i<

»•' " 'o.

NBS TECHNICAL NOTE 1018

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

An Assessment of

the Backscatter Technique

as a Means for Estimating Loss

in Optical Waveguides
NATIONAL BUREAU OF STANDARDS

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An Assessment of the

Backscatter Technique as a Means

for Estimating Loss in Optical Waveguides

rationaCburI^u

o? standards

UBRAEY

APR 1 7 1980

/?C>^ ace -on:

auco

no- loio

B.L. Danielson

Electromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

:^^.

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

6 NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1980
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1018

Nat. Bur. Stand. (U.S.), Tech. Note 1018, 88 pages (Feb. 1980)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Stock No. 003-003-021 60- 1 Price $3. 75 (Add 25 percent additional for other than U.S. mailing)
CONTENTS

Page

1

.

INTRODUCTION 1

2. THEORY OF THE OPTICAL TIME DOMAIN REFLECTOMETER 10

3. ATTENUATION RELATIONS BASED ON A DISCRETE MODEL 11

3 . 1 Direct Attenuation 15

3 . 2 Two-Point Backscatter Attenuation 16

3 . 3 Least-Squares Backscatter Attenuation 20

3.4 Piecewise Least-Squares Backscatter Attenuation 24

3 .

5

Algorithm for Backscatter Comparisons 24

4

.

COMPUTED EXAMPLES 25

4.1 Point Defects 25

4 . 2 Loss Discontinuities 37

4.3 Distributed Loss Effects 46

5. EXTENSION OF ANALYSIS TO PULSES OF ARBITRARY SHAPE 62

6

.

DISCUSSION OF RESULTS 6 7

7. COMPARISON WITH EXPERIMENT 69

8. CONCLUSIONS 72

9

.

REFERENCES 74

111
LIST OF TABLES

Page

TABLE I . LIST OF SYMBOLS NOMENCLATURE AND UNITS V

TABLE II . ATTENUATION COMPARISONS, NBS 70

TABLE III. ATTENUATION COMPARISONS, OTHER LABORATORIES 71

LIST OF FIGURES

Figure 1. Experimental apparatus for backscatter measurements 4

Figure 2. Backscatter signal as a function of time. Graded

index fiber, 1300 m 5 Figure 3. Logarithm of backscatter signal in figure 2 6

Figure 4. Residuals of least-squares fit to signal in figure 2 7 Figure 5. Examples of nonreciprocal waveguide 9

Figure 6. Geometry for discrete waveguide scattering model 12

Figure 7. Logarithm of backscatter power as a function of scattering

element for a fiber with three distinct loss regions 22

Figure 8. Complementary loss signature for fiber in figure 7 23

Figure 9. Least-squares error for a point absorption-loss imperfection

as a function of its location along the fiber 26

Figure 10. Least-squares error for a point scattering-loss imperfection

as a function of its location along the fiber 30

Figure 11. Least-squares error for a point capture fraction

imperfection as a function of its location along the fiber. 34

Figure 12. Temporal dependence of the input pulse considered

in section 5 61

Figure 13. The impulse response function for a fiber with

three loss regions 63

Figure 14. The logarithm of backscatter power as a function of time

for a square pulse input with the impulse response

given in figure 13 64

Figure 15. The logarithm of backscatter power as a function of time

for a pulse with Gaussian-shaped trailing edge 65

Figure 16. The logarithm of backscatter power as a function of

time for a Gaussian-shaped input pulse 66

TV
TABLE I. LIST OF SYMBOLS NOMENCLATURE AND UNITS

SYMBOL NOMENCLATURE UNITS

A2(n) Bulk absorption loss for n^" element

for forward traveling pulse

A2(n) Bulk absorption loss for n^" element

for back traveling pulse

A^Cn) Total forward loss for n^" element =

A-,^(n)+S-,^(n)

A4(n) Total back loss for n^" element = A2(n)+S2(n)

P Detector bandwidth Hz

Cj^ ( n ) Capture fraction for n*-" element

EgCn) Energy of forward traveling unit input

puise at n " element, measured at

time nAT J

R|^(n) Energy of back traveling pulse from n element measured at time 2nAT J

F Effective noise figure

i„ Noise current in detector A

i Signal current in detector A

k Assumed number of elements in waveguide

model

L-j^ Direct (insertion, two-length, cut-back

or conventional) attenuation dB

L2 Least-squares backscatter attenuation dB
TABLE I. LIST OF SYMBOLS NOMENCLATURE AND UNITS (Continued)

L3 Two-point backscatter attenuation dB

L^ Piecewise least-squares backscatter

attenuation dB

m Dumihy index for element in sampled pulse

M Assumed number of elements in input pulse.

Also used to denote detector

multiplication factor.

n Dummy index for element in wavequide nodel

Ni Core index of refraction on fiber axis

N2 Cladding index of refraction

NA Numerical aperature (see text). Not to be

confused with the abbreviation N.A. for not

applicable

.

P. (n) Average backscatter power at fiber input

end measured at time 2nAT VJ

P^(n) Average forv;ard traveling pulse power

measured at time nAT W

p. (m) Average power m input pulse measured

at time 2mAT W

Electronic charge

Correlation coefficient for leastsquares fit

Random number

R^ Detector responsivity AW-1

VI
TABLE I. LIST OF SYMBOLS NOMENCLATURE AND UNITS (Continued)

S-j^(n) Bulk scattering loss for n " element

for forward traveling pulse

S2(n) Bulk scattering loss for n^" element

for back traveling pulse

S/N Signal-to-noise ratio dB

Time

Group velocity of pulse propagating in

waveguide structure ms

-1

Xq Fiber length m

a. Total distributed waveguide loss in

'-

I forward direction = a,X

1 o

a_ Total distributed waveguide loss in

back direction = a„X

2. o

a Rayleigh scattering loss = a X

ol Distributed waveguide loss per unit

length in forward direction m

a' Distributed waveguide loss per unit

length m back direction m

a' Rayleigh scattering loss per unit

length m

^

AT One way transit time of a narrow pulse

across a differential element s

Ax Length of differential element in

waveguide model m

vn

AN ASSESSMENT OF THE BACKSCATTER TECHNIQUE AS A MEANS

FOR ESTIMATING LOSS IN OPTICAL WAVEGUIDES

B. L. Danielson

Electromagnetic Technology Division

National Bureau of Standards

Boulder, CO 80303

This technical note addresses some of the problems

associated with determining the accuracy of the backscatter

technique as it is applied to the estimation of attenuation

in optical waveguides. The basic theoretical assumptions

involved in optical time domain ref lectometry are reviewed;

the effect on calculated loss values resulting from a departure from these assumptions is then examined. The

approach taken is to employ computer modeling of the various

scattering and other loss mechanisms using the bulk material

properties of optical fibers. Computer responses permit a numerical comparison between the direct (insertion) method

of measuring attenuation and several methods of estimating

attenuation from analysis of backscatter data. Numerous

examples are given of physical effects which can produce

discrepancies in attenuation values calculated from

backscatter signals. Also, some experimental comparisons

are made between backscatter-derived and directly measured

attenuation values in step and graded-index optical

waveguides. Finally, the conditions necessary for good

agreement between the direct and backscatter methods are discussed and suggestions for minimizing these errors are made

Key Words: Backscattering ; fiber attenuation; fiber loss;

fiber scattering; optical time domain ref lectrometry

Rayleigh scattering.

1. INTRODUCTION

There are basically two different experimental techniques for the

determination of attenuation in optical waveguides. The direct, or insertion, method measures the actual transmitted power loss of a long length

of fiber. The second method involves the calculation of attenuation from

principles of optical time domain ref lectometry (OTDR), and is based on the

Rayleigh backscatter impulse response of an optical fiber. It has been

observed experimentally that the two approaches do not always exactly

agree. It is the purpose of this report to examine, by means of computer

modeling, some of the errors and measurement problems associated with

estimating attenuation from OTDR.

The attenuation of an optical waveguide at a given wavelength, between
two cross-sections and 1 separated by a length of fiber X^ is usually

defined as:

P^(0)

L^ = 10 log [p^Yxyl dB. (1-1)

where P^ (0) is the optical power traversing cross-section 0, and P^ (1) is

the power traversing cross-section 1. To help insure that the conditions of

excitation remain constant for the two required measurements, the fiber is

cut near the input end and P^ (0) is the measured power output of this short

length of fiber. We will refer to L^ measured in this way as the direct

attenuation; also referred to as the insertion, two-length, or cut-back

method. We will also include as part of the attenuation of the fiber such

environmentally-induced effects as fractures and radiation at micro-bends

even though, strictly speaking, these are not intrinsic properties of the

fiber in question. In general, the attenuation measured according to the

prescription of eq (1.1) will be dependent on the exact optical power

launching conditions [1,2] and, consequently, fiber length. Nevertheless, it

is common practice to characterize fibers in terms of a loss rate, or loss

per unit length given by L-^/X-^. Most commercial fibers are specified in this

way with units of dB/km.

The backscatter technique for estimating attenuation in fibers was suggested by Kapron et al . [3] and implemented by Barnoski [4] and Personick

[5] . Here, loss is determined from an experimental fit to the exponential

decrease as a function of time of the Rayleigh-backscatter power from a fiber

excited by a short pulse of radiation. The usual method reported in the

literature [4-6] is to use a least-squares curve-fitting routine to fit an exponential function to the experimental backscatter signal which is a function of time. This gives a "best fit" decay constant which is simply

related to the attenuation under certain restrictive conditions. Where the

fiber consists of two or more regions of different, but uniform, loss

characteristics (such as two dissimilar fibers spliced together) then a piecewise least-squares fitting can be done over each region separately. The

backscatter power impulse response can be converted to a length dependence of

the form

The expression "insertion loss method" has been used to refer to the

technique whereby the power P^ (0) measured through a short length of fiber,

and the power P^ (1) through the test fiber are made independently. In each

case adjustments are made in the launch conditions to maximize power

throughput. This procedure avoids repeatedly cutting the fiber, but

introduces the possibility of errors due to lack of alignment precision. In

this note we will make no distinction between "direct" and "insertion"

measurements, o
I

]

I

P, (x) = P, (0) exp (-2<a>X,). This is shown later to give

b D i

L2 = 5 (log e) <a>Xj^dB (1-2)

where <a> is the best fit (in the least squares sense) to the exponential

decay and L2 is our designation for fiber attenuation measured from a least

squares fit. It is shown in sections 3 and 4 that the two methods for

determining loss are identical for fibers that are uniform throughout their

length and have identical properties in the forward and back directions. If

these conditions are not satisfied then in general the results of the two

approaches do not agree

.

There are other data reduction techniques for estimating fiber

attentuation from the backscatter signal. One of these we will examine in

detail and refer to the two-point method. This attenuation, denoted by L3 is

given by the expression

Pb(0)

L3 = 5 log p^jxy clB (1-3)

where V^ (1) is backscatter signal from the cross-section labeled 1 and P^

(0) is the backscatter signal from the input end at x = . We shall see

that, under some circumstances, this approach gives closer agreement with the

direct attenuation than is obtained from least-squares curve fitting.

Although the direct method is generally preferred for loss measurements,

the backscatter technique has definite advantages in certain applications.

For example, the power level, and loss, in the fiber can be determined as a function of length. Mode coupling effects may be observed. Regions of high

absorption loss or scattering loss can be identified and located

nondestructively as a function of position along the fiber. The fiber does

not need to be cut to determine loss. Another important application is in

field measurements where there is access to only one end of the fiber. Also,

this technique provides a convenient way to measure the additional loss due

to splices and devices inserted into the optical waveguide such as couplers

and connectors

.

In this report we will assume that scattering in the waveguide is a linear function of pulse power. In fact, non-linear effects such as stimulated Raman and stimulated Briliouin scattering have been observed in some single mode fibers. The threshold for these effects has been calculated

[6] . Experimental evidence [7] indicates that 2 kW peak power can be
Polarizing

Beam Splitter

Collimating Optics

Pulsed

Laser

(Polarized)

Capillary 8. X-Y-Z

Translation

Polarizer <=i \ Mode

<3> \. Stripper

Pinhole t=7

^^^

Fiber

PMT Detector n

Trigger Box Car

Integrator

X 10 Microscope Objectives

Data

Processing

Figure 1. Experimental apparatus for backscatter measurements

Polarization is indicateci by arrows.
1 .

1

t)

TIME (us)

Figure 2. Backscatter signal as a function of tine

1300 m. Graded index fiber.
-7

Log P|3(t)

-14

-2 1

-28

-35

3.4 5.1

TIME (us)

10.2

Figure 3. Logarithm of backscatter signal in figure 2.
29.8 ^.

2U5

-20.0

Figure 4. Residuals of least-squares fit to signal in figure 2.
injected into a multimode waveguide of 75 pm core diameter with no non-linear

attenuation. However, the thresholds decrease for decreasing fiber core

area, so for single-mode operation the possibility of these scattering

processes must be considered.

A typical backscatter experimental setup is illustrated schematically in

figure 1. Some of the data generated at NBS with this type of apparatus is

depicted in figure 2. These data were chosen to illustrate the fact that the

backscatter response is not always an exact exponentially decaying

function. This work was done with a 10 mW He-Ne laser operating at .6328ym.

The cw laser was pulsed externally by means of an acousto-optic modulator,

producing pulses of about 80 ns full width at half maximum. This type of

laser was chosen because properties of a TEMqq beam allow for careful control

of launch conditions and observations and the effect of different mode energy

distributions on loss measurements. The polarizing beam splitting cube used

in conjunction with the polarized source allows discrimination between one component of the unpolarized Rayleigh backscatter and the polarized Fresnel

reflection from the launching optics at the front end of the fiber. This is

possible since most fibers will largely depolarize the input pulse within a few centimeters . Further rejection of unwanted reflections can be

accomplished with a pinhole spatial filter. As can be seen from the raw data

in figure 2, the reflection spike at the input end can be eliminated almost

completely. The fiber itself is held in a 6 cm long holder consisting of a precision-bore capillary tube with a microscope cover slide (thickness .1 mm) on the output end. The device is constructed so that the fiber is in contact

with an immersion oil which minimizes the effect of fiber surface

irregularities as well as performing the function of a mode stripper.

Realignment is also facilitated if the fiber needs to be removed. However

dust in this type of device can pose problems. The detector used in these

experiments was an eleven stage photomultiplier tube with enhanced S-20

response. The boxcar integrator is a signal averaging device producing an analog output which is then digitized and punched out on paper tape. The

numerical analysis is done on a digital computer.

The input pulse for the data of figure 2 had a beam waist of

approximately 8 \im which is near the optimum value for minimum beam

divergence in the graded index fiber [8] . The beam was launched near the

center of the 60 pm diameter core. The length of the fiber was 1.38 km with

a measured direct attenuation of 13.3 dB/km at .6328 ym.

It is clear from figure 3 that the decay of the backscatter is not

perfectly exponential? that is, log P5(t) is not a straight line. This

departure is also graphically demonstrated in the plot of the residuals to

the least-squares fit to the backscatter data (figure 4). Several end points

have been deleted in this analysis. Other launch conditions can produce a

8
(a) 02 >n^

(b)

(c)

Forward Direction of PropagationFigure 5. Examples of nonreciprocal waveguide structures
more nearly linear response. A likely explanation for the observed curvature

in this case is that energy is initially coupled into relatively low

scattering loss modes and this energy is redistributed into higher scattering

loss modes as the pulse propagates down the fiber. This then is an example

both of the power of the backscatter technique to probe the details of the

loss mechanisms as well as the possibilities for errors in the attenuation

determinations outlined in this section.

2. THEORY OF THE OPTICAL TIME DOMAIN REFLECTOMETER

The basic physical process underlying the operation of our time domain

ref lectometer is Rayleigh scattering which is due to scattering centers that

are small compared with a wavelength of light. In optical fibers these

centers arise from variations in dopant concentrations or fluctuations in the

density (and, consequently, index of refraction) of the glass matrix which occurs when the glass is cooled from the melt. Rayleigh scattering is

approximately isotropic and is proportional to the inverse fourth power of

the wavelength. This is often the most important source of loss in practical

fibers in the wavelength region around .85 ym [9]

.

Consider a waveguide whose loss and scattering properties are constant

along its length to be excited by a pulse whose power is in the form of a delta function P^(t) = E 6(t) and with unit energy E^ . The pulse propagates

down the fiber with group velocity v, so that at a distance x, corresponding

to the time t = x/v, the energy in the pulse is

E-(x) = E exp [-a^x]

,

(2-1)

where a is the total loss per unit length in the forward direction. In a distance dx the energy removed from the pulse by the Rayleigh scattering

loss a' is E a'dx exp [-a|x] . Of this energy, a fraction C-^ is trapped by

the waveguide and is guided in the reverse direction. We will refer to C^^ as the capture fraction. Personick [5] has shown that it is approximately given

by

C, = i^' (2-2)

for isotropic scattering where NA is the fiber numerical aperture and H^ is

10
the index of refraction of the fiber on the core axis. For Rayleigh

scattering C-i must be multiplied by a factor of 3/2 [10] . The energy backscattered from the element dx, and trapped by the

waveguide dEu^, then arrives at the input end of the fiber after undergoing an

I additional attenuation exp(-a'x] due to the total loss per unit length ot^ in

the reverse direction. The energy at the input end due to scattering in dx

is then

I I dE, = E C, a dx exp [-(ot, + a^)x] (2-3) b o 1 s i 2.

Expressed as a function of time at the input t = 2 x/v, the power Pj^Ct) = dEj^(t)/dt becomes

E C, a 'v , , . ^b^^^ " °

2 ^ exp[-2(aj^+ o.^)^\ (2-4)

We have assumed a fiber with uniform properties. With the further

assumption that the total bulk loss in the forward direction is the same as

t I in the reverse direction, a, = a„, then the attenuation of the fiber is

easily obtained from the backscatter response of eq (2-4). The attenuation,

in decibels, is just 5 <aj> X log,^e where <a,> is the best fit decay

constant to the experimental backscatter power data. It is the purpose of

this report to examine the magnitude of the derived backscatter attenuation

under conditions where the fiber loss exhibits departures from the above

assumptions. In order to do this, we will develop a discrete model whose

responses can more easily be determined by digital computation methods.

3.. ATTENUATION RELATIONS BASED ON A DISCRETE MODEL

In this section we will develop a model for obtaining the backscatter

response of an optical fiber which permits comparison of the direct

(insertion) loss with the loss estimated by various means from the

backscatter signal. We will employ a phenomenological approach using the

bulk properties of the fiber. Multiple scattering is neglected. Rather than

work with the continuous functions of the previous section, we will find it more convenient in our computer modeling to use a discrete representation for

backscatter simulation. Also, our experimental data, discussed in section 7,

is analyzed with this type of digital format; the same least-squares program

11
UJ jX<

CO 1 jr ± J

h- 5 < —

< c ^ o CVJ CVJ C\J

*"'-^ - !- ^^^ X3 x> ^ ^

UJ UJ UJ UJ

0) •coe

enc

-H

<u JJ

4-) mo

CO

<u 'O-H3

D-. 0)>^

0) 4J

0)o

01 -H

>JO

4Jeo

0)

0)D

12
can be used with the computer-generated data.

The fiber loss may be estimated from the backscatter in several ways.

The usual approach is to employ a least-squares fit to an expontential decay

which we have seen is to be expected from a uniform fiber. This fit may be

made to the backscatter signal from the entire fiber, or to successive

regions of the fiber which are expected to possess uniform properties.

Another approach is to identify the loss with the backscatter power ratio of

the first and last measurement points corresponding to the input and output

of the fiber. A model which permits the comparison of these

backscatter-derived attenuations with the direct attenuation will now be

examined. In all respects the model is intended to simulate experimental

conditions as discussed in the previous section.

Referring to figure 6, we will find it convenient to sample our fiber of

length X^ with k differential elements. A narrow pulse of radiation moves

from left to right and the backscatter signal travels in the return direction

right to left. The transverse intensity distribution is assumed constant for

propagation in both directions. Each of the k elements is considered to be

uniform in itself and characterized by bulk parameters as follows:

A-j^(n) = absorption loss of the n element in the forward direction.

A2(n) = absorption loss of the n " element in the reverse (back)

direction

.

S2^(n) = scattering loss of the n^" element in the forward direction.

S2(n) = scattering loss of the n^" element in the reverse (back)

direction

.

C2^(n) = capture fraction of the n^" element, assumed constant in both

directions

.

The absorption loss terms, Aj^ ( n ) and A2(n), as used here include all

contributions to attenuation which do not return a signal in the direction

opposite to the direction of propagation. These include not only intrinsic

absorption which converts optical power into heat, but such effects as radiation from bends. Intrinsic absorption has its origins in the extremities

of the uv and ir absorption bands as well as components due to impurities and

the 0H~ and transition metal ions. The scattering loss terms, S-]^(n) and

S2(n), are taken to include not only the Rayleigh scattering, but other

contributions such as Mie scattering and reflections from cracks and

imperfections which return a signal in the direction opposite to the original

direction of propagation. For example, a perfect reflection at a fiber break

would be represented by S = (N^ - l)^/(Nj^ + 1)^ and C^ = 1 • Here, Nj^ is the

index of refraction of the step-index fiber core. Also, we have allowed for

the possibility that the fiber is nonreciprocal in the sense that its

13
properties in the forward propagating direction may not be identical to the

corresponding properties in the reverse direction. Such a situation can arise

since it is well known that intrinsic loss and scattering loss are mode

dependent quantities [2,9]. For example, launching conditions can be such

that relatively low loss modes are excited in the forward direction.

However, Rayleigh scattering, being approximately isotropic, can excite a different set of modes whose average loss is somewhat higher. Some

additional examples of nonreciprocal behavior are illustrated in figure 2.

Effects of this sort have also been observed experimentally [11]

.

Nonreciprocity, as we shall see, can result in significant discrepancies in

backscatter estimates of loss in multimode fibers.

In the following discussion we will label the quantities traveling in

the forward direction with the subscript f and the corresponding quantities

traveling in the backward direction with the subscript b. We then have

E, (2nAT) = Energy of the back traveling pulse scattered from n^*^ differential element measured at input end of fiber at

time 2nAT.

E^(nAT) = Energy of the forward traveling input pulse measured at the

center of the n^" differential element at time nAT . P (2nAT) = Average power in time interval 2AT in backscatter signal at

time 2nAT.

P^(nAT) = Average input power in n*-" differential element,

where AT is the one way transit time of a narrow pulse across an element.

For reasons of convenience, all the elements in figure 6 of

length Ax = vAT except for the first (n=0) and last (n=k) which are of

length vAT/2. Here v is the pulse group velocity. This choice is arbitrary,

but it allows for the backscatter energy E, (2nAT) and forward propagating

pulse energy E^(nAT) to be sampled at the same physical point.

For simplicity, the waveguide is excited with a pulse whose power is in

the form of a delta function with unit energy

Pf(0) = E^6(t), (3-1)

E^(0) = 1. (3-2)

The extension to pulses of arbitrary length is considered in section 6,

14
3 .1 Direct Attenuation

We may now calculate the energy in the interrogating pulse as a function

of time as it propagates through each of the n discrete elements of the

waveguide. If the absorption loss and scattering loss parameters A^^Cn),

A2(n), S2(n), and S2(n) are small, the progressive energy of the pulse is

given approximately by the relations

E^(0) =1, (3-3)

A,(l) A,(0) S,(l) S,(0)

e^(At) = Ej(0) [1- -^4—' ^^" ~^—^ t^" ^4—^ f^" "4—^' ^^~^^

with the recurrence relation

A, (n) A, (n-1)

Eg(nAT) = Ej ((n-l)AT) [1 ^—] [1 ^ ]

S,(n) S,(n-1)

[1- ^V-] [1- -^

]r (3-5)

and for k elements

E^(kAt) k A, (n) , S, (n) „ —

= n ri- — 1 ri- — 1 E^(0) ^^^

^^ 2

J ^-^ 2^

A,(0) S,(0) A,(k) S,(k)

[1- -V-] [1- -V-] [1- -^^—] [1- -^ ] . (3-6)

The direct attenuation L^r in decibels, for the present pulsed system is most

conveniently represented in terms of pulse energy, rather than power, and for our purposes is defined as

E,(kAT)

.^ = -10 log [ g ^op dB (3-7)

or finally

15
L, k-1 A, (n) S, (n) ^ = 2 I (log [1- ^ty—] + log [1- -^^""^ ^ n=l

Aj^(k) S^(k)

+ log [1 ^—] + log [1 ^—

]

A,(0) S,(0)

+ log [1 ^2—1 + 1°9 fl S—

^^"^^

3 .2 Two-Point Backscatter Attenuation

We now examine the relations for the energy in the backscattered pulse

as a function of time. Each contribution to the backscatter signal E. (2nAT)

originates at the center of the respective scattering element. With the same

small loss approximations as before

E.(0) S,(0) C,(0)

Ej^(O) = -i i^ ± (3-9)

A (1) A (0)

E^(2AT) = E^(AT) S^^d) C^(l) [1- -~ ] [1 ^

]

S,(l) S,(0)

[1- -^^^r—] [1- -Aj—] (3-10)

A2(2) A2(l)

2 E^(4AT) = Eg(2AT) S^(2) C^(2) [1- -^ ] [1- -^ ]

A-(0) S-(2) S-(l) , S,(0)

[1- -V-] [1- -^o—] [1- -^o—] [1- ^V-l (3-11)

and

n-1 ^2^^) 2

Ej^(2nAT) = Ej(nAT) Sj^(n) Cj^(n) n [1 ^—

1

J— J.

S-(j) - A (n) A (0)

[1- A—1 [1- -V-] [1- -v-1

S,(n) S,(0)

[1- ^2—^ f^" -^2~^ ^^"^^^

76
From eq (3-12) we may derive the following approximate recurrence relation;

E,(2nAt) S,(n) C, (n) A, (n) A,(n-1) ^ ~ ^ ^ E^(2(n-1)AT) S^(n-l) C^(n-l) [1- -^—] [1- -^

A„(n) A„(n-1) S, (n) S,(n-1)

[1- -^5—] tl- -^-o ] [1- -^—] [1 ^-o 1

S„(n) S^{n-1)

[1- -^o—] [1- -^-o ] (3-13)

and for the final k element

E,(2kAT) S, (k) C, (k) A, (k) ^ ^ ^ [1- Ar-] Ej^(2(k-1)AT) 2 S^(k-l)C^(k-l)

A (k-1) A (k) A (k-1) S (k)

[1 ^ ] [1 ,r-] tl ^ 1 [1 ^3

S,(k-1) S-(k) S-(k-l)

[1- -^ ] [1- -^-^r—'\ [1- -^-o ] (3-14)

Successive applications of the above equations give

E. (2kAT) S, (k) C, (k) k 1 A, (n) _ A„(n) _ — = — = n n- — 1 n—= 1 Ej^(O) S^(0) C^(0) ^^^

^^

2

J ^-^ 2

J

S,(n) S,(n) , A,(k) A^(k)

[1- ^V-] [1- -^-^r-] [1- -^o—] [1- -^o—

S (k) S (k) A (0) A (0)

[1- % ] [1- -^o—] [1- -^—] [1 ^—

5,(0) S^(0)

[1- -Ar—] [1- -V-] (3-15)

The two-point backscatter attenuation L-j, in decibels, can now be defined for

k elements as

17
-10 E {2kAT)

L3 =— log [^-(oyl. (3-16)

'h

With the aid of eq (3-15) this may be written as

-L^ S^(k) C^(k)

-5- = -^"^ ^i7m"5^ro) ^

A^(k) A,(0)

+ loq [1- -^2 1 + 1°^ fl S ^

S,(k) S (0)

- log [1 ~—] + log [1 ^2—

1

A^(k) A^(0)

+ log [1- —2—^ •" ^^^ fl~

S^{k) S^(0)

+ log [1- ——] + ioq [1 2—

1

k=l A (n) S (n) + 2 I (log [1 ^—

]

+ log [1 ^—

]

n=l

k=l A,(n) S,(n)

+ 2 I (log [1- -^—

]

+ log [1- -^—

]

(3-17)

n=l ----_

If the optical waveguide has reciprocal bulk properties, that is

Aj^(n) = A2(n) (3-18)

and

S^(n) = S2(n) (3-19)

for all n, then from eq (3-8), we nay put eq (3-17) in the form

18
S(k) C '(k)

L3 = L^ + 1^ log [ s(0) c.^(0) ^ • (3-20)

We have dropped the subscripts on S since there is now no directional

dependence of the scattering. Under the additional condition that the

product of the scattering loss and capture fraction is the same at the first

and last observation points, or

S(k) C^(k) = S(0) C^iO), (3-21)

the two point backscatter attenuation agrees with the insertion method; L^ = L^. Some computer generated examples which illustrate this important result

are presented in section 4.

In cases where eq (3-21) is not valid (but the waveguide is reciprocal),

for example where two dissimilar fibers are joined, an exact identification

between L^ and L3 can still be made. This is done by interchanging the input

and output ends of the fiber and repeating the backscatter measurements.

This can be shown as follows: We denote by a prime the second set of

measurements where input and output ends are interchanged from the first set

of measurements (unprimed) given in eq (3-20).

S(k) C,(k)

L3 = L, +l^log [ s(0) c)(0) ^ (3-22)

S ' (k) C (k)

L- = L- + lA log [ 3.(0) c'(0) ^ (3-23)

but

L[ = L^ (3-24)

and, by definition.

S'(k) = S(0) (3-25)

19
S'(0) = S(k) _ (3-26)

q(k) = C^(0) (3-27)

By taking the mean of eqs (3-22) and (3-23) we have

L^ = ^/2 (L3 + L^) (3-28)

The arithmetic mean of the two backscatter measurements is equal to the

attenuation. This relation is expressed in decibels. The corresponding

equation in terms of the signals Ej^(t) is

_,f. E. (2kAt) E'(2kAT)

h - V 1°^ t E.(o) EMO) ^ <3-29)

In all the foregoing discussion we have used energy variables rather

than power vairiables in order to avoid computational complexities resulting

from the singular nature of our assumed interrogating

pulse P-:(t) = E_6(t). Henceforth we will approximate the backscatter power

due to the n " element, and observed at

time 2nAT, as P. (2nAT) = E (2nAT)/2AT, and will delete the explicit

labeling 2AT. With this understanding we write Pu^(n) = E, (n).

3 .3 Least-Squares Backscatter Attenuation

We have seen in section 2 that the expected temporal dependence of the

backscatter signal for a uniform fiber is an exponential decrease with decay

constant 2 av . It is reasonable, therefore, to estimate the attenuation from

experimental data by classical discrete curve-fitting techniques. The usual

approach [12], is to use a least-squares fit. The techniques of

least-squares curve-fitting are discussed in standard references [13] and

only some of the basic concepts will be mentioned here. The least-squares

program minimizes the sum of squares of deviations of the backscatter

variable. The correlation coefficient r is a measure of the goodness of fit

of the given data to the assumed exponential dependence. The correlation

coefficient varies between 1.0, for a perfect association between calculated

and experimental values, and 0.0 for a complete random relation. The

20
magnitude of r has rather little significance here except to demonstrate the

fact that low values of the correlation coefficient imply that other methods

(in particular, piecewise fitting) may be more appropriate. The residuals

may also be used in this connection as a diagnostic. Residuals represent the

difference between the calculated least-squares value of the scattering

variable and the experimental value. Examination of the residuals often

provide information on the type and location of imperfections.

In using the least-squares approach in the present discrete context, it

is necessary to disregard the first and last points in order to obtain exact

agreement with the direct attenuation. This is a result of the details of our scattering model which assigns a backscatter power value midway between

adjacent values at points where discontinuities occur. The relation between

the best fit of the exponential decay constant to the backscatter signal in our discrete model is obtained from eq (3-15). Assume all elements have

identical, but not necessarily reciprocal, loss values: A2(n) = A^ , A2(n) = A2, S-]^(n) = S^, S2(n) = S2, C-j^(n) = C-j^ for all n. Then eq (3-15) becomes,

for a constant (unity) fiber length and increasing number of elements, k,

PK(k) A, ^^ A. 2^ S, S„ ^^ m Pr(OT=iiS tl-2^] [1-/] [1-2^1 tl-/] , (3-30)

b

but the loss values per element are defined to be

Aj^ + S^ = ctj^/k (3-31)

and

A2 + S2 = a^/k (3-32)

Equation (3-30) then becomes

Pjj(k) = P^{0) exp [-(a^ + a^] (3-33)

by definition of the exponential function. The least-squares method yields

the best fit <a>, to this e? considered a, = a-, so thatthe best fit <a>, to this exponential. For the reciprocal case usually

21
5.0

2.5

® (D ©

2 log

Pb(n)

Pb(n-I)

log

vcf^i...

X3

CL

o

2 log

-2.5

5.0-

-7.5

-10.0

Pb(n-I ) p, (0) "•i'ii:

Pb(n+1) C=S

2 '°9 ^I^TT) + '°5 cfif

= A

2 log

*'*^*;-^

P|3(n+1)

1-

r 50

Scattering Element

100

Figure 7. Logarithm of backscatter power as a function of scattering

element for a fiber with three distinct loss regions.

22
13.0

10.5

© ® ®

I

8.0

CL

O

5.5

3.0

2 log

Ph(n)

log

C^S3^3

Pb(n+1) "^^CgSs

0.5 '""^^^^

Pu(n + 1) C5S5

A = 2 log s^ttttn + log ^

^

'^^'^^He...,,^ i

Pb(n-I) qsi

-2.0

50 100

Scattering Element

Figure 8. Complementary loss signature for fiber in figure 7 corresponds to reversing the fiber end for end.

This

23
<a>j^ = 2(a^) . (3-34)

The least-squares method used with our model is therefore expected to agree

with the theory given in section 2 for sufficiently large number of points

and uniform reciprocal fiber. Departure from uniformity produces an effect

on attenuation values that is most conveniently demonstrated by our computer-generated examples given in section 4.

3 .4 Piecewise Least-Squares Backscatter Attenuation

When the least-squares technique is employed for estimating loss in

fibers which consist of more than one uniform region connected sequentially

along the length of the guide, accuracy is improved if the curve fitting is

done in a piecewise manner. This is illustrated in figure 7. Here is an example of two dissimilar regions, A and C, joined at an interface, region B,

which has scattering and capture properties different from the bulk

properties at either end. The loss values chosen for this example are given

in run 650 (section 4.2). Curve fitting to the overall guide in this case will generally give rather poor agreement with the insertion loss value.

However, taking a least-squares fit to regions A and C successively giving,

L2(A) and L2(C), and estimating the total attenuation in region B, A, as indicated by the arrows in figure 7, gives a more accurate value of

attenuation. In this approximation, the total piecewise attenuation L^ is

L^ = L2(A) + L2(C) + A. (3-35)

We will show, however, that these approximations may contain significant

errors in cases where the scattering parameters and capture fractions are not

identical in the two uniform regions (A and C) . When using this approach to determine the insertion loss of couplers or splices, it should be kept in mind that at such points the pulse can be

relaunched with a different mode distribution of energies. Non-equilibrium

conditions can introduce significant complexities in the interpretation of

the backscatter signals.

3 .5 Algorithm for Backscatter Comparisons

The discrete representation for fiber attenuation developed in the

previous sectons is suited for calculations on a digital computer. A program

24
was developed which allows the transmitted power P^(n) and backscatter power

P|^(2n) to be calculated for each successive element n in the fiber of figure

6 when selected values are chosen for Aj^(n), A2(n), S-|^(n), S2(n), and

Cj(n). This was intended to simulate various types of imperfections,

defects, fractures, and general departures from a uniform fiber. Under these

conditions the different values for total attenuation, denoted by L-^, L2, Lj,

and L^ can be compared. This is done by evaluating eqs (3-8), (3-17),

(3-35), as well as the least squares value. We have assumed that the direct

attenuation, L-,, yields the "correct" value since it can be measured directly

(under specified launch conditions and for a given length). The differences

L-]^-L2, L-j^-L^, and L-^-h^ are then considered to be "errors" in estimating this

attenuation

4 . COMPUTED EXAMPLES

This section contains examples of several types of loss pathologies and

the effect they have on backscatter attenuation estimates.

4.1 Point Defects

Examples in this section simulate the condition in which defects occur

in a region of the fiber AX which is smaller than or comparable

to vAT where AT is the assumed input pulse duration. The fiber is otherwise

uniform. This type of perturbation could include such inhomogeneity sources

as bubbles, isolated contamination regions, particulates, and cracks.

25
0.15

LpL2

CJB

0.0

-.085

FIBER LENGTH

Figure 9. Least-squares error for a point absorption-loss imperfection as a function of its location along the fiber. Details are given in run 500.

26
RUN 500

VARIABLE 1

K^ = 0.004 n i^

A-,^ = 0.040 n = 4

Aj = A3_

PARAMETER VALUES:

0.5 Fiber Length

^1 Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004 to 0.040

0.004 to 0.040

0.005 to 0.041

0.005 to 0.041

0.001

0.001

0.005

100

COMPUTED RESULTS;

L, Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L,

^1"^2

L^ Piecewise attn.

L1-L4

L, Two-point attn.

^1"^3

2.309676 dB

2.191924 dB

0.9978675

0.117752 dB

2.309676 dB

0.0 dB

2.309676 dB

0.0 dB

REMARKS ;

Run 500 simulates a fiber with a point defect possessing excess absorption loss, the other parameters remaining unaffected. The values of

the least-squares attenuation L2 will depend on the location of the defect,

as shown in runs 501, 503 and figure 9. If the loss is sufficiently large

that its presence is indicated on the backscatter signature, then improved

accuracy may be obtained from a piecewise least-squares fit. The two-point

method yields the correct attenuation.

27
RUN: 503

VARIABLE:

Aj^ = 0.004

A-,^ = 0.040

Aj = Aj_

PARAMETER VALUES:

n ^ 95 n = 95

0.5 Fiber Length

.0

K

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004 to 0.040

0.004 to 0.040

0.005 to 0.041

0.005 to 0.041

0.001

0.001

0.005

100

COMPUTED RESULTS:

L-]^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L- ^1" ^2

L^ Piecewise attn.

L1-L4

L^ Two-point attn.

Lj_-L3

2.309677 dB

2.191924 dB

0.9978674

0.117753 dB

2.309676 dB

0.000001 dB

2.309676 dB

0.000001 dB

REMARKS i

See run 500,

This example illustrates for effect of reversing run 500 end for end.

The value of L2 in the two directions is the same.

28
RUNi 501

VARIABLE:

A^ = 0.004 n ?t 50

A-,^ = 0.040 n = 50

A2 = A^

1 - • - 0.040

0.004

1 Q5 1.0 Fiber Length

PARAMETER VALUES:

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004 to 0.040

0.004 to 0.040

0.005 to 0.041

0.005 to 0.041

0.001

0.001

0.005

100

COMPUTED RESULTS!

L-j^ Direct attn.

L2 Least sq. attn.

r Corr. coeff. for L^

Lj^-Lj

L^ Piecewise attn.

L1-L4

L, Two-point attn.

Lj_-L3

2.309677 dB

2.386327 dB

0.9969112

-0.076650 dB

2.309677 dB

0.0

2.309676 dB

0.000001 dB

REMARKS

See run 500

29
0.5

LpL2

dB

0.0

-0.35

FIBER LENGTH

Figure 10 . Least-squares error for a point scattering-loss imperfection as a function of its location along the fiber. Details are given in run 507,

30
RUN! 507

VARIABLE

S-^ = 0.001 n j^ 4

Sj^ = 0.010 n = 4

Sj_ = S2

0.5 Fiber Length

PARAMETER VALUES;

Aj^ Forward absorption loss

A2 Back absorption loss

A2 Total forward loss

A^ Total back loss

Sj^ Forward scattering loss

S2 Back scattering loss

Cj^ Capture fraction

K Number of points

0.004

0.004

0.005 to 0.014

0.005 to 0.014

0.001 to 0.010

0.001 to 0.014

0.005

100

COMPUTED RESULTS;

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L,

^1"^2

h. Piecewise attn.

L1-L4

L^ Two-point attn.

^1~^3

= 2.190782 dB

= 2.431885 dB

= 0.6744755

= -0.241103 dB

= 2.190781 dB

= 0.000001 dB

= 2.190781 dB

= -0.000001 dB

REMARKS ;

Run 507 is analagous to run 500 with the exception that the point defect

involves excess scattering loss, other parameters being held constant. The

effect on L2 differs in sign and magnitude from the previous example. Runs

506, 505 and figure 10 show the variation of L2 as a function of defect

location. As before, if the perturbation is sufficiently large that it can

be detected in the backscatter signature, the piecewise least-squares fit may

be employed. The two-point method yields the correct attenuation.

31
RUN: 505

VARIABLE

!

Sj^ = 0.001 n ^ 95

Sj^ = 0.010 n = 95

PARAMETER VALUES:

0.5 Fiber Length

.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.005 to 0.014

0.004 to 0.014

0.001 to 0.010

0.001 to 0.010

0.005

100

COMPUTED RESULTS:

Lj^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L^

^1~^2

L4 Piecewise attn.

L1-L4

Lg Two-point attn.

L]_-L3

2.190782 dB

1.891291 dB

0.5588888

0.2994910 dB

2.190782 dB

0.0

2.190781 dB

0.000001 dB

REMARKS ;

See run 507.

This run corresponds to reversing the fiber end for end over run 507.

The average value for L2 in the two directions is 2.161588 dB, which is a much improved value over the one-way results, but is still not exactly equal

to L-j^ . As in other cases involving point defects, Lg and L^ give correct

results. We will see in other examples that increased scatter near the

output end of the fiber will yield too low a value for Lj.

32
RUN: 506

VARIABLE

S-^ = 0.001 n ?t 50

S-j^ = 0.01 n = 50

PARAMETER VALUES;

0.5 Fiber Length

H 0.010

0.001

1.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.005 to 0.014

0.005 to 0.014

0.001 to 0.01

0.001 to 0.01

0.005

100

COMPUTED RESULTS:

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L^

L-L-L2

L^ Piecewise attn.

L1-L4

L, Two-point attn.

^1~^3

2.190782 dB

2.212755 dB

0.6270513

-0.021973 dB

2.190781 dB

0.000001 dB

2.190781 dB

0.000001 dB

REMARKS

See run 507

,

33
0.37 +

FIBER LENGTH

Figure 11 . Least-squares error for a point capture fraction imperfection as

a function of its location along the fiber. Details are given in run 660.

34
RUN: 660

VARIABLE

C^ = 0.005 n ?t 4

C-,^ = 0.050 n = 4

0.5 Fiber Length

- 0.050

0.005

.0

PARAMETER VALUES;

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

= 0.004

= 0.004

= 0.005

= 0.005

= 0.001

= 0.001

= 0.005 to 0.050

= 100

COMPUTED RESULTS:

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L^ L^-L2

L^ Piecewise attn.

L1-L4

Lo Two-point attn.

L-^-Lg

= 2.151587 dB

= 2.421884 dB

= 0.6739156

= -0.270297 dB

= 2.151587 dB

= 0.0 dB

= 2.151587 dB

= 0.0 dB

REMARKS :

Run 660 illustrates the effect on L2 of a point defect involving

increased capture fraction. An example is a small crack or bubble which

reflects radiation in the back direction. In practice a defect of this sort

would occur in conjunction with other loss mechanisms. The C-j^ effects shown

are similar to and slightly larger than corresponding S, defects of a comparable relative magnitude change. Runs 661 and figure 11 show the effect

of the location of the C-^ defect as a function of fiber length.

35
RUN: 661

VARIABLE

;

C-^ = 0.005 n 5^ 95

C-j^ = 0.050 n = 95

PARAMETER VALUES

0.5

Fiber Length

i0.050

0.005

.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

= 0.004

= 0.004

0.001

= 0.00^

= 0.005

= 0.001

0.005 to 0.050

= 100

COMPUTED RESULTS:

L-j^ Direct attn.

L2 Least-sq. attn.

Corr. coeff. for L^

Lj_-L2

L^ Piecewise attn.

L1-L4

L^ Two-point attn.

^1"^3

2.151587 dB

1.881290 dB

0.554970

0.270297 dB

2.151587 dB

0.0

2.151587 dB

0.0

REMARKS ;

See run 660.

Run 661 corresponds to reversing the fiber end for end. In this case

the average value of L2 in the two directions is equal to L-^.

36
4 .2 Loss Discontinuities

The examples in this section simulate the condition in which different

fiber parameters can be assigned to two distinct regions of the fiber. Two

different fibers joined together at a splice or connector would give a similar response. The difference in loss and scattering parameters used for

illustration purposes are generally exagerated over the values normally

encountered in practice. This was done in order to more clearly observe the

effects on fiber attenuation values. Representative backscatter responses

have been shown in figures 3 and 4. Many of the results obtained in the

examples here can be anticipated from the discussion of section 3.4.

37
RUN: 650

PARAMETER VALUES

:

^1 < n <_ = 0.004 59

^1 = 0.001 n = 60

^1 = 0.002 61 < n _< 99

^1 = 0.001 <_ n < 59

^1 = 0.002 n = 60

^1 = 0.0005 61 _< n _< 99

^1 = 0.005 < n _< 59

^1 = 0.010 n = 60

^1 = 0.0025 61 < n < 99

K ^ 100

COMPUTED RESULTS:

L-,_-L2

L1-L4

^1~^3

1.724346 dB

5.992639 dB

0.8010197

-4.268293 dB

4.734645 dB

-3.010299 dB

4.734645 dB

-3.010299 dB

1 1

0.004

• -

1 1 0.001

0.6 1.0

1 1 • - 0.002

0.0005

1 1 0.6 1.0

1 1 • — 0.010

0.0025

1 1

0.6 1.0 Fiber Length

REMARKS ;

Runs 650 and 651 are complementary; they correspond to reversing the

fiber end for end. From a comparison of these two runs we observe that the

least-squares average attenuation obtained with the ends interchanged

is L_ = 1.776 dB . This is rather surprising agreement with L^ = 1.724 dB,

considering the extreme pathology of the overall fiber. Other examples in

this series exhibit this same close association. As expected, L3 and L4

yield correct values of attenuation only when averaged with the ends

interchanged. Data is plotted in figure 7.

38
RUN; 651

PARAMETER VALUES

Ai = Al =

At =

Si = Si =

C, =

Cl = K =

0.002

0.001

0.004

0.0005

0.002

0.001

0.0025

0.010

0.005

100

£ n £ 38

n = 39

40 <^ n <^ 99

_< n < 38

n = 39

40 _< n _< 99

£ n £ 38

n = 39

40 < n < 99

COMPUTED RESULTS:

L-,_-L2

L1-L4

^1"^3

1.724346 dB

-2.440969 dB

0.3702988

4.165315 dB

1.285854 dB

0.438492 dB

1.285854 dB

0.438492 dB

II

- •

1 1 Q4 1.0

0.004

11

- • 0.010

0.0025

1 1

0.4 1.0 Fiber Length

REMARKS ;

The least-squares value of attenuation is negative here, indicating that

a positive exponential is the best fit to the backscatter data. This effect

is due to the large values of scattering loss and capture fraction near the

output end of the fiber. See run 650, and figure 8. Note the apparent gain

at the discontinuity.

39
RUN: 656

VARIABLE

A-^ = 0.002

Aj^ = 0.004

A2 = Ai

PARAMETER VALUES;

_< n _< 59

60 < n < 99

"1 r

J L

0.004

0.002

0.6 Fiber Length

1.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.002 to 0.004

0.002 to 0.004

0.003 to 0.005

0.003 to 0.005

0.001

0.001

0.005

100

COMPUTED RESULTS:

L, Direct attn.

1^2 Least-sq. attn.

r Corr. coeff. for L,

^1"^2

h. Piecewise attn.

L1-L4

Lo Two-point attn.

^1~^3

1.634001 dB

1.593529 dB

0.9841132

0.040472 dB

1.634000

0.000001 dB

1.634000 dB

0.000001 dB

REMARKS :

Runs 656 and 655 are complementary; they correspond to reversing the

fiber end for end. We note that the least squares attenuation L2 and

correlation coefficient r are the same with the ends interchanged. This is

true for all absorption loss perturbations, but is not the case for

scattering or capture fraction perturbations. Also, changing the absorption

loss by a factor of two has a much smaller effect on backscatter attenuations

than a corresponding change in scattering loss or capture fraction (compare

with runs 657 and 660). The piecewise attenuation h. and two-point

attenuation L^ values are correct.

40
RUN 655

VARIABLE;

A-^ = 0.004 < n < 39

A-j^ = .002 40 <^ n <^ 99

A2 = Ai

PARAMETER VALUES

r -T 0.004

0.002

1 1

D 0.4 1.0 Fiber Length

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.002 to 0.004

.002 to .004

0.003 to 0.005

0.003 to 0.005

0.001

.001

0.005

100

COMPUTED RESULTS:

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr . coef f . for L- LJL-L2

L^ Piecewise attn.

L1-L4

L^ Two-point attn.

^1~^3

1.634001 dB

1.593529 dB

0.9841132

0.040472 dB

1.634000

O.OOOOni dB

1.634000 dB

0.000001 dB

REMARKS

See run 656 for discussion,

41
RUN 657

VARIABLE:

S-^ = 0.0005

Sj^ = 0.001

S2 = Si

_< n < 59

60 < n < 99

0.0005

0.6 Fiber Length

PARAMETER VALUES:

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.0045 to 0.005

0.0045 to 0.005

0.0005 to 0.001

0.0005 to 0.001

0.005

100

COMPUTED RESULTS

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr . coeff. for L^ L]^-L2

L^ Piecewise attn.

L1-L4

L-j Two-point attn.

LJL-L3

2.022336 dB

0.1337266 dB

0.01012404 dB

1 .888609 dB

0.5171862 dB

1.505149 dB

0.5171862 dB

1.505149 dB

REMARKS :

Runs 657 and 658 are complementary; they correspond to reversinq the

fiber end for end. Scattering loss changes are seen to have a much greater

effect on backscatter attenuation values than corresponding absorption loss

changes. Also, increased scattering loss near the output end of the fiber

decreases all of the backscatter-derived attenuations, L2 , L3 , and L4 . The

average least-squares attenuation L„ with the fiber ends interchanged is

2.145956 dB, which as in other examples of this kind, is in fair agreement

with the direct attenuation considering the magnitude of the S-j^ changes.

Also, Lj and L^ yield correct values only when averaged with the ends

interchanged

.

42
RUN: 658

VARIABLE:

S-^ = 0.001 £ n < 39

Sj^ = .0005 40 _< n _< 99

S2 - S-^

PARAMETER VALUES

1 1

0.001

0.0005

1 1 0.4 1.0 Fiber Length

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

.004

0.0045 to 0.005

.0045 to 0.005

0.0005 to 0.001

0.0005 to 0.001

0.005

100

COMPUTED RESULTS

L-^ Direct attn.

L2 Least-sq. attn.

Corr . coef f . for L- L3_-L2

L^ Piecewise attn.

L1-L4

L3 Two-point attn

L1-L3

2.022336 dB

4.158186 dB

0.9039554 dB

-2.135850 dB

3.527486 dB

•1.505150 dB

3.527486 dB

1.505150 dB

REMARKS

See run 657 .

43
RUN; 660

VARIABLE:

Ci = 0.0025

0.005

< n < 59

60 < n < 99

PARAMETER VALUES

1—

r

J L

0.6 Fiber Length

0.005

0.0025

1.0

A4

Si

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

.004

0.005

.005

0.001

.001

0.0025 to 0.005

100

COMPUTED RESULTS:

L-j^ Direct attn

.

L2 Least-sq. attn.

r Corr . coeff. for L2

'^1~^2

L4 Piecewise attn.

L1-L4

L-j Two-point attn.

L1-L3

= 2.151587 dB

= 0.005630 dB

= 0.000017 dB

= -2.145957 dB

= 0.646437 dB

= -1.499519 dB

= 0.646437 dB

= -1.499519 dB

REMARKS ;

Runs 660 and 661 are complementary; they correspond to reversing the

fiber end for end. The effect of capture fraction changes on backscatter

results is very similar to scattering loss changes (in this connection see figures 10 and 11). There is one difference, however. The average (in

decibels) least-square attenuation with the fiber ends interchanged, L» , is

exactly equal to the direct attenuation L-j^ (also see runs 660 and 661). This

is not true in general for perturbations involving absorption loss and

scattering loss. Also, the piecewise attenuation L-^ and two-point

attenuation L. yield correct values only when averaged with the ends

interchanged

.

44
RUN: 659

VARIABLE

C-^ = 0.005 £ n £ 39

C^ = 0.0025 40 j< n j< 99

- 0.005

0.0025

0.4 Fiber Length

1.0

PARAMETER VALUES

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.005

.005

0.001

0.001

0.0025 to 0.005

100

COMPUTED RESULTS

L^ Direct attn

L2 Least-sq. attn.

r Corr . coeff. for L^ ^1~^2

L^ Piecewise attn.

L1-L4

L3 Two-point attn.

L1-L3

= 2.151587 dB

= 4.297544 dB

= 0.9116329 dB

= -2.145957 dB

= 3.656737 dB

= -1 .505150 dB

= 3.656737 dB

= -1.505150 dB

REMARKS ;

See run 660 for discussion. This type of splice cannot be realized

physically without increased radiation loss which in our model would

correspond to increased A-j^ values at the interface region.

45
4 .3 Distributed Loss Effects

Distributed effects in our context are defined to be slowly varying

changes in absorption, scattering or capture fraction as a function of

distance along the fiber. These effects have two origins. The physical

properties of the fiber may change due to the way it was drawn, or the

scattering and absorption properties may change due to mode coupling effects

as the pulse propagates down the fiber. Both of these effects are modeled

here

.
RUN: 30 2

VARIABLE;

S-^ = 0.0005 exp [.00701 n]

S2 = .001

PARAMETER VALUES:

0.001

-0.0005

0.5 Fiber Length

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.0045 to 0.005

0.005

0.0005 to 0.001

0.001

0.005

100

COMPUTED RESULTS

Lj^ Direct attn

L2 Least-sq. attn

Corr . coeff. for L' Lj_-L2

L4 Piecewise attn.

L1-L4

L-j Two-point attn

L1-L4

2.031732 dB

0.5853181 dB

0.9994468 dB

-1.4464139 dB

N.A.

N.A.

0.5853181 dB

-1.4452224 dB

REMARKS :

This run simulates the case where the pulse is launched into a multimode

fiber in which most of the energy initially is carried in low order (and low

scattering loss) modes. Near the fiber end the energy has reached its final

distribution where we assume that S^ = S2. As in other examples where S^^ is

not constant, we note that drastic departures are introduced into the

backscatter-derived attenuation values, even though the scattering loss is a small fraction of the total loss . This type of effect yields low values for

L2, and L4 . Note also that the correlation coefficient is near unity, even

though attenuation errors are very large. This latter result is

characteristic of distributed defects.

47
RUN: 311

VARIABLE: SS-j^ = .002exp [-.00701 n]

S2 = .001

PARAMETER VALUES:

0.5 Fiber Length

0.002

0.001

1.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.005 to 0.006

0.005

0.001 to 0.002

0.001

0.005

100

COMPUTED RESULTS:

Lj^ Direct attn.

L2 Least-sq. attn.

r Corr . coeff. for L- ^1"^2

L^ Piecewise attn.

L1-L4

L^ Two-point attn

.

Lj_-L3

= 2.342053 dB

= 3.842431 dB

= 0.9997911 dB

= -1.5003780 dB

= N.A.

= N.A.

= 3.847202 dB

= -1.505149 dB

REMARKS :

This run is similar to run 302 except here the simulation applies to the case where the pulse is launched into relatively high order (and high

scattering loss) modes, with the final scattering value the same as before.

Not surprisingly, the errors in L2 and L^ are opposite in sign.

48
RUN; 310

VARIABLE: Si

SjL = .005 exp [.007001 n]

S2 - S-^

PARAMETER VALUES

0.5 Fiber Length

0.001

0.0005

1.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.0045 to 0.005

0.0045 to 0.005

0.0005 to 0.001

.0005 to 0.001

0.005

100

COMPUTED RESULTS;

Lj^ Direct attn.

L2 Least-sq. attn.

r Corr . coef f . for L^ ^1"^2

L^ Piecewise attn.

L1-L4

L-j Two-point attn.

Lj_-L3

2.031732 dB

-0.5241989 dB

0.9972125 dB

2.555930 dB

N.A.

N.A.

-0.526582 dB

2.558314 dB

REMARKS ;

This run simulates the case where the intrinsic scattering loss

increases toward the output end of the fiber. We note that, with the

parameters chosen, a positive exponential is the best fit to the backscatter

signal which results in a negative value for L2 . Runs 310 and 303 are complementary; they correspond to reversing the fiber end for end. As we have observed before, the average value of the two-point attenuation L- when

the ends are interchanged yields the correct attenuation. The average value

for the least-squares attenuation in runs 310 and 303 is L_ = 2.0293489 dB,

which is approximately correct.

49
RUN; 303

VARIABLE: Si

S-|^ = .001 exp [-.00701 n]

S2 - S3_

PARAMETER VALUES:

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

1

0.001

Si

_

1 0.0005

0.5 1.0 Fiber Length

0.004

0.004

0.0045 to .005

0.0045 to .005

0.0005 to .001

0.0005 to .001

0.005

100

COMPUTED RESULTS:

L-j^ Direct attn

.

L2 Least-sq. attn.

r Corr . coef f . for L^ ^1"^2

L4 Piecewise attn.

L1-L4

L-j Two-point attn

.

^1"^3

2.031732 dB

3.534499 dB

0.9999385 dB

-1.502767 dB

N.A.

N.A.

3.536882 dB

-1.505150 dB

REMARKS

:

See run 310 .

50
RUN: 315

VARIABLE: A^

A-j^ = .002 exp [.00701 n]

A2 = .004

PARAMETER VALUES

0.004

- 0.002

0.5 Fiber Length

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.002 to 0.004

0.004

0.003 to 0.005

0.005

0.001

.001

0.005

100

COMPUTED RESULTS

L-^ Direct attn.

L2 Least-sq. attn.

r Corr . coeff. for L. L]_-L2

L^ Piecewise attn.

L1-L4

L-j Two-point attn.

L1-L3

= 1.671574 dB

= 1.906803 dB

= 0.9991511 dB

= -0.235229 dB

= N.A.

= N.A.

= 1.911580 dB

= -0.240006 dB

REMARKS :

This run simulates the case where the pulse is launched into a multimode

fiber in which most of the energy initially is carried in low order (and low

absorption loss) modes. Near the fiber end the energy has reached its

equilibrium distribution where we assume that A-^ = A2 . Even though the

absorption is the dominant component of the total loss, the effect of

absorption changes on backscatter values is much smaller than comparable

scattering loss changes. This type of effect yields high values for L2 and

L^ with a near-unity correlation coefficient.

51
RUN: 316

VARIABLE : A-^

A^ = .008 exp [-.00701 n]

A2 = .004

PARAMETER VALUES:

0.5 Fiber Length

0.008

0.004

.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004 to 0.008

0.004

0.005 to 0.009

0.005

0.001

0.001

0.005

100

COMPUTED RESULTS:

Lj Direct attn.

L2 Least-sq. attn.

r Corr . coef f . for L^ ^1~^2

L4 Piecewise attn.

L1-L4

L-j Two-point attn.

L3_-L3

2.914955 dB

2.523689 dB

0.9980565 dB

0.391266 dB

N.A.

N.A. ^

2.53327 dB

0.381685 dB

REMARKS ;

This run is similar to run 315 except here the simulation applies to the case where the pulse is launched into relatively high order (and high

absorption loss) modes, with the final absorption loss as before. The

backscatter attenuations L2 and L^ are too low in this case.

52
RUN 306

VARIABLE: Ai

A-^ = .002 exp [.00701 n]

A2 = hi

PARAMETER VALUES

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

-

I

0.004

Al

^^^^^ —

1 0.002

0.5 1.0 Fiber Length

0.002 to 0.004

0.002 to 0.004

0.003 to 0.005

0.003 to 0.005

0.001

0.001

0.005

100

COMPUTED RESULTS

L-j^ Direct attn

L2 Least-sq. attn.

r Corr . coeff. for L2

Lj^-L2

L4 Piecewise attn.

L1-L4

L^ Two-point attn.

L1-L3

1.671574 dB

1 .662020 dB

0.9955431 dB

0.0095540 dB

N.A.

N.A.

1.671573 dB

0.000001 dB

REMARKS

This run simulates the case where the intrinsic absorption loss

increases toward the output end of the fiber. No large errors are encountered with this type of perturbation. Runs 306 and 307 are complementary; they correspond to reversing the fiber end for end. The least

squares attenuation L2 is the same for both cases, and both are slightly

low. The two-point attenuation L3 yields the correct value in each case.

53
RUN: 307

VARIABLE: h^

A-^ = .004 exp [-.00701 n]

Ad = Ai

PARAMETER VALUES

:

0.5 Fiber Length

1.0

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattermq loss

Rack scattering loss

Capture fraction

Number of points

0.002 to n.004

.002 to .004

0.003 to 0.004

.003 to .004

0.001

.001

0.005

100

COMPUTED RESULTS:

L-]^ Direct attn

.

L2 Least-sq. attn.

r Corr. coef f . for L^ L]^-L2

L^ Piecewise attn.

L1-L4

L^ Two-point attn.

Lj_-L3

1.671573 dB

1.662020 dB

0.9955432 dR

0.0095530 dB

N.A.

N.A. ^

1.671573 dB

0.0

REMARKS

:

See run 306

.

54
RUN: 800

VARIABLE: Ci

C-^ = 0.005 exp [-. 00701 n]

PARAMETER VALUES;

0.5 Fiber Length

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.004

0.005

0.005

0.001

0.001

0.0025 to 0.005

100

COMPUTED RESULTS:

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr . coeff. for L- Lj^-Lj

L^ Piecewise attn.

L1-L4

L^ Two-point attn

^1"^3

2.151587 dB

3.656738 dB

1.000000 dB

-1.505151 dB

N.A.

N.A.

3.656737 dB

-1.505149 dB

REMARKS ;

This run simulates the case where the intrinsic capture fraction is

lengh dependent. Variations in C-|^ produce errors in L2 and Lo which are very

similar to those produced by S-^ changes. That is, the backscatter-derived

attenuations are very sensitive to these changes. Runs 800 and 801 are complementary; they correspond to reversing the fiber end for end. The

average value of L^, and L^ when the ends are interchanged agree exactly

with the direct attenuation h-^ .

55
RUN: 801

VARIABLE: Ci

C^ = .0025 exp [.00701 n]

PARAMETER VALUES;

K

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

-

1 0.005

Ci

=^

1 0.0025

() 0.5 1.0 Fiber Length

0.004

0.004

0.005

0.005

0.001

0.001

0.0025 1to 0.005

100

COMPUTED RESULTS:

Lj^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L^

^1"^2

L^ Piecewise attn.

^1~^4

L2 Two-point attn.

Lj_-L3

2.151587 dB

0.646437 dB

1.000009 dB

1.505149 dB

N.A.

N.A.

0.646437 dB

1.505149 dB

REMARKS

:

See run 800.

56
RUN: 326

VARIABLE: Ai^A2

PARAMETER VALUES;

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

0.004

0.008

0.005

0.009

0.001

0.001

0.005

100

0.008

0.004

0.5 1.0 Fiber Length

COMPUTED RESULTS:

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L^ ^1"^2

Piecewise attn.

L1-L4

Lo Two-point attn.

L1-L3

= 2.151587 dB

= 3.014078 dB

= 1.000000 dB

= -0.862500 dB

= N.A.

= N.A.

= 3.014078 dB

= -0.862500 dB

REMARKS ;

In this example the absorption loss in the forward direction is not

equal to the reverse direction due to different modal energy distributions.

We may conclude from an examination of runs 326 and 327 that both

backscatter-derived attenuation values L>2 ^^'^ ^4 give the arithmetric mean in

decibels of the forward and back direct attenuation. That is, both L2 and L^

depend only on the total round trip loss, independent of interchange of

forward and reverse absorption components. These results are to be expected

from the defining equations given section 3. Also a unity correlation

coefficient does not insure good agreement of L^^ and L2.

57
RUN: 327

VARIABLE; Aj^^Aj

PARAMETER VALUES;

K

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

Ai

0.008

0.004

0.009

0.005

0.001

0.001

0.005

100

0.5 Fiber Length

0.008

0.004

1.0

COMPUTED RESULTS;

L-j^ Direct attn.

L2 Least-sq. attn.

r Corr . coeff. for L,

L^-Lj

L^ Piecewise attn.

L1-L4

Lj Two-point attn.

L1-L3

3.876569 dB

3.014078 dB

1.000000 dB

0.862491 dB

N.A.

N.A.

3.014078 dB

0.862491 dB

REMARKS

See run 326.

58
RUN: 312

VARIABLE ; Sj^^Sj

PARAMETER VALUES:

K

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

'

0.001

0.0005

1 0.5 1.0 0.004 Fiber Length

0.004

0.0045

0.005

0.0005

0.001

0.005

100

COMPUTED RESULTS:

L-]^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L^ ^1"^2

L^ Piecewise attn.

L1-L4

Lg Two-point attn.

Li —L-i

1.936531 dB

2.044060 dB

1.000001 dB

-0.107529 dB

N.A.

N.A.

2.044059 dB

-0.107528 dB

REMARKS :

In this example the scattering loss in the forward direction is not

equal to the scattering loss in the reverse direction which could possibly be

due to different modal energy distributions. By comparison with run 313 we

see that L3 is equal to L2 and both depend only on the round trip loss,

independent of interchange of forward and reverse scattering components. The

unity correlation coefficient does not insure good agreement of Lj^ and 1,2'

59
RUN: 313

VARIABLE ; S-^^Sj

PARAMETER VALUES;

Forward absorption loss

Back absorption loss

Total forward loss

Total back loss

Forward scattering loss

Back scattering loss

Capture fraction

Number of points

""1

0.004

0.004

0.005

0.0045

0.001

0.0005

0.005

100

0.001

0.0005

0.5 Fiber Length

1.0

COMPUTED RESULTS:

Lj^ Direct attn.

L2 Least-sq. attn.

r Corr. coeff. for L^

Lj^-Lj

L^ Piecewise attn.

L1-L4

L^ Two-point attn.

Lj_-L3

2.151587 dB

2.044059 dB

1.000001 dB

0.107528 dB

N.A.

N.A.

2.044059 dB

0.107528 dB

REMARKS

;

See run 312.

60
cf^

<uo

CL

0) jn

3

3

Q.C

Time

Figure 12. Temporal dependence of the input pulse considered in section 5.

The pulse is truncated so that it extends from t=€ to t=W . The

uniformly spaced time intervals are of duration 2AT = W /M. The

pulse propagates in the sampled waveguide labeled in figure 1.

61
5. EXTENSION OF ANALYSIS TO PULSES OF ARBITRARY SHAPE

Up to this point we have considered the response of a discrete-waveguide

model to an input pulse whose temporal power dependence is approximated by a delta function. In this section we will indicate how the present analysis

can be extended to a more general input pulse shape. The results of the

foregoing computer modeling will not be altered significantly so long as the

pulse length expressed in meters is small compared to the regions of the

waveguide over which the scattering and loss properties vary appreciably.

Nevertheless, it is desirable from an experimental point of view to obtain as much energy in the pulse as possible, and this often is practical only for a longer pulse. We will therefore give a few examples of the effect of a nonzero width pulse on the backscatter response.

In order to simplify the following analysis, we will consider a discrete

pulse which is sampled at twice the time interval previously defined in

connection with the waveguide of figure 6. This factor of two comes about as

a result of the fact that the backscatter signal must travel down the fiber

and return, while we have taken the unit of time AT to be the transit time across an element in one direction only.

Referring now to figure 3, we have a input pulse P- (t) whose amplitude

is sampled at intervals 2 AT = 2X /vk where v is again the group velocity of

the pulse in the fiber, X the assumed fiber length and k the number of

scattering elements in the waveguide model. The input pulse is taken to have

M intervals, that is we consider P. (t) to be composed of a sum of discrete

pulses of energy P. (2mAT)2AT, with _< m _< M.

We will now examine the response of our discrete fiber to this sampled

pulse. In section 3 it was shown that the response of our sampled fiber to

an impulse input P.(t-t ) = 6(t-t ) at time t^ is given by the normalized

quantity B^ (t-t^). Since we are considering a linear system, the response

to the input P. (t )2AT6(t-t ) is P. (t )2ATE^{t-t ) and the overall output

backscatter power

in

of the fiber

n as inbn a function of time is given by

P^{t) = I P.{tr)E^(t-t^) (5-1)

Making the substition t-t = t' and using the fact that the time intervals

are digitized, t = 2mAT, and t' = 2nAT. eq (5-1) becomes

m

I n=o P^(2mAT) = P^[2AT(m-n)] E^(2ATn) (5-2)

62
5.2 ^

2.5 ^

-.2

-2.9

o

-5.6

-8.3

-1 1 .0

Figure 13. The impulse response function for a fiber with three loss

regions. The logarithm of the backscatter power is plotted as a function of time, with the response normalized to unity time.

63
QO

-12

Figure 14 The logarithm of backscatter power as a function of time for a square pulse input (inset) with the impulse response given in

figure 13.

64
.a

Q_

O

-12

Figure 15. The logarithm of backscatter power as a function of time for a pulse with Gaussian-shaped trailing edge (inset). The impulse

response function is given in figure 13

65
Q_

CDO

TIME

Figure 16 . The logarithm of backscatter power as a function of time for a Gaussian-shaped input pulse (inset). The impulse response

function is given in figure 4.

66
where the summation is taken over all non-vanishing terms as indicated. With

the understanding that each time interval is of duration 2AT, we have finally

m

I n=o

%irn) = i P^(m-n) E^(n) (5-3)

Figures 14, 15, 16 and 17 show the logarithm of the backscatter power,

expressed in decibels, for several input pulse shapes to a fiber with three

loss regions, the n=60 region consisting of a single point. Figure 14 is the

impulse response function E^(t). Figure 15 shows the corresponding output

power P^(m) for the same fiber excited by a square input pulse of unit height

as shown in the inset of the figure. The location of the discontinuity is

still obvious. Also, the optical backscatter power is increased by about

10 dB, as expected. Figure 6 shows the response to a pulse with an abrupt

leading edge and Gaussian trailing edge. Here again the location of the

discontinuity is evident. However, in figure 7 we see that the response to a Gaussian-shaped pulse tends to obscure the precise location of the fault as well as distort the impulse response function signature. Accurate loss

measurements under such conditions require the deconvolution of P^(m) with

the exciting pulse shape. While such techniques are straightforward in

principle, they often produce results that are difficult to interpret when

the signals are noisy. Such procedures will not be pursued here.

6. DISCUSSION OF RESULTS

The principal effects responsible for the discrepancy between the

attenuation values determined from the insertion method and the backscatter

methods are

(1) Different properties of the fiber in the forward and reverse directions. This nonreciprocity can be due to a different

distribution of modal energy in the two directions, or to physical

properties affecting the localized numerical aperture of the

fiber.

(2) Non-uniformity of loss parameters along the length of the fiber.

Scattering variations particularly can produce large differences in

the calculated backscatter value.

(3) Excessive pulse durations. The source pulse lengths (expressed as a length in the fiber) should be much less than the length of the

region over which appreciable loss (or capture fraction changes)

occur. Narrow pulses permit identification of the limits of uniform

67
loss regions, but at the expense of degraded SNR. The SNR is

effected in two ways: the backscatter signals are proportional to

pulse energy, or pulse width for constant peak power, and in

addition increased electronic bandwidth is required for narrower

pulses. Obviously, some kind of experimental tradeoffs are required

as to input pulse width.

(4) Signal-to-noise considerations. Backscatter signals are inherently

small due to the nature of the scattering involved, and SNR

limitations may be the determining factor in the utility of this

approach.

From the discussion of the OTDR theory and examination of sections 4 and

5, we can infer the following:

(1) In the case of reciprocal waveguides whose loss and scattering

properties as well as localized capture fractions are not length

dependent, we can expect the direct attenuation values to agree

exactly with backscatter-derived attenuation values (ignoring SNR

considerations and any non-zero width of the interrogating pulse).

(2) In cases where the fiber is uniform, but the loss properties differ

in the forward and reverse directions, both the least-squares

backscatter attenuation and the two-point backscatter attenuation

will yield the average (in decibels) of the total attenuation (sum

of absorption + scattering loss) in the two directions.

(3) Where lossy point defects are encountered and where fibers of

different properties are joined, the piecewise least-squares method

is preferred to a least-squares fit over the entire fiber. In such

cases, the piecewise fit agrees with the insertion value only if the

scattering loss and capture fraction are the same in the two regions

to be fit.

(4) The two-point backscatter attenuation in some cases can agree with

the insertion attenuation even if the fiber is not uniform. The

conditions for equality are: the impulse response function must be

known; the back/forward losses are identical? and the scattering

loss and capture fractions are the same at the two measuring

points. If the scattering loss-capture fraction criterion is not

met, the two-point method still gives the correct value if the

average (in decibels) of the attenuation values is taken from the

two ends of the fiber.

(5) The correlation coefficient is not always a good indication of

agreement between the least-squares and insertion attenuation

values. This is graphically demonstrated in the examples of section

4.4 where distributed scattering losses as a function of length

produced large L-j^-L2 differences, but correlation coefficients near

68
unity. However, the correlation coefficient is a good indication of

agreement in cases involving reciprocal, uniform fibers where errors

are introduced due to random noise.

(6) A short rise time on the exciting pulse is important in the

identification of regions of differing uniform loss so that the

piecewise fit method can be applied. The trailing edge of the pulse

is less significant (section 5)

(7) The residuals to the least-squares fit, the backscatter signal as a function of time, and, to a lesser extent, the correlation

coefficient, in combination often provide a signature which can be

used as a diagnostic to identify a particular class of perturbations

as well as their location. In some cases this information may be

employed to judiciously pick the regions for a piecewise fit. A

detailed analysis of these defect signatures will be the basis for a future NBS internal report.

(8) Departures from scattering-loss uniformity and capture fraction

uniformity produce much larger Ij-^-'L2 discrepencies than departures

from absorption-loss uniformity.

(9) The two-point backscatter method is often more accurate than a least-squares fit. In addition, the two-point method is preferable

from the practical standpoint that it requires less time for an attenuation measurement. However, there is the drawback that this

approach does not provide a defect signature.

7. COMPARISON WITH EXPERIMENT

The inherent accuracy of the backscatter approach for making loss

determinations is difficult to assess from published experimental comparisons

with insertion loss measurements [2,12,17,18]. Usually a detailed knowledge

of the materials properties of the fiber in question, such as index profile,

uniformity and loss characteristics, as well as the precise experimental

conditions under which the probing pulse is launched, is not known. About

the only conclusion which can be drawn from these comparisons is that good

agreement is at least possible with some fibers under some conditions. A few

of these experimental comparisons, at NBS and other laboratories, are summarized in tables II and III. In some of these examples the agreement is

very good.

The data shown in table II represent preliminary results (subsequent

experimental refinements have resulted in improved precision [25]). The

input pulses of about 80 ns full width at half maximum, were launched under

conditions where the waveguide was overfilled.

69
TABLE II. ATTENUATION COMPARISONS, NBS

FIBER

TYPE

MEASUREMENT

ATTENUATION

dB/KM

NUMBER OF

MEASUREMENTS

GRADED

FIBER (1365)

BACKSCATTER

(L2)

10.21

±0.19

7

.63iiin

DIRECT

(L^)

10.41

±0.16

3

STEP

FIBER (1006)

BACKSCATTER

(L2)

15.99

±0.28

5

.63ym

DIRECT

(L^)

16.75

±0.21

3

STEP

(FIBER 1006)

BACKSCATTER

(L2)

13.7

±0.4

24

.87um^

DIRECT

15.2

±0.3

4

^These measurements made by D.L. Franzen on a waveform digitizer with a laser

specifically constructed for backscatter studies [16].

70
TABLE III. ATTENUATION COMPARISONS, OTHER LABORATORIES

(L2) (L^)

LABORATORY FIBER BACKSCATTER, dB/KM INSERTION, dB/KM DIFFERENCE, dB/KM

CSELT^ STEP 5.026 5.9 -.9

±.04

HUGHES^ 7.9 8.2 -.3

^Reference [17]

^Reference [12]

71
Mode strippers were used to eliminate radiation propagating in the

cladding. The scatter is indicated by the one sigma limits.

In addition to the results presented in the tables, some recent data

from the CSELT group [18] show backscatter-direct loss comparisons on sixteen

different fibers with an average difference of about 0.01 dB/km on fibers

with nominal attenuation in the 3 to 4 dB/km range. Excellent agreement of

this sort inspires confidence in the possibilities of backscatter

measurements when good fibers are under test and measurement conditions are carefully controlled.

8. CONCLUSIONS

We have shown by means of computer modeling that backscatter-derived

attenuation values can agree exactly with the direct attenuation values under

certain ideal conditions. We have also examined a number of fiber

perturbations which could be encountered in practice to upset these ideal

conditions, and noted their effect. The perturbed backscatter attenuation

values can be either higher or lower than those determined from direct, or insertion, methods. The precise agreement is a function of many factors; the

quality of the fiber employed, location and magnitude of defects, type of

waveguide excitation assumed, duration of the input pulse, signal-to-noise

ratios, and the details of the backscatter data analysis.

It would appear that there are two distinct aspects relating to the

problem of accurate backscatter-derived attenuation measurements. The first

is the inherent quality of the fiber. Errors will be minimized if absorption

loss, scattering loss and localized numerical aperture are not functions of

fiber length. The second consideration involves measurement procedures. We

feel that, before meaningful experimental backscatter-direct attenuation

comparisons can be made, it will be necessary to establish standard

measurement conditions and procedures. This is a consequence of the well

known fact that absorption and scattering loss in multimode optical

waveguides are both mode dependent quantities and that mode coupling produces

complex and length-dependent loss characteristics. In order for attenuation

results to be reproducable, and comparisons to be significant, measurements

must be made under conditions that initially excite that set of modes at the

input which most closely approximate the equilibrium or steady state modal

72
2 • . energy distribution. This can be done, in principle at least, by carefully

controlling launch conditions such as radiation spot size and NA. Another

approach is to employ devices which can be used with fibers to simulate an equilibrium mode distribution (EMb). They are variously referred to as mode

filters, mode scramblers, mode eliminators, or, the term we prefer,

equilibrium mode simulators (EMS). The problems associated with EMS as well

many practical arrangements, such as long pigtails and S-shaped channels,

have been discussed by numerous authors [19,20,21]. These devices do not in

themselves produce a steady state power distribution, but in combination with

a set of standard excitation conditions they can produce an environment

which approximates the desired steady state. At the present time no universally accepted EMS has been established. However, efforts are currently underway by the Electronic Industries Association (EIA) , NBS, and

other organizations to reach agreement on the precise definition of

equilibrium mode distribution, as well as suitable EMS and other measurement

procedures. If, as seems likely, EMS devices are adopted for use in

conjunction with standardized direct attenuation procedures, we can expect

some inherent discrepancy between this and backscatter-derived attenuation.

This is a result of the fact that the OTDR signal originating from the

Rayleigh scattering in a given fiber element is not launched in the back

direction with the same sort of EMD conditions; rather, this signal

approximates overfilling the waveguide (exciting all modes equally). As we have seen, nonreciprocal behavior of this sort produces attenuation

variations

When a consensus on EMS, EMD specifications, and measurement procedures

is achieved the author intends to pursue further backscatter-direct

attenuation comparisons. Based on our computer modeling and on preliminary

results at this, and other laboratories, it would appear that, with modern

high-quality fibers and standard techniques, good agreement is still possible

between direct attenuation measurements and attenuation values obtained

indirectly from backscatter signals.

2 . . An equilibrium distribution may, m fact, not exist for some optical fibers

with extremely small mode coupling. Fibers have been observed in which the

output NA continually decreases even after two to three km, indicating that

no steady state condition has been reached.

73
9 . REFERENCES

[1] Marcatili, E.A.J. , Factors Affecting Practical Attenuation and

Dispersion Measurements (Digest of Technical Papers Presented at

the Topical Meeting on Optical Fiber Transmission II, Williamsburg,

VA, Feb. 22-24, 1977)

[2] Stone, F.T., Launch Dependent Loss in Short Lengths of Graded-index

Multimode Fibers, Appl . Opt., 17, no. 17, Sept. 1978, 2825-2830.

[3] Kapron, F.P., R.D. Maurer, and M.P. Teter, Theory of Backscattering

Effects in Waveguides, Appl. Opt., 11, no. 6, Dec. 1972, 1352-1356.

[4] Barnoski, M.K., M.D. Rourke, S.M. Jensen, and R.T. Melville,

Optical Time Domain Reflectometer, Appl. Opt., 16, no. 9, Sept.

1977, 2375-2379.

C51 Personick, S.D., Photon Probe-An Optical Time Domain

Reflectometer, B. S.T.J. , 56j no. 3, Mar. 1977, 355-366.

[6] Smith, R.G., Optical Power Handling Capacity of Low Loss Optical

Fibers as Determined by Stimulated Raman and Brillouin Scattering,

Appl. Opt., llj no. 11, Nov. 1972, 2489-2494.

[7] Crow, J.D., Power Handling Capability of Glass Fiber Lightguides,

Appl. Opt., 13 , no. 3, March 1974, 467-468.

[8] Yariv, A., Introduction to Optical Electronics, [Holt, Rinehart

and Winston, New York, New York, 1971], 38.

[9] Olshansky, R., and Susan M. Oaks, Differential Mode Attenuation

Measurements in Graded-index Fibers, Appl. Opt., 17 , no. 11, June

1978, 1830-1835.

[10] Neumann, E.G., Optical Time Domain Reflectometer : Comment, Appl.

Opt., 17, no. 11, June 1978, 1675.

74
[11] Hillerich, B. On Site Location of Fiber Defects and Evaluation of

Transmission Loss, (Proceedings of the 26 International Wire a Cable Symposium, Cherry Hill, N.J., Nov. 15-17, 1977, 373-379).

[12] Barnoski, M.K., and S.M. Jensen, Fiber Waveguides: a Novel

Technique for Investigating Attenuation Characteristics, Appl

Opt., 15j no. 9, Sept. 1976, 2112-2115.

[13] Draper, N.R., and H. Smith, Applied Regression Analysis [John

Wiley and Sons, Inc., New York, New York, 1966].

[14] Melchoir, H. , M. Eisher, and F. Arams, Photodetectors for Optical

Communication Systems, Proc. IEEE, 5S, Oct. 1970, 1466-1486.

[15] Webb, P.P., R.J. Mclntyre, and J. Conradi, Properties of Avalanche

Photodiodes, RCA Review, 35j June 1974, 234-278.

[16] Franzen, D.L., B.L. Danielson, and G.W. Day, A Simple First

Positive System Nitrogen Laser System for Use in Optical Fiber

Measurements, IEEE J, Quantum Electron., QE-14 , no. 6, June 1978,

402-404.

[17] Costa, Bruno, and B. Sordo., Experimental Study of Optical Fibers

Attenuation by a Modified Backscatter Technique, (Digest of

Technical Papers Presented at the Third European Conference on Optical Communication. Munich, Germany, Sept 14-16, 1977).

[18] Costa, Bruno, C. DeBernardi, and B. Sordo, Investigation of

Scattering Characteristics and Accuracy of the Backscatter Technique

by Wavelength Dependent Measurements, (Digest of Technical Papers

Presented at the Fourth European Conference on Optical

Communication, Geneva, Italy, Sept. 12-15, 1978).

[19] Love, W.F., Novel Mode-Scrambler for Use in Optical Fiber Bandwidth

Measurements, (Digest of Technical Papers Presented at the Topical

Meeting on Optical Fiber Communication, Washington, D.C. Mar. 6-8,

1979).

[20] Stone, F.T., and P.H. Krawarik, Mode Elimination in Fiber Loss

Measurements, Appl. Opt., 18, no. 6, Mar. 1979, 756-758.

75
[21] Tokuda, M. , S. Sikai, K. Yochida, and N. Uchida, Measurement of

Baseband Frequency Response of Multimode Fibre by using a New Type

of Mode Scrambler, Electron. Lett. 13 , Mar. 1977, 146-147.

[22] Keck, D.B., Optical Fiber Measurements, (Digest of Technical

Papers Presented at the Topical Meeting on Optical Fiber

Communication, Washington, D.C., March 6-8, 1979).

[23] Hillerich, B. , Measurement of the Attenuation of Optical Fibers by

Pulse Reflection, Technisches Messen Atm no. 9, 1976, 269-270.

[24] Nelson, M.A. , T.J. Davies, P.B. Lyons, J.E. Golob, and L.D. Looney,

A Fiber-optic Time Domain Ref lectometer. Proceedings of SPIE, 139,

(1978), 93-97.

[25] Chamberlain, G.E., and G.W. Day, NBS Technical Note, in preparation.

75 liuS QOVERNMENT PRINTING OFFICE 1980-677-096/1233
U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-1018

2,Gov't Accessrw No,

TITLE AND SUBTITLE

An Assessment of the Backscatter Technique as a Means

for Estimating Loss in Optical Waveguides

iliiiife^

5. Publication Date February 1980

'I' I ' Mi i i iiii im wii nwmn^n Performing Orpnization tk^ -

AUTHOR(S)

B. L. Danielson

8. Performing Organ. Report No.

PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

16. Pro|8ct/Tas!</Worl< Unit No.

11. Contract/Grant No.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fSfree(, city. St^te, ziP)

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Station, Newark, OH 43055 and

Communications Systems Ctr. , U.S. Army Communications

Command, Fort Monmouth, New Jersey 07703

13. Type of Report & Period Covered

14. ^3ons<wtng AgeRcy Code

SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached. ABSTRACT (A 200-word or Jess tactual summary of most significant information. If document includes a signi ficant bibliography or

literature survey, mention it here.) This technical note addresses some of the problems associated with determining

the accuracy of the backscatter technique as it is applied to the estimation of

attenuation in optical waveguides. The basic theoretical assumptions involved in

optical time domain ref lectometry are reviewed; the effect on calculated loss values

resulting from a departure from these assumptions is then examined. The approach

taken is to employ computer modeling of the various scattering and other loss mechanisms using the bulk material properties of optical fibers. Computer responses

permit a numerical comparison between the direct (insertion) method of measuring

attenuation and several methods of estimating attenuation from analysis of backscatter data. Numerous examples are given of physical effects which can produce

discrepancies in attenuation values calculated from backscatter signals. Also,

some experimental comparisons are made between backscatter-derived and directly

measured attenuation values in step and graded-index optical waveguides. Finally,

the conditions necessary for good agreement between the direct and backscatter

methods are discussed and suggestions for minimizing these errors are made.

KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Backscattering; fiber attenuation; fiber loss; fiber scattering; optical time

domain ref lectrometry ; Rayleigh scattering.

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