s

2

) V

2 = —p(R)(ds2 dR2 - ds2 ds2

). (2.58)

The total power at position 6 is then

P(s)ds2 = J

l ^-p(R)ds2 dR2

= V

2 sds f

p(R)RdR (2.59)

The term with ds 2 ds 2 in Equation (2.58) has no contribution to this integral

over R

2

. All this power is distributed along the annular element of radius 5

and width ds (Figure 2.9). Thus the nearfield intensity at 6 is

I(s) =

P(s)ds‘

2irsds

V

2z r

1 — I

P(R)RdR. (2.60)

Taking derivatives with respect to 6 on both sides of this equation,

we get

P(R)

2ir 1 dl(s)

\s=R (2.61) V2 df/ds ds

This same expression was obtained earlier from different approaches

[27, 31, 36, 37, 38]. Since the measured nearfield pattern is always in a normalized scale, the constant factors in these two equations are never used in

practice.

Similarly, we can derive the relation between the farfield pattern and

the modal power distribution. In phase space representation, parameters t and
25

Figure 2.9. Depiction of derivation of nearfield power distribution I(s) from

modal power distribution p(R) in phase space.

t and get

and

or in terms of 6

and

can then in Equations (2.60) and (2.61) replace s with

/(*) = 7T f'pimdR Z7r Jt (2.62)

2?r 1 dIW , P( ’ V2 df/dt dt

(2.63)

V

2 f'

1(6) = — / p(R)RdR

2,7T Jrii sin 9/N A

(2.64)

2tt 1 dl(0)

V2 df/d6 d6

• (2.65)

For overfilled launch, p(R) = 1 and Equation (2.64) yields

1 ( 6 )

-- 2tt ' NA2 /nl

sin 2 0 x = P( ° )(1 ~

2n[A ] ' ( 2 . 66 )

which was first derived by Gloge and Marcatili in Reference [24].
26

In phase space, an incident light spot with uniform intensity and a

Lambertian angular distribution will create a uniform modal power distribution. This kind of launch is called an overfilled excitation. The argument is as

follows.

The solid angle element of the incident light at the angle 0 is

= 2x sin OdO. (2.67)

The area element of the annulus at radius r is

dA = 27rrdr. (2.68)

The incident Lambertian light has irradiance

1(0) = 1(0) cos 0. (2.69)

Thus the total power received at r and 0 is

P(r,9) = 47r

2 /(0)rdr cos 0 sin OdO = 7r 2 a

2 (NA)2 l(0)ds 2 dt2

. (2.70)

Since ds 2 dt 2 is the area element in the phase space, Equation (2.20) means that

the incident light distribution is uniform in phase space, or p(R) is constant.

The equations relating the nearfield and farfield patterns and the

modal power distribution (Equations 2.60, 2.61, 2.64, and 2.65) were derived

using the mode continuum assumption. However, when the light source has

a narrower bandwidth, or when the input light is well collimated, as in the

case of plane wave launch (Section 3.4), the light will have relatively higher

degree of temporal and/or spatial coherence and the measured nearfield or

farfield pattern will have ripples due to interference (these ripples are significantly different from noise fluctuations). Using Equation (2.61) or (2.65) to
27

determine modal power distribution will result in negative values of p(R) at

certain values of R. The appearance of these negative values can be treated in

the following way. In the continuum limit, p(R) at a single value of R indeed

carries no power. Only the integral of p(R) in a certain range of R will give

us the power of some discrete mode, which should be positive. In the partially

coherent situation, modes will not be completely continuous; that is, there will

be some spread of R but no overlapping. Thus these negative values are an

indication of a certain degree of coherence. Since the actual nearfield pattern

can be obtained from the modal power distribution with these negative values

by using Equations (2.61) and (2.65), we will call this kind of modal power

distribution a generalized modal power distribution. The generalized modal

power distribution can formally be treated the same as the normal (positive

definite) modal power distribution. Experimental results presented in Chapter 5 show that when the transfer function is applied to the generalized input

modal power distribution, we can get the right output modal power distribution. Various authors have studied the generalized radiance, a quantity which

roughly corresponds to a free space version of the generalized modal power

distribution (Walther [39, 40], Wolf [41, 42]).
CHAPTER 3

MODE TRANSFER MATRIX (MTM) AND MODE TRANSFER

FUNCTION (MTF)

3.1 Introduction

In this chapter, we will discuss the mode transfer matrix method and

the mode transfer function method. We will use the mode transfer function

to analyze the precision of the mode transfer matrix. Expressions have been

derived which show how the transfer matrices can deviate from the ideal matrix due to the finite discretization and nonideal modal power distribution.

Relations between the measured matrix and the matrices corresponding to the

launch conditions involved in the measurement are also derived. With this

analysis the measurement results can be better understood. The experimental

setup and procedures are discussed. Experimental results of transfer matrices,

especially those of a round-robin measurement test conducted in collaboration

with the NIST and TIA are presented. Measured transfer functions will be

shown in Chapter 5, where details of transfer functions for different devices are

discussed.

3.2 MTM Technique

In 1981, G. T. Holmes proposed using a mode transfer matrix to model

modal power redistribution in optical fiber devices [14]. The basic tenet of his

technique is that all the modes in an optical fiber component can be divided

into n mode groups. The power incident on a component is represented by a
29

vector P* and the power exiting the component by a vector P°, both being

n-dimensional. A component can be represented by a n x n matrix T. The

diagonal elements of this matrix represent proportions of power remaining in

each mode group after going through the component, and all the other elements

represent proportions of power transferred from one mode group to another.

For example, element A'j represents the proportion of power in mode group

j transferred to mode group i. The transfer matrix is defined by the matrix

relation

P° - TP\ (3.1)

where

tu tl 2 tin

^21 1 22 tin

t Til t-n.2 tnn

Pol Pi1

o II Pol

,PZ - Pi2

Pan

1 IP3

Using Equation (2.53), we can make the identifications

(3.2)

(3.3)

Pj = It, P°{R)TM(R)dR for 1 < j < n,

P] = It P'(R) m(R)dR for 1 <j<n,

where R0 = 0 and Rn = 1. Matrix T has n

2 elements while P° and P* have

only n elements each. Therefore, although Equation (3.1) defines matrix T,

it is not sufficient to solve for T. We need n

2 equations to determine the n2

(3.4)
30

components of T. This can be done by making n independent excitations.

What we mean by ‘independent excitation’ is that the n input power vectors

we get from these n excitations are linearly independent. We can thus get n

input and output power vectors and form P° and P

l according to the rules

P° = [pj, ps ,

•

, (3.5)

p

1 pi nz T>t r li r2 »

’ ' ‘ rn (3.6)

where P°(PJ) is the output (input) vector for the jth independent excitation.

With these matrices, we can now write the analog of Equation (3.1)

P° - TP*', (3.7)

which can be solved for T to yield

T = P°[P‘]

_1

. (3.8)

Though modal power distribution cannot be measured directly, it can

be determined from the nearfield pattern by Equation (2.61). Thus the experimental determination of the n x n mode transfer matrix of a particular fiberoptic component requires the measurements of n input and output nearfield

patterns of the n independent launches. Input and output power values also

need to be measured in order to determine the relative values of P° and Ph

Since the mode transfer matrix takes power coupling between different

modes into account, it represents a more accurate approach to the characterization of multimode fiber-optic components than a single loss value. Indeed, the

tool has been successfully applied to the prediction of loss in fusion splice [43],

microbends [44], connectors [45], power splitters [46], and fibers [47], as well as
31

to the prediction of system bandwidth [48]. It is also used in the loss prediction of step index LAN link [19]. The MTM method has been adopted by the

industry as a candidate for the standard method of optical fiber components

and systems characterization.

3.3 Definition of MTF

In the mode transfer matrix approach of the last section, the choice of

the discretization of Equation (3.4) is somewhat arbitrary, because there is no

rule on how to discretize the modes. But in fact, if we have a mode continuum,

it is not necessary to discretize the modes. Instead of input and output modal

power vectors, we have input and output modal power distributions in terms

of the mode parameter R. The output distribution can be represented as an

integral operator operating on an incident distribution [49]:

P°(R) = TP\R), (3.9)

where P°(R) and P

l (R) are the total output and input power in mode group

R and are given by

P°{R)=p°(R)m(R),

P'(R) =pi {R)m(R).

The kernel of this operator is the transfer function T(R. R') defined as

(3.10)

p°(R)m(R) = I T(R,R')p, (R')m(R’)dR'.

Jo

(3.11)

This transfer function can therefore be taken as the object which represents

given fiber components, at least in practical systems where the mode continuum

approximation applies.

The main task of the transfer function approach is to find specific
32

forms of the transfer function for different kinds of devices. The transfer function would have some free parameters that need to be determined for individual

components. The form of the transfer function characterizes the kind of device,

while the values of the free parameters characterize the individual components.

It is left for Chapters 4 and 5 to derive the transfer functions for some of the

most commonly used devices in optical fiber communications.

From the definition of the transfer function of Equation (3.11), it

is possible that one single launch measurement could determine the transfer

function of a particular component, which is a great practical advantage over

the transfer matrix approach.

3.4 MTM as a Truncation of MTF

Though the MTM method has achieved some success in characterizing

optical fiber systems, it has long been known that the MTM is dependent on

model power distribution incident on the component [18, 19]. However, detailed

analysis has never been given. In the following, we will show how the measured

matrix is dependent on the launch conditions.

Equation (3.9) defines the effect of a component in a rigorous sense.

To find a matrix corresponding to the operator T of Equation (3.9), we choose

an orthogonal basis of functions d>k(R) and find that [49]

t]k = 4- f

1 dRct>3 (R)T<Pk(R), (3.12)

tk 4o

where

Ek = C dR<j>l(R), (3.13)

Jo

with the power vectors of Equation (3.4) now given by

P° = [

l dRp0

(R)m(R)<t>3 (R),

Jo

(3.14)
33

P\ = f dRpi {R)m(R)(j)j (R). (3.15)

J Jo

Whereas in Equations (3.1 - 3.4), it was assumed that the power

vectors could be chosen of an arbitrary dimension n, here it is evident that

the power vectors are infinite-dimensional because it takes an infinite number

of functions p(R) to span the space 0 < R < 1. In the two mode group case,

however, we could choose

|

a2j+i cos{R -

_ a 2j+i s\a(R —

0 < R < Rc , j = 0,1,2,..., (3.16)

|

b2jCOs{R _l±^

)T^.

|

b2j sin(R- 1±^)Tz+;

0 < R < Rc , j = 0,1,2,..., (3.17)

which is just a double Fourier series, one on [0,i?c ] and one on [i?c , 1]. The

first terms of <j>i(R) and (f>2 (R) are rectangle functions which have the value

1 on the two intervals, respectively. They represent the amount of power in

the two mode groups, incident or exiting. The rest of the terms of these two

vectors describe the shape of modal power distribution in the two mode groups.

The terms t

3 k for j.k = 1,2, purely represent modal power transfer of the two

mode groups regardless of the variation of model power distribution in the two

modes, which is portrayed by all the other terms. Therefore, the values of t

3 k

for j, k = 1,2, and the first two elements of the power vectors are exactly those

which are defined for a 2 x 2 case from Equations (3.1 - 3.4). Unfortunately,

there are still a countable infinity of terms left over. To clarify what these

4>2j —

m(R)

92j+i — m(R)
34

other terms do, we pick a set of n excitations. We can then choose input and

output matrices as in Equation (3.6). These matrices will now have dimensions

of oc x n. Clearly, Equation (3.8) will still hold except that now the transfer

matrix will have dimensions of oc x oo. If we had the correct power matrices,

we could perform the operations of Equation (3.8) and then truncate to n x n to

obtain a transfer matrix we will denote by T

e

, meaning the exact T. However,

in practice we do not determine the exact power matrices but use truncated

n x n versions P

oT and P*T

to determine the measured transfer matrix T^771 ).

This operation is not the same as calculating directly, as can be denoted by

writing

__ poT^piT

^ -1

= T

e + T

+

P’+(P't

) -1

, (3.18)

where T+is the transfer matrix with the first n columns replaced by zero

vectors and P

2+

is the excitation matrix with its first n rows replaced by zeros,

a situation illustrated in Figure (3.1).

A major point of this argument is that the truncation of the power

matrices causes an added error. The only way this error can be made to go

away is if the P

t+

is identically zero. Equations (3.14) and (3.15) show that

this could be the case only if

T dRpl {R)m{R)(j)3 {R) = 0 ,Vj > n, and p

l (R)- (3.19)

Jo

that is, if all the n excitations were orthogonal to all the basis functions. In the

double Fourier series exampleof Equations (3.16) and (3.17) as applied to a 2x2

case, the two excitations would have to be perfectly flat modal distributions,

the first flat between 0 and Rc and the second flat between Rc and 1. This
35

P'

P

oT = T*P iT + T* P

”

T

(ml= P

oT

(P iT)'’

= T

e + T *P'*(P iT

)

Figure 3.1: Truncation of the infinite-dimensional matrices.

is what is meant by orthogonal excitations. They cannot overlap and must

be orthogonal to all higher-order terms. Unfortunately, the diffraction limit

precludes us from actually generating such launches.

3.5 Dependence on Launch Conditions of the Measured Matrix

In the last section, we explained that the measured transfer matrix

depends on the input modal power distribution. Since, as was described in Section 3.2, n independent launches are required for determining an n x n matrix,

the question naturally arises, What matrix is determined by the measurement

of n outputs, given n conditions? For simplicity, we consider only the 2x2

case. However, the conclusion can be generalized to the n x n case. Clearly,
36

the power exiting a component can be expressed in the form

P° Pi + P°2 — tllPl + ^ 12-^2 + 1 21 P\ + ^ 22 -^ 2

- (3.20)

A second expression for the power can be found by writing Equation (3.4) in

the form

P° =

+

+

+

f

Rc

dR [

Rc

dR'T(R.R')m(R')pl (R')

Jo Jo

[

Rc

dR I' dR!T(R,R’)m(R!)p\R')

Jo Jrc C dR [

Rc

dR!T(R,R!)m(R')p\R!)

JRc J0

[

l dR [l dR'T(R,R')m(R')pl (R').

Jr, Jr, (3.21)

Each of the corresponding terms in Equations (3.20) and (3.21) have

the same physical meaning; that is. the first term represents power coupled from

the input lower-order mode group to the output lower-order mode group; the

second term is power coupled from the input higher-order mode group to the

output lower-order mode group; the third and fourth terms are power coupled

to the output higher-order mode group from the input lower and higher mode

groups, respectively. We can thus identify the corresponding terms in these

two equations. Using Equation (3.4) to express the incident power vectors, we

obtain

/0

R‘ dRJ0

Rc dR'T(R, R')m(R')p'(R')

f0

Rc dR'm(R')pi (R')

(3.22)

dRJl d_RT(R

1

Irc dR'rn i R!)pl (RJ)

(3.23)
37

fl dR Jg dKT(R, )m(flV( R)

/0

fie dR'm(R')p'(R')

(3.24)

22

5

lRJR'm(R')p'{R')

Equations (3.22 - 3.25) show that the measured transfer matrix is

simply a weighted average of the transfer function, where the weighting is

given by the incident modal power distribution. In practice, however, several

launches are needed to get one measured matrix. Different launches create

different incident modal power distributions and each has a transfer matrix.

Therefore, it is not clear what information the measured matrix gives us.

In the following, we will derive the relations between the measured

matrix elements and the elements of each matrix corresponding to the different

launch conditions. We will use superscripts (1) and (2) to denote power vectors,

transfer matrices and matrix elements corresponding to the first and second

launch, and superscript (m) for the measured results.

When the two launches are independent, from Equation (3.1) we have

for the two launch conditions the following expressions:

(3.25)

P,

(i)

o 1 p(l)

Ro2

/

^ n(2) ^ rol

p(2 ) V

^

I

t

[l)

L t ll i

( 1

) \

12

t

\

t

t

(1) hi)

21 22 J

^ p(l) ^ ri\

p( 1)

V

^

)

(3.26)

(2) i (2) 11 12

( 2 ) .( 2 ) L <

\

/

/

ri\

( 2 ) \

p( 2 ) \ /

(3.27)

'21 b 22

We can write explicitly the expressions for the output power vector

elements from the above equations:

p(i) r0\ „ j.\(1) d(1) i j.(1) d(1) l ) pv 1 ; , n-1 — -; p>

L 11 ril "T l 12 ri1 2 (3.28)
38

5(1)

o2

h

1

) p(l)

'21 ril P

( 2 ) _

o 1

p(2)

-*02 — t(2) p(2)

'21 rtl

, hl)p(l)

i ‘22 -ri2 i (3.29)

, y(2) p(2)

i ‘12 ri2 5 (3.30)

1 y(2) p(2)

1 ‘22 r%2 ' (3.31)

However, for the measured matrix we should use Equation (3.7) and

write in the present notation the following expression:

\ ( .(ml . ( m ) \ ( () ^( 2.1 ^

(3.32)

( p(!) p( 2 ) rol r'

\

rP

o2

(!) rp

o 1

( 2 ) o 2 /

Am) Am)

‘'ll ‘12

V

Am) Am)

l’ 21 22 7

P\ L

) UK

-in iii

v

(1); p(2) r P i2 i2

Inserting Equations (3.28 - 3.31) into Equation (3.32) and solving for the measured matrix elements, we obtain the following relations:

ii

±m

l \2

j.m

l 21

4\

] kik2 - 4V -f [4V - t^]k2 hk2 — 1

4V M2 ~ 4V + [4V ~ 4\

] }k2 kik2 — 1

4V + [4V - 4V]*2

/(!)l

C 21 M "'2

Ei A:

— 1

(3.33)

(3.34)

(3.35)

t m _ 4VMs

22 —

4V + [4V - 4V]

where Ei and k2 are defined as

k\

k2

kik2 — 1

(i)

rPi l p(i)’ ri2 p( 2)

-ril

( 2 )’ Pi2

(3.36)

(3.37)
39

which are power ratios of the two mode packets for the two launch conditions.

If the launches become orthogonal, that is, if ki and k2 approach oo, then

4-m j ( 1 ) Hi — Hi J

-i-m y. (2

)

l \2 ~ L \2 5

J.7TI .l(l) Hi ~ Hi i

4.m y( 2 ) l 22 ~ l 22 *

The measured matrix thus gives us some meaningful results; that is, it tells

us at least two exact matrix elements for each launch condition. The other

four missing elements are indeed useless because their values would not affect

the power transfer results since P and Pji\ on which they act, are zero.

However, , 4V? an-d t^22 stiff depend on modal power distribution within

the mode packets. When, in addition, the modal power distributions are flat,

the measured matrix becomes the truncated version of the exact matrix, and

the measured matrix represents exactly that for the ideal overfilled launch

where the modal power distribution is all flat, that is, p(R) = 1 . We might

be able to create launch conditions that are almost orthogonal; that is, k\

and k2 are very large in a specially designed measurement system [50, 51].

However, in real application cases, the launch conditions can never be ideal and

controlled. The measured matrix will never be completely accurate. Though

the transfer matrix for any launch condition can never be measured except

for the overfilled excitation, it is possible to create matrices for any input

modal power distribution p

l (R) by using Equations (3.22 - 3.25), if the transfer

function of the component is known. It is also possible to create matrices of

any n x n dimensions in the same way by using equations similar to (3.22 -

3.25).
40

Table 3.1. Measured matrix and matrices corresponding to the two excitations

of one connector.

X^ 1

) t(2)

( 0.8779 0.1982 \ ( 0.7415 0.2274 \ / 0.9508 0.0128 \

\ 0.0976 0.3921 J { 0.2099 0.4014 ) ^ 0.0158 0.6005 )

Table 3.2. Measured matrix and matrices corresponding to the two excitations

of a second connector.

rp(i) rj>(2) 'J'(m)

( 0.9026 0.1206 \ l 0.8783 0.1207 \ f 0.9259 0.0619 \

^ 0.0401 0.6899 ) { 0.0501 0.6910 ) \ 0.0296 0.7164 )

To give some idea of how the measured matrix is different from the

matrices pertinent to the two excitations, the resultant matrices calculated

from the transfer functions of two components are shown in Tables 3.1 and 3.2.

Details about the transfer functions will be shown in Chapter 5, which deals

specifically with transfer functions for different kinds of devices.

It is obvious from Tables 3.1 and 3.2 that the measured matrix is quite

different from either of the matrices corresponding to the two excitations. In

Table 3.1, one can see how the values of each matrix element in is different

from the values of corresponding elements in and T^ 2 h In Table 3.2, though

4V and ty} have almost the same value, t

is still very different from them.

This is because the measured matrix element depends not only on the values

of corresponding elements in and T^2

), but also on other elements, as can

be seen in Equations (3.34 - 3.37). In some cases, we even get negative values

for t jTM) or 47^ and values greater than 1 for t[TM\ The measured matrix can
41

be meaningless.

3.6 Experimental Setup

The experimental apparatus is depicted in Figure 3.2. A 100-W halogen lamp with an ellipsoidal reflector was used as the source. The light source is

focused onto a variable field stop by means of a 5x microscope objective. The

combination of the lens of 100-mm focal length and input microscope objective of 20 x magnification produces a demagnification of 10.96, since the focal

length of the 20 x lens is 9.12 mm. To select the range of the wavelengths, an

infrared bandpass filter with center wavelength of 850 nm and a passband of 50

nm was employed. The variable field stop consists of three patterns. The first

is a small hole with a diameter of 100 //m which is used for central excitation;

the second is used for off-central excitation and is an annulus which has an

outer diameter of ~ 688 fi m and an opaque center circle with a diameter of

525 //m; and the third is a large hole with a diameter of ~ 1.5 mm which is used

for overfilled excitation. The variable aperture stop provides a second control

of launch conditions. The numerical aperture can be changed from 0.06 to 0.4.

In the receiving system, the fiber output is imaged with a 60 x microscope lens onto a TV camera. The beam splitter allows 90% of the output

to be transmitted to the camera channel and 10% to the power meter. This

configuration allows most of the energy to be transmitted to the camera channel, since the power meter channel does not require a large signal with the

lock-in amplifier sensitivity. The camera is a charge-coupled device (CCD)

array camera which produces resolution in the horizontal direction of 280 TV

lines and in the vertical direction of 350 TV lines. The nearfield intensity is

then digitized with the use of a solid state video memory with high speed A-D
42

Lges t iw: Beangptitter T /*n«

Figure 3.2: Schematic diagram of measurement set-up.
43

and D-A converters and capable of digitizing, storing, and displaying a single

frame of video information. This unit uses standard 2:1 interlace and dot interlace to increase horizontal resolution. An I/O module is installed in the video

frame store, which provides interfacing to a personal computer. Power measurements are critical in obtaining the transfer matrices. To obtain accurate

power measurements, the software is written so that 100 power measurements

are averaged. Also, the nearfield intensity measurement requires some caution. Here, the software is written so that the total nearfield intensity can be

averaged for up to 80 measurements. The technique is much superior to the

one used in mechanical scanning setups, since the high accuracy is achieved

by simple and fast averaging. The chopper with the lock-in amplifier will improve the signal-to-noise ratio in power measurement. The speed of the system

minimizes the noise-producing effects of thermal variation.

The variable field stop can provide three kinds of launches: central,

overfilled, and off-central excitations. Central excitation is the kind of launch

where the lower-order modes are excited with more power than higher-order

modes. In the off-central launch, the higher-order modes have more power

at input than the lower-order modes. When all the modes have the same

input power, that is, when the input modal power distribution is uniform, the

excitation is called overfilled. The practical overfilled launch may not be so

ideal that the modal power distribution is completely uniform, but as long as

the modal power distribution shows no significantly biased concentration it can

be considered as an overfilled launch. Any two of these three excitations are

independent launches. Figure 3.3 shows typical modal power distributions of

the three kinds of excitations.
44

p(IU

Figure 3.3. Typical modal power distributions of (a) central excitation, (b)

off-central excitation, and (c) overfilled excitation.
45

Figure 3.4. Schematic diagram of the incident part of plane wave launch setup.

Since the 2x2 matrix is the least measurement-intensive, measurements were made with two different launch conditions for each component. In

most cases, we use a combination of central and overfilled excitations. The

other combination used is the overfilled and off-central excitations.

Another kind of launch, which is called plane wave launch [51], is used

by some laboratories for the measurement of transfer matrix. The incident

setup of the plane wave launch is sketched in Figure 3.4. A source with a

small emitting area is used, which is usually a multimode laser diode or a light

emitting diode (LED). The source is put at the focal point of a lens and the

light beam is thus well collimated after going through the lens. The size of

the beam can be varied by means of a variable aperture stop. The collimated

beam is directed through the center of a rotating stage where the input fiber

of the test component is laid. The end of the input fiber is placed right at

the center of the stage. By rotating the stage and selecting the size of the
46

Figure 3.5: Phase space illustration of plane wave launch.

variable aperture stop, we can select the modes to be launched into the fiber.

For example, when the beam is parallel to the optical axis of the input fiber

and the aperture stop is relatively small, lower-order modes will be launched,

whereas higher-order modes can be launched if the stage is rotated so that the

incident beam is at an angle slightly less than the numerical aperture of the

fiber with the optical axis of the fiber. Figure 3.5 illustrates these selective

launches in phase space.

The plane wave launch has the advantage of selective launch. However, if the size of the aperture stop is not properly selected, high leaky mode

power will be launched into the component. If the input nearfield pattern is

not measured far enough from the input end, the input nearfield and modal

power distribution will be distorted and hence error will be introduced. A

second problem is that since the launch is also highly selective in azimuthal

angle, the degenerate modes with the same mode number R may not attain

equal power distribution at the measurement point. A third problem is related
47

to the second and involes the high spatial coherence due to the collimation.

The nearfield pattern may exihibit fluctuations which will make it difficult to

calculate the modal power distribution and the generalized modal power distribution (Section 2.5) may be necessary to be adopted. Figure 3.6 shows the

measured input nearfield pattern of a low-order mode and a high-order mode

plane wave launch.

There are two ways to calculate modal power vectors P° and P

z from

the measured nearfield patterns. The first is to calculate the elements in the

vectors directly from the nearfield. If the lower- and higher-order mode packets

are divided at mode parameter Pc , element Pi, which is the power in the

lower-order mode packet in either the input or the output power vector, can

be expressed as

Pi = [ p(R)m(R)dR. (3.39)

Jo

Inserting the expressions for p(R) (Equation 2.61) and m(R) (Equation 2.54)

into Equation 3.39 and using integration by parts, we have

7r r Rc 0 Pi = —J R

2 dl

cr Jo

jr Ott pRc = —-R2

C I(RC ) + — / RIdR. (3.40)

cr or Jo

The integration in the second term of Equation (3.40) can be done numerically

from the measured nearfield, and the first term can be immediately obtained

from the nearfield value at Rc

. The total power Pt can be obtained in a similar

manner by changing the upper limit of the integration to 1 and omitting the

first term in Equation (3.40) because 7(1) = 0. The value of P2 , the power in

the higher-order mode packet, can be obtained from

P2 = Pt~ Pi- (3.41)
48

Figure 3.6: Input nearfield patterns of plane wave launch.
49

This way we have avoided calculating the modal power distribution from the

derivatives of the nearfield pattern. The derivatives of the measured nearfield

could show very large random oscillations because of the noise in the measurement. The integration in the last equation averages out the effect of the

noise. However, the value of I(RC ) is still affected by the noise. We can do

curve fitting in the neighboring section of that point and determine the value

of I(RC ) from the fitted curve. Because the input and output nearfield patterns are normalized, they carry no information of the loss. We have to use

the measured loss to modify the values of the elements of one of these vectors

relative to the other.

The second method is to curve fit the measured nearfield. Curve

fitting can be done using Chebyshev polynomials or other empirical forms [43].

The modal power distribution p(R) is then obtained by taking the derivative

of the fitted curve, and the vector power components are then obtained by

Equation (1.20). Empirical curve fitting can obtain a smooth modal power

distribution but lacks generality because the form of the curve may not fit for

all launch conditions. Chebyshev fitting can be used for all launch conditions,

but sometimes unusually sharp changes in the measured nearfield caused by

faults in the measurement or in the fiber structure may affect the fitting result.

For example, we may get negative values in the modal power distribution. Some

modifications might be necessary to improve the fitting accuracy. The modal

power distribution needed for the calculation of transfer function is obtained

in the same way.

Test components are usually prepared in two ways. One kind of component has long pigtail fibers at both ends, one of which is usually longer than
50

the other. These components are used for the cut-back method. Other components are connectorized at both ends and are used for the dummy input

method. The measurement procedures for these two kinds of measurements

are described as follows.

Cut-back method:

The test component is placed in the measurement setup. The longer

fiber end is used as the input end. The input light is focused onto the center

of the input fiber. The output end is focused onto the CCD camera through

the beamsplitter. The nearfield pattern and output power are then measured.

They are the output nearfield and power of the component for the first excitation. Then the input pigtail fiber is cut about one or two meters from the input

end. The test component is taken out and the output end of the remaining

short piece of fiber is focused onto the CCD camera through the beamsplitter.

The nearfield pattern and power output are then measured. They are the input

nearfield and power of the test component for the first excitation. The launch

condition is then changed to the second excitation and the above steps are

repeated to obtain the output and input nearfield patterns and power values

for the second excitation. The length of the short piece of fiber cut from the

component is usually 2m long for overfilled and off-central excitations and lm

long for central excitation. The 2m length is sufficient for the leaky modes

excited in these two launches to dissipate.

Dummy input method:

Two pieces of short fiber about 2m long and connectorized at one

end are used for this method of measurement. They are used as the input

and output connectors. First we connect the input and output connectors.
51

The loose fiber ends are placed onto the input and output stages. The light

is focused onto the input end. and the output end is focused onto the CCD

camera through the beamsplitter. The nearfield pattern and the power output

are measured. These are the input nearfield pattern and power of the first

excitation. The input and output connectors are then disconnected and the

test components inserted. The connector at the input end of the component

is connected to the input connector and the connector at the output end is

connected to the output connector. The nearfield pattern and the power are

measured. These are the output nearfield and power of the first excitation.

The launch condition is then changed to the second excitation. The nearfield

pattern and power, which are the output nearfield pattern and power of the

second excitation are then measured. The test component is then disconnected

from the input and output connectors and the input and output connectors

reconnected. The input nearfield and power of the second excitation are then

measured.

The cut-back method is usually more accurate than the dummy input

method because the dummy input is not the true input. However, the cut-back

method is destructive to the component because the input pigtail fiber will

become shorter and shorter with repeated measurements.

3.7 Round-Robin Test and Experimental Results

As was mentioned in Section 3.2, the TIA Subcommittee on Fiber Optic Measurements has discussed using the mode transfer matrix as a possible

standard method for the characterization of optical fiber devices and systems.

It is the desire of the TIA to test the repeatability of measured matrices obtained by different laboratories. Our purpose is to direct this exercise for the
52

TIA and simultaneously test the premise that the measured matrix is a complex

mixture of the matrices corresponding to the two excitations. So, in collaboration with TIA and the NIST, we conducted a round-robin measurement

test. Four laboratories participated in the round-robin. No mandatory launch

conditions were stipulated: that is, the participants used whatever excitations

they normally use in their laboratories. This provided us with a good variety

of excitations for the measurement of the same component. Three participants

used more or less the same kind of measurement setup described in the first

part of the last section, whereas the fourth participant used the plane wave

launch.

Artifacts for the round-robin test include connectors which have pigtailed fibers at both ends, short pieces of fiber which are about 2m long and

connectorized at both ends, two long pieces of cable about two hundred meters

long, of which one is connectorized at both ends, and two power splitters which

are connectorized at the input and output ports. Not all the participants measured all the test artifacts. Results will be presented here of those artifacts that

were measured by most of the participants; thus a comparison can be made

and meaningful statistical results be calculated. Results presented here are of

these components:

• ccl: a short piece of fiber connectorized at both ends;

• ccml: a connector with pigtailed fibers at both ends;

• cpml: a connector with pigtailed fibers at both ends;

• canc: a piece of cable about 200 m long;

• caec: a piece of cable about 200 m long connectorized at both ends;

• psg: a power splitter connectorized at all the input and output ports.
53

Participants are named as PI, P2, P3, and P4, respectively, in the

presentation of the results. Participant PI made four or five measurements on

most of the components, so it is possible to compare the statistics of measurements made by one participant using the same launch condition and that by

participants using different launch conditions.

The input modal power distributions used by the participants are

shown in Figures 3.7 to 3.10. These modal power distributions show that

we have a good variety of launch conditions. Modal power distributions used

by participant P4 are especially different from those of the other participants

because it used the plane wave launch.

Details of the measured matrices of cable canc, connector ccml and

power splitter psg are listed in Tables 3.3 to 3.8. Matrices of each of these components measured by participant PI are listed in Tables 3.3, 3.5, and 3.7 and

those measured by different participants are listed in Tables 3.4, 3.6, and 3.8.

Average values and standard deviations are calculated and listed at the bottom

of each table. That the standard deviation of matrix elements measured made

by participant PI is significantly lower than that by different participants. This

fact is summarized in Table 3.9, where standard deviations of seven components (the two output ports of a power splitter are listed as two components)

are listed. The large differences between these two standard deviations convincingly shows that the measured transfer matrix is seriously dependent on

modal power distribution. Also, for most of the components listed, the matrices made by P4 are more different from those of the others, a fact which

strengthens the above conclusion because the launch condition used by P4 is

the most different from the others.
54

Figure 3.7: Input modal power distributions used by PI
55

Figure 3.8: Input modal power distributions used by P2
56

Figure 3.9: Input modal power distributions used by P3
57

Figure 3.10. Input modal power distributions used by P4, wbo used plane wave

launch, (a) is low-order mode launch; (b) is high-order mode launch; and (c)

is generalized modal power distribution for high-order mode launch.
58

Besides the launch conditions, causes for the deviation of the measured matrices include the imperfections in the fibers, measurement error, and

error due to data processing. Almost all the participants used Chebyshev fitting to obtain the modal power distribution from the measured nearfield pattern. However, they used different order of Chebyshev polynomial. In order

to show the effect of different data processing, Table 3.10 lists the matrices of

four components calculated from the same raw measured data yet processed

by two participants using three different processing methods. Participant PI

processed the data twice using two different programs, one of which is written

by another participant. Comparing the average standard deviations listed at

the bottom of this table with those listed in Table 3.9, we conclude that the

standard deviations due to different measurement and different data processing

are roughly the same.

Table 3.3. Measured transfer matrices of cable canc. Three measurements were

made by participant PI using the same launch conditions.

in i\2 ^21 1 22

Pl-l 0.8852 0.0200 0.0187 0.7056

Pl-2 0.8510 0.0670 0.0496 0.6703

Pl-3 0.8463 0.0597 0.0543 0.6638

Average 0.8608 0.0489 0.0409 0.6799

Standard Deviation 0.0212 0.0253 0.0193 0.0225
59

Table 3.4. Comparison of measured transfer matrices of cable canc. Measurements were made by four participants using different launch conditions. Data

of PI are the average values in Table 3.3.

in tl2 hi ^22

PI 0.8608 0.0489 0.0409 0.6799

P2 1.1385 -0.2729 -0.0472 0.7037

P3 0.9635 0.0487 0.0702 0.5871

P4 0.6910 0.1214 0.2117 0.3470

Average 0.9134 -0.0135 0.0689 0.5794

Standard Deviation 0.1874 0.1763 0.1075 0.1629

Table 3.5. Measured transfer matrices of connector ccml. Four measurements

were made by participant PI using the same launch condition.

in tl2 hi h2

Pl-1 0.8810 -0.0169 0.1012 0.6336

Pl-2 0.8818 0.0212 0.0913 0.6354

Pl-3 0.8778 0.0439 0.0912 0.6615

Pl-4 0.9184 -0.0188 0.0717 0.6842

Average 0.8897 0.0073 0.0888 0.6537

Standard Deviation 0.0192 0.0305 0.0124 0.0240

Table 3.6. Comparison of measured transfer matrices of connector ccml. Measurements were made by different participants using different launch conditions.

Data for PI are the average values in Table 3.5.

ii2 hi h2

PI 0.8897 0.0073 0.0888 0.6537

P2 0.9380 0.0779 0.1519 0.4414

P3 0.8357 0.1087 0.1426 0.4120

P4 0.5715 0.1951 0.1526 0.3222

Average 0.8087 0.0972 0.1340 0.4573

Standard Deviation 0.1635 0.0778 0.0305 0.1404
60

Table 3.7. Measured transfer matrices of power splitter psg. Five measurements were made by participant PI using the same launch condition.

Port 1 hi tl2 hi h2

pm 0.4022 0.0227 0.0371 0.3060

Pl-2 0.4059 0.0312 0.0371 0.3183

Pl-3 0.4120 0.0284 0.0304 0.3231

Pl-4 0.4300 0.0538 0.0289 0.3636

pl-5 0.4250 0.0145 0.0291 0.3488

Average 0.4150 0.0301 0.0325 0.3320

Standard Deviation 0.0120 0.0147 0.0042 0.0236

Port 2 hi tl2 hi

PM 0.4796 0.0372 0.0363 0.3393

Pl-2 0.4586 0.0548 0.0271 0.3705

Pl-3 0.4491 0.0517 0.0463 0.3197

Pl-4 0.4784 0.0328 0.0308 0.3772

pl-5 0.4765 0.0186 0.0305 0.3499

Average 0.4684 0.0390 0.0342 0.3513

Standard Deviation 0.0138 0.0147 0.0075 0.0234

Table 3.8. Measured transfer matrices of power splitter psg. Measurements

were made by three participants using different launch conditions. Data of PI

are the average values in Table 3.7.

Port 1 hi tl2 hi h2

PI 0.4150 0.0301 0.0325 0.3320

P2 0.5762 -0.0640 0.0033 0.2951

P4 0.3689 0.0883 0.0936 0.2766

Average 0.4534 0.0181 0.0431 0.3012

Standard Deviation 0.1088 0.0768 0.0461 0.0282

Port 2 hi hi 1 22

PI 0.4684 0.0390 0.0342 0.3513

P2 0.5157 0.0030 0.0129 0.3313

P4 0.3675 0.0894 0.0981 0.2788

Average 0.4505 0.0438 0.0484 0.3205

Standard Deviation 0.0757 0.0434 0.0443 0.0374
61

Table 3.9. Comparison of standard deviations of measured matrices made by

participant PI using the same launch condition and by different participants

using different launch conditions.

component tl2 ^21 ^22

ccml SD-P1 0.0192 0.0305 0.0124 0.0240

SD-A11 0.1636 0.0778 0.0305 0.1404

cpml SD-P1 0.0071 0.0409 0.0094 0.0159

SD-A11 0.1082 0.1039 0.0196 0.1337

ccl SD-P1 0.0275 0.0261 0.0132 0.0542

SD-A11 0.0941 0.0492 0.0509 0.0896

canc SD-P1 0.0212 0.0253 0.0193 0.0225

SD-A11 0.1404 0.1857 0.0611 0.0616

caec SD-P1 0.0037 0.0471 0.0141 0.0078

SD-A11 0.1176 0.0665 0.1172 0.1551

PSg

port 1 SD-P1 0.0120 0.0147 0.0042 0.0236

SD-A11 0.1088 0.0768 0.0461 0.0282

PSg

port 2

SD-P1 0.0138 0.0147 0.0075 0.0234

SD-A11 0.0757 0.0434 0.0443 0.0374

Average SD-P1 0.0149 0.0285 0.0114 0.0245

SD-A11 0.1155 0.0861 0.0528 0.0922
62

Table 3.10. Average values of measured transfer matrices of four components.

Averages are taken on four or five measurement results. Results are calculated

by two participants using three different data processing methods. The average

values and standard deviations of the results of the three methods are then

calculated and also listed in this table. Listed at the bottom of this table are

the average values of the standard deviations of the four components. This

average represents the uncertainty due to data processing methods.

ccl hi tl2 hi h2

PM 0.765 0.013 0.076 0.740

Pl-2 0.7939 0.0290 0.0400 0.7138

P2 0.7900 0.0334 0.0352 0.6985

Average 0.7830 0.0251 0.0504 0.7174

Standard Deviation 0.0157 0.0107 0.0233 0.0210

caec hi ti2 hi h2

Pl-1 0.826 0.023 0.058 0.808

Pl-2 0.8801 0.0242 0.0218 0.7708

P2 0.8989 0.0435 0.0109 0.7742

Average 0.8683 0.0302 0.0302 0.7843

Standard Deviation 0.0378 0.0115 0.0246 0.0206

port 1 of psg in tl2 hi h2

Pl-1 0.394 0.017 0.070 0.374

Pl-2 0.4082 0.0324 0.0396 0.3323

P2 0.4150 0.0301 0.0325 0.3320

Average 0.4057 0.0265 0.0474 0.3461

Standard Deviation 0.0107 0.0083 0.0199 0.0242

port 2 of psg hi tl2 hi h2

Pl-1 0.452 0.031 0.077 0.392

Pl-2 0.4645 0.0405 0.0413 0.3436

P2 0.4684 0.0390 0.0342 0.3513

Average 0.4616 0.0368 0.0508 0.3623

Standard Deviation 0.0086 0.0051 0.0229 0.0260

Average of

Standard Deviation 0.0182 0.0083 0.0199 0.0242
CHAPTER 4

COUPLING MECHANISMS AND TRANSFER FUNCTIONS

4.1 Introduction

In this chapter, we will discuss the basic power coupling processes

occurring in fiber-optic devices and derive the mode transfer functions corresponding to these coupling mechanisms. Two coupling mechanisms can cause

modal power transfer in fiber-optic devices: scattering and geometrical transition of a ray.

Scattering can occur when light passes through a medium. Whenever

there are discrete variations in the refractive index due to the presence of

particles or because of small scale density fluctuations, part of the radiation

will be scattered in all directions [52]. Scattering in optical fibers can also be

caused by irregularities of fiber geometry and index inhomogeneities, stress,

and microbending, etc. [53]. Scattering coupling in optical fibers has been

discussed extensively in the literature [20, 21, 54, 55], though it is oftentimes

called coupling. It is the coupling referred to by Marcuse when he discusses

coupled mode equations and coupled power equations [20]. In the coupled

mode equation,

dav _ A r

q — 2J (4.i)

M=1

where a u {v = 1,2,...) and Cv

M

denote the amplitude of the z/th mode and

the conversion coefficient from the ^th to the z/th mode, respectively. The
64

amplitude coupling coefficients are complex-valued and satisfy

Cvv

(4.2)

The solution of the coupled mode equation gives the amplitude and phase of

each mode as a function of position along a fiber section. However, this information is generally unnecessary and, due to the randomness of the fluctuations

that cause the coupling, it is impossible to determine the phase difference between modes along the fiber length. In addition, the use of incoherent sources

for multimode fiber excitation in practical cases leads to excitation of a continuum of modes (Section 2.3), so the phase information is averaged out. Marcuse,

therefore, proceeds to derive the coupled power equations [56], which take the

form

N

dP^/dz = -2 + Y Kn(Pu ~ P?)- (4.3)

j/=

The coupled power equations have the simple meaning that the variation of

power in mode [i equals the total power coupled from other modes into this

mode minus the total power coupled out of this mode to other modes. The

first term on the right is the differential mode loss. Our derivation of the

transfer function for scattering coupling in Section 4.2 is based upon the same

assumption, but in the mode continuum limit and for a case where the coupling

coefficients have a specific expression which is derived from analysis of the

physical processes that cause scattering.

Total power coupling due to the geometrical offset of two fibers was

studied by Gloge [29] and Di Vita et al. [57]. Although in his calculation,

Gloge used for the first time the concept of modal power distribution p{R),

details of modal power conversion due to geometrical offset have never been
65

studied. In devices comprising two pieces of fiber such as splices and connectors,

geometrical offset between the two fibers is unavoidable. When a ray of a

certain mode exits a first fiber and enters a second, it may enter a different ray

path in the second fiber relative to the ray path in the first fiber. We call this

kind of coupling overlap coupling, because it is caused by the overlap of the

two rays in the spatial position at the crossing plane. Evidently, the transfer

function for overlap coupling would be quite different from that for scattering

coupling, and it will be derived in Section 4.3.

4.2 Transfer Function for Scattering Coupling

For discrete modes, it is assumed that mode coupling takes place between adjacent modes [20, 21, 58]. This assumption is based on an experiment

of Gloge [59], who found that the power in a mode moved gradually to neighboring modes along the fiber length. With this assumption, the coupled power

equation can be rewritten as [21]

dPm

7TL ~ — TYlOLrn PTn -)- TTld-jqn ( Pm+\ Pm) T 1 )^m—l{Pm— 1 Pm)' (4.4)

dz

This equation was represented in matrix form in Reference [44], where good

experimental results were obtained with overfilled excitation. In the mode

continuum limit, the assumption of adjacent mode coupling can be generalized

so that the scattering coupling operator is proportional to a Gaussian function

of the mode difference. This assumption is plausible because a great amount of

random factors contribute to the coupling, so the central limit theorem applies.

The scattering coupling function then takes the form

a(R,R') = Q0 e

_|fi2_fl2|/T A'(i?, R

1 ), (4.5)
66

where Qo is a parameter used to determine the amount of scattering and

K(R, R!) is a factor representing mode-dependent effects. The cause of these

effects can be described as follows. Because scattering alters only the ray’s

propagation direction and not its position, this can be represented in phase

space by a vertical line that runs through the point representing the mode at

the place where scattering takes place (Figure 4.1). It can be assumed that

scattering coupling between modes R' and R is proportional to [min(R, i?')]

2

,

that is, proportional to the common area traversed by the two modes. Since

mode R\ from which power is coupled out, traverses an area proportional

to R' , the relative scattering coupling is then [min(i?, R')] / R' (Figure 4.2).

Rayleigh scattering is always present in the medium and the scattered power

is proportional to the density of the medium. Since refractive index, or the

dopant concentration, is higher in the center than at the edge of the fiber,

Rayleigh scattering will be stronger in the center than at the edge [20, 21, 60].

Conversely, stress and deformation caused by external forces will be greater at

the edge than in the center and would cause the extraneously induced scattering to be greater at the edge than in the center.

These last two facts suggest that scattering coupling coefficient is

proportional to (1 — R'

T

-f R

/y

), where x and y determine the shape of the

variation with the mode parameter. Using this assumpion, we write that

I<(R,R') = (l - R’

x + R

,y

). (4.6) R

f

Scattering coupling takes place along the propagation path. The variation of power per unit length in a particular mode R at position z must be

equal to the total power coupled into this mode from all the guided modes

minus the total amount of power coupled out of this mode to all other modes
67

Figure 4.1: Phase space diagram of scattering coupling,

guided or radiated. These effects can be expressed in the form

=

/' a(R,R')P(R',z)dR'

dz Jo -

[°° a(R",R)P{R,z)dR".

Jo

(4.7)

The first term on the right side of Equation (4.7) represents the amount of

power coupled into mode R from all the guided modes, which have mode

parameter in the range from 0 to 1. The second term represents the total

amount of power coupled out of mode R into all the guided or radiated modes,

the latter having mode parameters from 1 to oo. Marcuse's coupled power

equation (Equation 4.3) contains a differential mode attenuation term, which

is not included in Equation (4.7). There are two reasons. First, present high

quality fibers have very low absorption loss, which is the sole major source of

differential mode attenuation. Rayleigh scattering loss is the dominant loss

mechanism and is manifested in the typical spectral loss behavior of a fiber

(Figure 4.3). If a fiber had a high absorption loss, we would observe a significant
68

Figure 4.2. Common area traversed by two mode groups involved in the scattering coupling.

difference between the measured loss and the loss calculated from the transfer

function. Further, for usual devices, absorption loss is negligible because only

very short pieces of fibers are used for pigtailing those devices. In a device,

extrinsic scattering could be stronger than Rayleigh scattering. As a result of

these considerations, in general, absorption loss can be ignored. The first-order

solution to Equation (4.7) is then

J

r oo

1 a{R",R)dR"

o + z

[' a(R.R')P{R';0)dR!. (4.8)

Jo

Rewriting the definition of the transfer function of Equation 3.11, which is z

independent, in a r-dependent form,

P(R; z) = [' P{R';0)T{R,R';z)dR',

Jo

(4.9)
69

Figure 4.3: Loss spectrum of a typical high quality fiber,

we find the following expression for the transfer function for scattering coupling

T(R. R'-.z) = [1 -

[°° za{R",R')dR"]6(R' - R) + za{R,Rt ). (4.10)

Jo

In the first-order solution, parameters Qo and 2 always appear as a

product, so q0 2 can be treated as one parameter. In a piece of fiber or cable,

scattering coupling is the sole coupling mechanism. If the fiber is of high

quality and in an almost stress free state, the scattering per length should be

small. This will be manifested in a relatively small value of Qo2 . Only when

such a fiber is very long does q0c take on a significant value. In devices like

power splitters, where short pieces of fiber are severely twisted and deformed,

scattering will be very strong and a0 z will be large in spite of the small value

of 2 . These effects will be illustrated in experimental results to be presented

in Chapter 5. At this moment, I have no idea how the internal and external
70

physical conditions of the devices will affect the other parameters.

4.3 Transfer Function for Overlap Coupling

Overlap coupling is caused by geometrical offset of the fiber ends

in devices consisting of two pieces of fibers. Connectors, splices, and power

splitters are such devices. Butt coupling is a typical case of overlap coupling.

There are three kinds of geometrical offset: lateral shift, longitudinal shift, and

tilt. From the relation R

2 = s2 + £

2

, we see that R is symmetrical in s and t,

the normalized radial position and the normalized numerical aperture of the

ray. This implies that lateral shift and tilt will have similar effects on modal

power conversion. The effect of longitudinal offset is less than lateral shift and

tilt; this will be shown in the last part of this section. We will, therefore, start

with the discussion of lateral shift.

Figure 4.4 illustrates the change of the radial position of a ray exiting

a fiber at r' and entering a second fiber at r. The coordinate r can be related

to r' by

r

2 = r'

2 + d

2 4- 2r'dcos ?/>, (4.11)

where d is the lateral shift of the two fibers, that is, the distance between

the centers of the two fibers; ip is the angle the radial r' makes with the line

connecting the two centers.

From Equation (4.11), we derive

R

2 = R!

2 + l 2 + 2s'

cos V> (4.12)

for the magnitude of the shift in mode number from the first to the second fiber,

where R' is the mode parameter of the ray in the first fiber and R the mode

parameter it takes when entering the second fiber, s = r/a is the normalized
71

Figure 4.4. Change of radial position of a ray crossing two fibers having lateral

displacement.

position, and l = d/

a

the normalized shift. A ray exiting the first fiber and

entering the second will experience a slight change of refractive index due to

the lateral shift of the fiber ends. Typical index difference between the core

and the cladding of the fiber is 1 to 2%. Thus a 100% relative shift, that is,

/ = 1, will cause 1 to 2% change of index profile, which will in turn result

in a similar variation of the ray's propagation angle and the value of t. This

variation of t can be ignored compared to the 100% variation of 6. When the

relative shift is less than 100%, the change of the propagation angle and t will

correspondingly become less and still can be ignored.

This mode transition relation is illustrated in the phase space diagram

of Figure 4.5, where the two sets of phase space coordinates representing the

two fibers are plotted. The first set of coordinates s'

2 — t'2 is represented by

the solid lines and the second set s

2 — t 2 by the dashed lines. The other solid

or dashed curves represent the spread of coordinates in the first and second

fibers accordingly. Power in mode R' of the first fiber lying on a solid line will
72

-0.25 X X

-0.50 -0.25 0.00

( 0. 00)

0.25 0.50 0.75

( 0. 50) (

1 - 00)

1.00 1.25

(

1 . 50 )

1.50 1.75

( 2. 00) 8

2.00 3

* 1

:

s- £

&

sin6

NA

Figure 4.5. Power spreading of a mode in one fiber into other modes in the

second fiber due to lateral displacement.
73

/2

Figure 4.6. Illustration of calculation of power transfer from mode group R' to

mode group R.

couple into the modes of the second fiber which are located in an area below

the corresponding dashed curve shown in the figure. The transfer function due

to overlap coupling can be derived from this picture.

Consider power coupling between mode group R' in the first fiber

and mode group R in the second fiber (Figure 4.6). Total power along the

line R

2 under the spread curve is the power coupled from mode group R

f to

n mode group R. Though the power distribution is uniform along R' , the power

distribution along R

2 is not uniform. This is because power at different points

on R' spreads to different numbers of mode groups in the second fiber. Since

a ray does not change its propagation direction, that is, its angle with the

optical axis of the fiber, in its transition between the fibers, power spread is
74

along the horizontal line across a certain point on R'

2

. The width within the

spread curve on this horizontal line is w(t 2

)

w(t2

) = RU-R^ = il\/Rf 2 -t2

. (4.13)

Power at that point on R'

2 is proportional to the power density p(R'

z

); thus /2

power intensity along R

2 at t

2 is

Cp(R’2

) I(t2

) = (4.14)

4IsjR'2 - t

2

We denote the total power coupled from mode group R'

2 to mode group R

2 by

P(R2

. R'

2

). which is the integral of I(t 2

) over t 2

\

P(R2 ,R!

2

) = /

Jo

R'2 -( r -g ~f ) 2 Cp[R'2

)

0 4iJr’2 - t

2

Cp(R’2 ),„ ,R2 -R'2 -P

dV

21

(R21

I)- (4.15)

The upper limit in Equation (4.15) is determined from Equation (4.12) by

taking ip — 0 or 7r. The proportionality factor C can be determined from the

relation of power conservation; that is, the total power beneath the curve must

equal the total power in mode group R' :

/*”“ P{R2 ,R'

2 )dR2 = p(R’2 )R'2

, Jr2 (4.16)

where i?mzn and Rmax denote the minimum and maximum order of mode groups

of the second fiber that have power coupled into them from mode group R' of

the first fiber and are determined at xp = 0 or 7T and t = 0:

RLn = {R' - If,

RL* = (R' + if-

(4.17)

Equation (4.16) yields C — 1.
75

The total power coupled from the first fiber into mode group R

2 in

the second fiber can be expressed as

P(R2 )= C P(R2 ,R'

2 )dR'2

. (4.18)

Jo

Inserting the relation P(R2

) = p(R2 )R2 into the above equation, we have

P{R2 )R2 = C P{R2 ,R'

2 )dR'

2

. (4.19)

Jo

Inserting Equation (4.15) into (4.19), we obtain

(R1 - | —— | )p(R')R,3 «*>* dR'. ' £

(4.20)

Comparing Equation (4.20) with the definition of transfer function of Equation

3.11, we finally get the overlap transfer function:

R ( nr I R2 —R' 2 —l2 Toi(R. R') =

IR

0.

h(R- 21 |), if

| R — l |< R' < min(l, R -f /),

otherwise.

(4.21)

When / approaches zero, T0i(R. R') will become a 8- function since the integration area in Equation (4.20) will shrink to zero.

As was stated earlier in this section, tilt will have a similar effect to

lateral shift. The discussion above can be generalized to include the effect of

tilt. We can define l as the combined relative offset:

/ =-r +

d

2 sin 62 (4.22)

a

2 ' ( NAf

Thus all the above expressions and discussions about lateral shift apply to tilt

also.

The variation of radial position of a ray due to longitudinal offset is

illustrated in Figure 4.7 and can be expressed as

r

2 = r

12 + d

2 tan2 0 + 2r'd tan 0 cos (4.23)
76

Figure 4.7. Variation of the radial position of a ray due to longitudinal displacement of the ends of two fibers.

Remembering that 6 is small, we can replace tan 6 by sin which equals t(NA).

We then obtain the mode conversion relation

R

2 = R'

2 + 2s’lt(NA) cos V> + l 2

t 2 (NA)2

. (4.24)

Since (N A) — 0.2 and for a practical device we can assume that / is no more

than 0.4 (which corresponds to a longitudinal displacement of 10 fim for a 50

/rm diameter fiber) and t < 1, the last term on the right side of Equation (4.24)

can be safely ignored and the equation becomes

R

2 = R!

2 + 2s lt(N A) cos rj,. (4.25)

The mode conversion area in the phase space is illustrated in Figure

4.8. Its behavior is similar to that of scattering coupling. The maximum spread

occurs at tp = 0 or 7r and at t = s' where ts' = R

’ 2 /2 because t

2 + s

' 2 = R'

2

.
77

Figure 4.8: Illustration of mode spread due to longitudinal shift.

Thus for maximum spread we have

R

2 = R

' 2 + l(NA)R/2

(4.26)

or

„ _ l(NA) _

R ~ R' +

V

' #

- i?' + 0.1/i?'. (4.27)

Devices such as splices and power splitters have no gap between the two pieces

of fiber that compose the device; that is, / = 0. The only device that has longitudinal displacement is the connector. Practical, high-quality connectors have

very small value of /, which means a very narrow spread. Since connectors also

exhibit scattering coupling (Chapter 5) and this localized coupling is similar

to scattering coupling, its effect could be absorbed into the scattering coupling

expression and thus be ignored.
CHAPTER 5

FIBER-OPTIC DEVICES AND THEIR TRANSFER FUNCTIONS

5.1 Introduction

In this chapter we will discuss some fiber-optic devices commonly used

in telecommunications systems. These devices are fibers and cables, connectors,

splices, and power splitters. To derive the transfer functions, it is necessary

to study the physical structure of these devices and find out how and where

the two basic coupling mechanisms discussed in Chapter 4 contribute to modal

power transfer in these devices. The devices can then be modeled in terms of

the transfer functions of the two coupling mechanisms. Section 5.2 discusses

fibers and cables while connectors and splices are discussed in Section 5.3. A

power splitter is more complicated in structure and power coupling. Section

5.4 is dedicated to this kind of device.

A fiber-optic system is a concatenation of these devices. Once the

transfer functions of the devices are known, the modal power transfer relation

of the system can be formed by applying these functions successively. This can

be expressed for a n-component system with the relation [61]

P

n (Rn

) = [

l dRn ~ l C dRn~ 2 • •

• ^ dR°

Jo Jo Jo

T

n (Rn ,Rn- 1 )Tn- 1 (Rn -\Rn-2 )--‘T1 (R\R0 )P°{R0

). (5.1)

This relation is equivalent to expressing the system matrix as the product

of matrices of constituent components when the transfer matrix is used to
79

characterize components and systems [61]:

rj\s

We would expect, according to the discussions of Chapter 3, that Equation

(5.1) will give a more precise description of the power transfer property of the

system.

Experiments were made on devices whose transfer functions are derived in this chapter. In the experiment, the input and output nearfield patterns of a device are measured. The input and output modal power distributions are then calculated by Equation (2.61). The transfer function with a

set of chosen values of free parameters is then used to operate on the input

modal power distribution to obtain the output modal power distribution, from

which a predicted output nearfield pattern is calculated by Equation (2.60).

This nearfield pattern is then compared with the measured output nearfield

pattern. The transfer function is determined in this way by best fitting the

measured and the predicted nearfield patterns using the least squares method.

Experiments with two different launch conditions were made on many components to determine whether the transfer function of a component is independent

of launch conditions. Experimental setup and procedures are exactly the same

as those described in Section 3.4. Experimental results for different devices are

shown in the relevant sections. Most of them are obtained from round-robin

data taken by different participants and calculated by the transfer function

method. Since parameters in the transfer function have physical meanings

different from the transfer matrix elements, it is meaningless to compare the

statistical results of the transfer function parameters with the matrix elements

calculated from the same set of data. Matrices are calculated by operating the

_ rjinrjm— 1 _ _ . rj-U

(5.2)
80

transfer functions determined from different measurements on the same input

modal power distribution using Equations (3.22-3.25) and (3.33-3.36). The

statistical results of these matrices are then compared with those of the correponding matrices directly determined from the same sets of measured data. In

this way, a meaningful comparison can be made between the two approaches

about their dependence on measurement launch conditions. Interpretations

of the resultant values of the parameters are discussed. A concatenation experiment for a link containing only splices is also presented. Based on the

experimental results, especially the comparison between results obtained from

the same measurement data by the transfer matrix method and the transfer

function method, some conclusions are made in Section 5.5.

5.2 Fibers and Cables

5.2.1 Transfer function Fibers and cables are the basic and

perhaps the simplest devices in a fiber-optic system. Scattering coupling is

the sole coupling mechanism in the fiber or cable itself. The transfer function

of a fiber or cable is therefore exactly the same as the transfer function for

scattering coupling. It is length dependent and is written

roc Tjb (R. R'\ z) = [1 - / za(R\R')m{R”)dR”]6{Rf - R) + za(R, R')m(R).

Jo

(5.3)

Important physical and propagation characteristics that are relevant

to the transfer function of a piece of fiber or cable are core diameter, length,

index profile, loss, and bandwidth (or dispersion). Two kinds of fibers with

different nominal core diameters are now commonly being used: 50 fim and

62.5 [im. In the experiment as well as in real systems, different components

must be of the same diameter. As long as this requirement is satisfied, core
81

diameter is not a critical parameter because radial position in our theory is

always normalized with respect to the core radius. Random deviations of core

diameter will cause scattering coupling, which was discussed in Section 4.2.

The length z of the fiber appears in the transfer funtion. It is always combined

with the coupling strength a0 . a0 z can be determined from experiment. If the

length is known, we can obtain the coupling strength a0 , and fibers of different

lengths can be compared. It is also possible to compare values of q0 of a fiber

before and after cabling and the see effect of cabling on the scattering of the

fiber.

We have assumed in the derivation of the transfer function that the

fiber has a parabolic index profile. This is appropriate because almost all

present-day fibers are approximately parabolic index. However, the index profile of a real fiber will more or less deviate from the ideal profile, which is one

source of error in the experimental results. The effect of random fluctuations

of index profile has already been included in the scattering transfer function.

There are two major causes of loss in the fibers: scattering loss and

absorption loss [62]. In high-quality fibers, absorption is reduced to a very

low level, and scattering loss, especially the intrinsic Rayleigh scattering, dominates. This is manifested in the Rayleigh scattering behavior of the spectral

loss charateristics of a good-quality fiber as is illustrated in Figure 4.3. The loss

of a fiber or cable calculated from the scattering transfer function should be

close to the measured loss if the fiber is of good quality. When the calculated

value is notably lower than the measured value, we should consider the difference to be the absorption loss. Details of the differential mode attenuation

could be obtained by including a differential mode loss term in the equation for
82

the length variation of modal power (Equation 4.7). This term is equivalent to

the first term in Marcuse's coupled power equation.

The transfer function can be used to solve the modal dispersion problem in multimode fibers [63]. We write the time dependent equation of transfer,

llT + -7mi)P( z > t ' R) = /' dR'a(R,R')P(z,t,R'), (5.4)

OZ Vg{K) ot Jo

where vg (R) is the group velocity of mode R and a(R, R') is the same as defined

in Equation (4.4). If absorption loss is significant, we could also include an extra

term for differential mode attenuation. The first-order solution of Equation

(5.4) is given by

R

2 = vg (R2

) dt' £

d(R'2 )a(R\ R'

2 )P(0,t'- — T #’)•

(5.5)

The function vg (R2

) is the group velocity of modes designated by R

2 and is

assumed to have the form

Vg(R2

) — VgQ “f VglR2

. (5.6)

Making some modifications in the nearfield measurement setup to include a

laser diode and a fast detector, we can also measure the pulse broadening of

the fiber in addition to the nearfield pattern. The modal dispersion v

g (R2

) as

well as the transfer function of the fiber can thus be determined. Details about

the modal dispersion are beyond the scope of this technical note.

5.2.2 Experimental results One of the artifacts in the roundrobin test is a piece of cable about 200m long. The fiber has a 62.5 fim core

diameter and an NA ~ 0.25. The cable is not connectorized, and the cut-back

method was used in its measurement. The fitting results of the output nearfield

measured by three participants are shown in Figures 5.1 - 5.3.
83

Figure 5.1. Curve fitting results of the nearfield pattern of cable canc measured

by participant PI for (a) central launch and (b) overfilled launch.
84

Figure 5.2. Curve fitting results of the nearfield pattern of cable canc measured

by participant P2 for (a) central launch and (b) overfilled launch.
85

X

Figure 5.3. Curve fitting results of the nearfield pattern of cable canc measured

by participant P3 for (a) central launch and (b) overfilled launch.
86

Table 5.1. Main parameters of the transfer function of cable canc determined

from three measurements made by participant PI using the same launch condition.

canc T OiZ

Pl-1 0.48 1.2

Pl-2 0.32 1.7

Pl-4 0.49 2.0

Average 0.43 1.5

Standard Deviation 0.09 0.4

Table 5.1 lists the values of the main parameters r and aoz determined

from three measurements made by PI using the same launch condition. Table

5.2 lists the values of the same parameters determined from measurements

made by PI, P2, and P3 using different launch conditions. Comparing the

standard deviations in these two tables, we can see no indication of dependence

of t and a0 z on launch conditions.

Since parameters of the transfer function have different physical meaning than the transfer matrix elements, transfer matrices are calculated by operating the transfer functions on the same input modal power distributions of

central and overfilled excitations using Equations (3.22-3.25). The ’measured’

matrices are then obtained from these matrices by using Equations (3.33-3.36),

Table 5.2. Main parameters of the transfer function of cable canc determined

from measurements made by three participants using different launch conditions. Data for PI are the average values in Table 5.1.

canc T OLZ

PI 0.43 1.5

P2 0.60 1.2

P3 0.42 1.25

Average 0.48 1.3

Standard Deviation 0.10 0.2
87

Table 5.3. Transfer matrices of cable canc calculated from transfer functions

measured by PI using the same launch condition.

canc tu tl 2 Pl-1 0.9918 0.0410 0.0032 0.8686

Pl-2 1.0003 0.0377 -0.0013 0.8956

Pl-3 0.9849 0.0383 0.0056 0.9067

Average 0.9923 0.0390 0.0025 0.8903

Standard Deviation 0.0077 0.0018 0.0035 0.0196

Table 5.4. Transfer matrices of cable canc calculated from transfer functions

measured by three participants using different launch conditions. Data for PI

are the average values in Table 5.3.

canc tn tl2

PI 0.9923 0.0390 0.0025 0.8903

P2 0.9259 0.0619 0.0296 0.7164

P3 0.9984 0.0352 0.0001 0.8550

Average 0.9722 0.0454 0.0107 0.8206

Standard Deviation 0.0402 0.0144 0.0164 0.0919

which are listed in Tables 5.3 and 5.4. Comparing the standard deviations in

the last two tables, we can see that the standard deviation for the same launch

condition is less than that for different launch condition, which indicates the

dependence on launch condition of the measured transfer function. However,

when this difference is compared with those in Tables 3.3 and 3.4, we can

see that, except for t 2 2 , the differences between the standard deviations for

matrices determined from the transfer functions are much less than those for

matrices directly calculated from the measurement results. This fact implies

that the transfer function depends less on launch conditions than the transfer

matrix. It might be too early to reach this conclusion from the result of only

one component where one matrix element does not satisfy this conclusion. A

more comprehensive comparison will be made in the conclusion section of this
88

chapter, where statistics of all the test components for the transfer function

and the transfer matrix are calculated and more convincing conclusions will be

reached.

5.3 Connectors and Splices

Connectors and splices are devices that link two pieces of fibers end

to end. There are different kinds of connectors and splices according to the

structure [64, 65, 66, 67]. The key portion of a connector of any structure is

a thin tube to hold the fiber. The coating of the fiber is stripped before it

is inserted into the tube. This is to avoid the effect of the eccentricity of the

coating and ensure good alignment. The thin tube is filled with epoxy to fix

the fiber with respect to the the tube.

There are two commonly used splices: fusion splices [68, 69, 70] and

rotary splices [71]. Fusion splices use an electric arc to fuse together two finely

aligned fiber ends. The completed fusion splice is put in a glass tube to protect

it. The splice is fixed inside the tube by epoxy. Rotary splices use three short

thin glass rods that hold the two pieces of fiber between them. Then epoxy

is used to fix the fiber ends. A copper clip keeps the glass rods together and

holds and protects the splice.

A connector or a splice is basically a butt coupler. However, both

devices hold a short piece of the fiber on each side of the actual connection and

the actual connection, either fusion or the employment of epoxy, will deform

or exert stress on sections of both fibers in the vicinity of the device, causing

a significant amount of scattering coupling in spite of the short length of the

fibers. Coupling can be characterized by a short scattering region, the overlap

region, and another scattering region. The output modal power distribution
89

can therefore be obtained by operating on the input modal power distribution

the scattering and overlap transfer functions in this order. Parameters in the

two scattering processes can be assumed the same because the two sections

are close to each other and are perturbed by the same operation. The factor

K(R.R') is the same as in Equation (3.20) for connectors. Since fusion deforms the outer region of the fiber more than the central area and will result in

scattering much stronger than the Rayleigh scattering, the parameter x, which

represents the effect of Rayleigh scattering, is much less than y, which represents the effect of fusion, and can be taken to zero. Thus the factor k(R.R')

in Equation (4.6) is reduced to

I<{R, R!)

[min(R-R!)] 2 R

,v R,2

(5.7)

Reduction of one parameter will ease the curve fitting in data processing.

5.3.1 Experimental results of connectors The results for one

connector, ccml, are presented here. It is a connector with pigtailed fibers at

both ends and was measured by using the cut-back method. The curve fitting

results of measurements made by four participants are shown in Figures 5.4 -

5.7. The main transfer function parameters are listed in Tables 5.5 and 5.6.

Standard deviations in these two tables show that parameters r and a0 z have

some dependence on launch conditions, whereas the parameter d seems independent of launch conditions. Transfer matrices calculated from these transfer

functions are listed in Tables 5.7 - 5.8. The standard deviations in Table 5.8,

compared with their counterparts in Table 3.6, show less dependence on the

launch condition of transfer matrices calculated from the transfer functions

than those calculated directly by transfer matrix method, a fact which implies

that for connectors the transfer function is less dependent on launch condition.
90

Table 5.5. Main parameters of the transfer function of connector ccml determined from four measurements made by participant PI using the same launch

condition.

ccml T az d

Pl-1 0.57 0.69 0.24

Pl-2 0.52 0.67 0.18

Pl-3 0.54 0.61 0.20

Pl-4 0.41 0.61 0.15

Average 0.51 0.64 0.19

Standard Deviation 0.07 0.04 0.04

Table 5.6. Main parameters of the transfer function of connector ccml determined from measurements made by four participants using different launch

conditions. Data for PI are the average values in Table 5.5.

ccml T az d

PI 0.51 0.64 0.19

P2 0.34 0.66 0.245

P3 0.26 1.14 0.23

P4 0.56 1.10 0.22

Average 0.42 0.88 0.22

Standard Deviation 0.14 0.27 0.02
91

Table 5.7. Transfer matrices of connector ccml calculated from transfer functions listed in Table 5.5.

ccml tl2 ^21 ^22

Pl-1 0.8873 0.0642 0.0493 0.6838

Pl-2 0.8585 0.0592 0.0673 0.5828

Pl-3 0.8860 0.0602 0.0516 0.6586

Pl-4 0.9266 0.0479 0.0331 0.7611

Average 0.8896 0.0579 0.0503 0.6716

Standard Deviation 0.0280 0.0070 0.0140 0.0735

Table 5.8. Transfer matrices of connector ccml calculated from transfer functions listed in Table 5.6. Data for PI are the average values in Table 5.7.

ccml in ^12 ^21 ^22

PI 0.8896 0.0579 0.0503 0.6716

P2 0.9137 0.0476 0.0501 0.5533

P3 0.9255 0.0563 0.0438 0.5450

P4 0.7867 0.0730 0.0936 0.5786

Average 0.8789 0.0587 0.0594 0.5871

Standard Deviation 0.0632 0.0105 0.0230 0.0581
92

Figure 5.4. Curve fitting results of connector ccml measured by participant

PI for (a) central and (b) overfilled launches.
93

Figure 5.5. Curve fitting results of connector ccml measured by participant

P2 for (a) central and (b) overfilled launches.
94

Figure 5.6. Curve fitting results of connector ccml measured by participant

P3 for (a) central and (b) overfilled launches.
95

Figure 5.7. Curve fitting results of connector ccml measured by participant P4

using plane wave launch, (a) is low-order mode excitation, and (b) is high-order

mode excitation.
96

5.3.2 Experimental results of splices and their concatenation A concatenation experiment of four fusion splices was carried out to

verify that the transfer function technique can be applied to concatenated links

of components and to compare the precision of the three methods, that is, loss

measurement, transfer matrix method, and transfer function method, in the

prediction of loss of concatenated components. A 20-m section of 125-fim outer

diameter, 62.5-pm core diameter, 0.26-NA multimode fiber was selected. The

fiber was cut 2m from the end and then respliced with a fusion splicing apparatus. The fiber was then cut 2m in from the splice and then respliced. This

procedure was repeated until the link contained four splices, each located 2 m

from its nearest neighbor(s). The end, located 12m from the nearest splice,

was chosen as the input end. The input and output nearfield patterns and

the loss were measured with central and overfilled excitations using the cutback method described in Section 3.6. The four splices were then cut from the

link and input and output nearfield patterns and loss for each were measured

with central and overfilled excitations. The measured data were processed to

get both the transfer matrix and the transfer function. The measured loss,

the transfer matrix, and the parameters of the transfer function for individual

splices are listed in Table 5.9. The curve fitting of the transfer function method

for the individual splices are shown in Figures 5.8 - 5.11.

The measured and predicted loss and matrices for the link are listed in

Table 5.10. The predicted loss for the loss measurement method is simply the

sum of the loss of each splice and no matrix can be obtained using this method.

The concatenated matrix is the product of the matrices of the four splices

using Equation (5.2) and the concatenated loss is calculated by operating with
97

Table 5.9. Measured loss for central and overfilled excitations, transfer matrices, and main parameters of transfer functions of splices spl, sp2, sp3, and

sp4.

Zc(dB) / 0 (dB) Matrix T az d

spl 0.062 0.491 0.9139

0.0838

0.0779

0.6804

0.03 1.5 0.16

sp2 0.274 0.475 1.0719

-0.0478

-0.0589

0.7905

0.18 1.1 0.16

sp3 0.126 0.751 0.9265

0.0820

0.1039

0.5793

0.10 2.0 0.19

sp4 0.215 0.836 1.1765

-0.1157

-0.3853

0.9209

0.01 0.3 0.25

Table 5.10. Measured loss for central and overfilled excitations and transfer

matrix of the concatenation of the four splices compared with those calculated

from the losses and matrices of the individual splices measured by the three

methods.

SP1-4 /c (dB) /„( dB) Matrix

Measured

Value

0.834 2.081 0.9071 -0.0841

0.0268 0.3376

Loss

Measurement

0.677 2.553

Transfer

Matrix

0.497 1.755 1.0294 -0.0080

-0.0191 0.2772

Transfer

Function

0.778 2.110 0.8665 -0.0829

0.0587 0.3062
98

Figure 5.8. Curve fitting results of splice SP1 for (a) central excitation, and

(b) overfilled excitation.
99

Figure 5.9. Curve fitting results of splice SP2 for (a) central excitation, and

(b) overfilled excitation.
100

X

Figure 5.10. Curve fitting results of splice SP3 for (a) central excitation, and

(b) overfilled excitation.
101

Figure 5.11. Curve fitting results of splice SP4 for (a) central excitation, and

(b) overfilled excitation.
102

this resultant matrix on the input modal power vector calculated from the

input modal power distribution. The concatenation of the individual transfer

functions are performed using Equation (5.1) and the resulting nearfield fits

well with the measured nearfield for both launch conditions as shown in Figure

5.12. The concatenated matrix and loss can be obtained at the same time,

although the concatenated matrix is not necessary in this case.

Table 5.10 shows clearly that the transfer function method can predict the concatenated loss better than the other two methods. The result of

the transfer matrix method is unexpectedly poor and is worse than most previous work reported in the literature [18, 19]. This is probably because in our

experiment the launch condition is not strictly controlled, as is manifested in

the variation of the values of ki and k2 listed in Table 5.11. The values of

k2 for overfilled launch are resonably uniform because overfilled launch has a

better repeatability. Previous experiments by other laboratories may have had

better control of the launch condition. However, this once more reveals the important disadvantage of the transfer matrix method: its dependence on launch

conditions.

Table 5.11. A list of ki and k2 defined in Equation 3.37, which shows how the

launch conditions are different for the measurement of each splice and their

concatenation.

spl sp2 sp3 sp4 spl4

k\ 19.127 2.434 7.762 3.814 5.262

h 0.776 0.774 1.059 0.815 0.860
103

I 


III!

Figure 5.12. Nearfied pattern predicted from the transfer functions of individual splices and the measured nearfield pattern for concatenated splices for (a)

central excitation, and (b) overfilled excitation.
104

5.4 Power Splitters

A fiber-optic power splitter belongs to a broader category of devices

called fiber-optic couplers [72, 73] , which are divided into two groups: power

splitters and wavelength division multiplexers (WDMs). Power splitters can

also be divided into 3-port couplers and star couplers, which are multiport couplers. Power splitters are used in local area networks (LANs). 3-port couplers

are used for fiber-optic signal tapping and distribution, and star couplers for

optical bus systems and multiport application. Light entering a coupler is split

out to two or more output ports. The splitting ratio depends on the structure

for the device and the fabrication process. There are three different structures

of 3-port couplers:

1. The offset structure is shown in Figure 5.13. This is evidently

an overlap coupling device, and the splitting ratio is mode dependent. This

structure has a high insertion loss.

2. The semi-transparent mirror is illustrated in Figure 5.14. The two

output ports collect the transmitted and the reflected light separately. The

splitting ratio is determined by the thickness of the semi-transparent mirror.

The splitting ratio is independent of mode order and wavelength.

3. The fused biconical taper [74], shown in Figure 5.15, is the most

common 3-port coupler. Two pieces of fiber are twisted around each other and

then heated in the middle until they fuse together. It has low insertion loss, but

the splitting ratio is mode dependent. Fused biconical tapered power splitters

have found wide application in optical fiber distribution networks because of

their versatility, high performance, relatively low cost, and ease of fabrication.
105

Figure 5.13: Schematic diagram of a coupler of the offset structure.

Figure 5.14. Schematic diagram of the semi-transparent mirror structure of

couplers.

Figure 5.15. Schematic diagram of fused biconical taper structure of couplers.
106

Figure 5.16: Structure of mixed-rod star couplers.

Star couplers have two different kinds of structures:

1. The mixer-rod structure, illustrated in Figure 5.16, has a thin

platelet of quartz glass in the middle with several pieces of fiber attached to

both ends. The platelet is embedded in a layer of adhesive of lower refractive

index and thus becomes a waveguide. The light coupled into the mixer platelet

from any of the input fibers will achieve a uniform spatial distribution and

produce almost the same power in all the output ports. This structure is

similar to the offset structure and also has a high insertion loss.

2. The fused biconical taper structure [75] is shown in Figure 5.17.

This is the multi-port version of the 3-port fused biconical coupler. The splitting ratio is mode dependent and the port-to-port output variation is relatively

wide. The insertion loss is low.

WDM couplers are essentially not power splitters. We will not discuss

them in this technical note.

Couplers with offset structure are overlap coupling devices; therefore,
107

Figure 5.17: Structure of fused biconical tapered star couplers.

their transfer functions are exactly the overlap transfer function. Because the

assembly may also affect the sections of fibers close to the coupling endface,

a scattering-overlap-scattering model similar to that of connectors and splices

may be employed.

Couplers of the semi-transparent structure are simpler devices regarding the power transfer process. The splitting ratio, which is independent

of mode parameter, describes the power transfer properties of these devices.

Power coupling in power splitters of the fused biconical taper structure

is much more complicated. The next sections are dedicated to the derivation of

the transfer function of 3-port power splitters with this structure. Experimental

results are presented in the section following the derivation. Multi-port power

splitters of the same structure can. in principle, be derived in the same way,

though it is even more complicated.

5.4.1 A model of 3-port fused biconical taper power splitters The structure of a 3-port fused biconical tapered power splitter can

be portrayed as in Figure 5.18. The twisting of the fibers is not drawn in the
108

me r-ge * sc afta r* split

Figure 5.18. Schematic diagram of the structure of a 3-port fused biconical

taper power splitter.

figure because the effect of the twisting will be built into the values of the

fitting parameters. The two output ports are often called the straight port and

the coupled port. The straight port and the input port are the two ends of one

of the two fibers. In the experimental results we shall call the straight port,

port 1, and the coupled port, port 2. The structure can be roughly divided

into three sections according to different physical processes and coupling mechanisms: the merging, scattering, and splitting sections. The transfer functions

for these sections will be derived separately in the following.

(i) Merging: Modal power coupling in the merging section is very

similar to that in butt coupling; that is, power is coupled from one piece of

fiber to the other due to geometrical overlap. The major difference in the two

cases is that power is completely transferred from the first fiber to the second

in butt coupling, while in the merging region, the two fibers coexist in the

overlapping section and power is shared by the two pieces of fiber. However,
109

Figure 5.19. Idealized index profile and mode composition before and after

merging

the modal power coupled to the second fiber as well as that remaining in the

first fiber are still proportional to what the overlap coupling transfer function

would predict.

At the completion of merging, the two pieces of fiber become one

entity. Its idealized geometry and index contours are depicted in the cross sectional diagram of Figure 5.19. The curves in the diagram are also boundaries of

different modes as in the simple case of a undeformed circular fiber waveguide.

Since the middle parts of the two fibers do not merge, the lower-order modes

in these unmerged regions are unaffected by the process. Higher-order modes

which had the same mode parameter in the two original fibers will be shared

by the two fibers and become joint modes. The modal power transfer relations

can be derived from these assertions as follows.

For R < 1 / 2 (separate lower-order modes)

r\ 1 roo Pim(R) = /[1-r/ Tol (R~,R')dK}P,m (R')6{R- R!)dR', (5.8)

Jo Z Jo

P*n(R) = 1 [' Tol (R,R')PUR')dR'.

Z Jo

(5.9)
110

For R > 1 / 2

(joint higher-order modes)

Plm(R) — P2m(R)

1 r

1 1

/'°° = o /

[! - o /

Tol (R” , Rr)dR”]Plm (R')6(R - R')dR'

Z Jo Z Jo

+ 1 f

1 Tol (R,R')Plm (R')dR!. (5.10)

4 Jo

Notice that the kernel function of both these relations is still the overlap coupling transfer function Tc/.

(ii) Scattering: Scattering coupling in a power splitter is basically

the same as in a simple piece of optical fiber. The difference is the mode

composition. In power splitters, the mode composition is characterized by

joint higher-order modes and separate lower-order modes as stated above. Thus

power in higher-order modes is shared by the two fibers throughout the entire

middle merged section of the power splitter. However, power is still coupled

between different modes of the same fiber. Since the fabrication of the power

splitter involves twisting and fusing of the fibers, scattering coupling is very

strong compared to that in a normal unstressed piece of fiber; this is manifested

in a large value of az in the experimental results in spite of the very small value

of the length 2 of the device.

(iii) Splitting: Splitting is the reverse of merging. The mode structure

experiences a transition as shown in Figure 5.20. The modal power coupling is

still characterized by overlapping and sharing of power by overlapping modes.

The power transfer relations are expressed as

r i

1 r°°

Pxm(R) = /[l--/ Tol (R\R!)dR'}Plm {R!)6(R~ R!)dR

Jo Z Jo

+ \ f Tol (R,R!)P2m (R!)dR!

Z Jo

(5.11)
Ill

Figure 5.20. Idealized index profile and mode composition before and after

splitting.

r 1 1 too

P2m (R) = / [l--/ Tol (R\R)dR')P2m {R!)S(R-R!)dR

Jo l Jo

+ 1 f

1 Tol (R..R!)Plm {R')dR!. (5.12)

L JO

The parameter /, the normalized lateral shift, is the same in merging as in

splitting.

5.4.2 Experimental results The results of one power splitter

psg are presented here. The fitting results are shown in Figures 5.21-5.26. The

main transfer function parameters are listed in Tables 5.12 and 5.13. Similarly

to the experimental results of connector ccml, the parameters r and az show

more dependence on launch condition than the parameter d. Transfer matrices

are calculated from these transfer functions and are listed in Tables 5.14 - 5.15.

Comparing the standard deviations for participant Pi and those for the three

participants for the corresponding matrices also show a certain dependence on

the launch condition of the transfer function. However, the standard deviations

in Table 5.15, as opposed to their counterparts in Table 3.8, indicate that the

transfer function is less dependent on the launch condition than the transfer

matrix.
112

Table 5.12. Main parameters of the transfer function of power splitter psg

determined from five measurements made by participant PI using the same

launch condition.

psg T OLZ d

Pl-l 0.13 8.0 0.16

Pl-2 0.13 7.8 0.22

Pl-3 0.13 7.5 0.18

Pl-4 0.13 7.6 0.18

Pl-5 0.13 7.4 0.19

Average 0.13 7.7 0.19

Standard Deviation 0.00 0.2 0.02

Table 5.13. Main parameters of the transfer function of power splitter psg determined from measurements made by three participants using different launch

conditions. Data for PI are the average values in Table 5.12.

psg T az d

PI 0.13 7.7 0.19

P2 0.15 8.5 0.25

P4 0.20 7.0 0.19

Average 0.16 7.7 0.21

Standard Deviation 0.04 0.7 0.03
113

Table 5.14. Transfer matrices of power splitter psg calculated from transfer

functions measuremed by participant PI using the same launch condition.

port 1 tu t\2 hi ^22

Pl-1 0.5011 -0.0047 -0.0186 0.3809

Pl-2 0.5026 0.0010 -0.0511 0.3553

Pl-3 0.5043 -0.0031 -0.0368 0.3623

Pl-4 0.5043 -0.0031 -0.0366 0.3612

Pl-5 0.5041 -0.0020 -0.0410 0.3627

Average 0.5033 -0.0024 -0.0368 0.3645

Standard Deviation 0.0014 0.0021 0.0118 0.0096

port 2 tn tl2 hi ^22

Pl-1 0.4918 0.0030 -0.0100 0.3737

Pl-2 0.4896 0.0085 -0.0392 0.3485

Pl-3 0.4935 0.0043 -0.0268 0.3555

Pl-4 0.4934 0.0044 -0.0265 0.3543

Pl-5 0.4927 0.0056 -0.0304 0.3558

Average 0.4922 0.0052 -0.0266 0.3576

Standard Deviation 0.0016 0.0021 0.0106 0.0095

Table 5.15. Transfer matrices of power splitter psg calculated from transfer

functions measuremed by three participants using different launch conditions.

Data for PI are the average values in Table 5.14.

port 1 tl2 ^22

PI 0.5033 -0.0024 -0.0368 0.3645

P2 0.4871 0.0103 -0.0207 0.3575

P4 0.4757 0.0207 -0.0179 0.2689

Average 0.4887 0.0095 -0.0251 0.3303

Standard Deviation 0.0139 0.0116 0.0102 0.0533

port 2 hi tl2 hi ^22

PI 0.4922 0.0052 -0.0266 0.3576

P2 0.4737 0.0190 -0.0085 0.3496

P4 0.4621 0.0292 -0.0056 0.2612

Average 0.4760 0.0178 -0.0136 0.3228

Standard Deviation 0.0152 0.0120 0.0114 0.0535
114

Table 5.16. Main parameters of the transfer functio of power splitter psg, dpi,

and dp2. dpi and dp2 were made by the same manufacturer while psg was

made by a different manufacturer.

T az d

psg 0.16 7.7 0.21

dpi 0.11 22.0 1.07

dp2 0.14 23.0 0.97

Two power splitters, dpi and dp2, which are not round-robin test artifacts, were measured before the round-robm measurement. These two power

splitters were fabricated by the same manufacturer, while the round-robin

power splitter was made by a different manufacturer. The transfer function

parameters for all three power splitters are listed in Table 5.16, where the values for psg are the average values in Table 5.13. The values of az and d for

dpi and dp2 are close to each other while significantly different from those

for psg. This fact implies that the transfer function parameters have physical

significance and may be used for the analysis of the structure or the fabrication

process of the devices in addition to the characterization of the components.
115

Figure 5.21. Curve fitting results of port 1 of power splitter psg measured by

PI for (a) central launch and (b) overfilled launch.
116

Figure 5.22. Curve fitting results of port 2 of power splitter psg measured by

PI for (a) central launch and (b) overfilled launch.
117

Figure 5.23. Curve fitting results of port 1 of power splitter psg measured by

P2 for (a) central launch and (b) overfilled launch.
118

X

Figure 5.24. Curve fitting results of port 2 of power splitter psg measured by

P2 for (a) central launch and (b) overfilled launch.
119

Figure 5.25. Curve fitting results of port 1 of power splitter psg measured by

P4 for (a) low-order mode excitation and (b) high-order mode excitation.
120

Figure 5.26. Curve fitting results of port 2 of power splitter psg measured by

P4 for (a) low-order mode excitation and (b) high-order mode excitation.
121

5.5 Conclusions

I have demonstrated the measurement results and the statistics of one

sample of each kind of the round-robin component and compared, for each of

them, the results of using the transfer matrix method and the transfer function

method. It is time now to make an overall comparison of the accuracy of the

two methods with more data. Table 5.17 lists the standard deviation of the

transfer matrix elements of seven round-robin components measured by one

participant of the round-robin test, listed as SD-P1, and those measured by

three or four different participants, listed as SD-A11.

Table 5.18 is a list of the standard deviation of the transfer matrix

elements calculated from the transfer functions determined from the same sets

of data measured by Pi and by all the participants. Let us first compare the

data in the last two lines of the two tables. For both methods, the standard deviation of different participants is greater than that of one participant, a fact

which indicates that both methods depend on launch conditions. However,

comparing the magnitude of this difference of the two methods shows that the

transfer function method is much less dependent on launch conditions than the

transfer matrix method. This is demonstrated in Table 5.19 where the differences are listed for the two methods. The improvement is significant. What is

more important is that the cause of dependence on launch conditions of the the

measured results of the two methods is fundamentally different. The cause of

the dependence of the transfer matrix method on launch conditions lies in the

method per se, as was analyzed in Chapter 3; whereas the dependence of the

experimental results on launch conditions for the transfer function method is

due to the fact that the theoretical model for the transfer functions of various
122

devices can not always be perfect. One cause might be that the parameters r

and az in the scattering transfer function are different for different scattering

mechanisms. Since different scattering mechnisms could have different effects

at different radial positions in the fiber, the values of r and az could be a

function of R. The values of the two parameters determined in the present

model represent some average effect of these mechanisms. To make these parameters vary with R will introduce more parameters and greatly complicate

the curve fitting process. We have to make a compromise between convenience

and accuracy.

Further, the standard deviation of t 22 calculated for PI by using the

transfer function method is greater than that by using the transfer matrix

method. Similarly, the net improvement of £22 of the transfer function method

over the transfer matrix method, that is, the difference between the values of

1 22 in the last line of Tables 5.17 and 5.18, is not large. One possible reason

is that the transfer function method uses the details of the input modal power

distribution and output nearfield pattern. Errors in the measurement or in data

processing may influence the result of the transfer function method more than

that of the transfer matrix method. The nearfield pattern is always affected at

the edge of the core by the diffraction limit and by leaky modes. This will in

turn result in errors in the modal power distribution, mainly the higher-order

modes. Most participants did not take any special precautions to reduce this

error. That is why £22 has the largest error in the transfer function method.

However, the overall accuracy of t 2 2 for the transfer function method is still

notably better than that for the transfer matrix method. By standardizing

the correction technique regarding those effects, we can anticipate that the
123

Table 5.17. Comparison of standard deviations of measured matrices by participant PI using the same launch condition and by different participants using

different launch conditions.

Component in tl2 ^21 ^22

ccml SD-P1 0.0192 0.0305 0.0124 0.0240

SD-A11 0.1636 0.0778 0.0305 0.1404

cpml SD-P1 0.0071 0.0409 0.0094 0.0159

SD-A11 0.1082 0.1039 0.0196 0.1337

ccl SD-Pl 0.0275 0.0261 0.0132 0.0542

SD-A11 0.0941 0.0492 0.0509 0.0896

caec SD-Pl 0.0037 0.0471 0.0140 0.0078

SD-A11 0.1176 0.0665 0.1172 0.1551

canc SD-Pl 0.0212 0.0253 0.0193 0.0225

SD-A11 0.1404 0.1857 0.0611 0.0616

PSg

port 1 SD-Pl 0.0120 0.0147 0.0042 0.0236

SD-A11 0.1088 0.0768 0.0461 0.0282

PSg

port 2

SD-Pl 0.0138 0.0147 0.0075 0.0234

SD-A11 0.0757 0.0434 0.0443 0.0374

Average SD-Pl 0.0149 0.0285 0.0114 0.0245

SD-A11 0.1155 0.0861 0.0528 0.0922
124

Table 5.18. Comparison of standard deviations of matrices calculated from

transfer functions measured by participant PI using the same launch condition

and by different participants using different launch conditions.

Component ^ii tl2 ^21 ^22

ccml SD-P1 0.0280 0.0070 0.0140 0.0735

SD-A11 0.0632 0.0105 0.0230 0.0581

cpml SD-P1 0.0544 0.0276 0.0126 0.0370

SD-A11 0.0414 0.0210 0.0189 0.0346

ccl SD-P1 0.0070 0.0025 0.0103 0.1326

SD-A11 0.0302 0.0188 0.0051 0.0376

caec SD-P1 0.0048 0.0201 0.0003 0.0343

SD-A11 0.0906 0.0166 0.0311 0.1300

canc SD-P1 0.0077 0.0018 0.0035 0.0196

SD-A11 0.0402 0.0144 0.0164 0.0919

PSg

port 1 SD-P1 0.0013 0.0021 0.0118 0.0096

SD-A11 0.0139 0.0116 0.0102 0.0533

PSg

port 2

SD-P1 0.0016 0.0021 0.0106 0.0095

SD-A11 0.0152 0.0120 0.0114 0.0535

Average SD-P1 0.0150 0.0090 0.0090 0.0451

SD-A11 0.0421 0.0150 0.0166 0.0656

Table 5.19. Difference of standard deviations for the MTF and MTM methods

between statistical results of measurements made by PI and of measurements

made by different participants.

tn tl2 ^21 ^22

MTF 0.0271 0.0060 0.0076 0.0205

MTM 0.1006 0.0577 0.0414 0.0678
125

accuracy of the transfer function method will further be improved.

In addition to the improvement in the accuracy and precision in predicting the loss of optical fiber devices and systems, the transfer function also

has the potential to improve the prediction of bandwidth of such systems because it gives details of mode coupling, which has a big impact on the pulse

broadening of an optical fiber system. The transfer function may also be used

to analyze the structure of the devices and the effect of assembly on the devices.

ACKNOWLEDGEMENTS

I wish to thank Professor Alan R. Mickelson for his guidance in this

research. My thanks also goes to Dr. Robert L. Gallawa, whose encouragement and support made it possible to start and complete this research project.

I would also thank Dr. Matt \oung, who made valuable comments and corrections on the original manuscript.

The discussion about the accuracy of mode transfer matrix using the

expansion of mode transfer function is mainly Dr. Dag R. Hjelme's contribution, which is very much appreciated. I would also thank him and Dr. Michael

Yadlowsky for many valuable discussions.

My thanks were also given to Igor P. Vayshenker, who built the original measurement setup, and to Indra P. Januar and Chon Howe Oon, who did

some of the measurements and data processing.
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