Nonlinearity in Fiber Transmission

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1232 PROCEEDINGS OF THE IEEE, VOL. 68 , NO.

10,

OCTOBER 1980

1661 E.

G.

Ramon and M. D. Bailey, “Bitaper star couplerswithup

tion, p. 636, Sept. 1978.

to 10 0 fibre channels,” Electron. Lett., vol. 15, no. 15, p. 432,

July 5, 1979.

[

671 T. Ito et al., “Bidirectional tapered fiber star couplers ,” in Proc.

4t h European Conf Optical Communication, p. 318, Sept. 1978.

1681 W. J. Tomlinson, ‘Wavelength multiplexing n multimode optical

fibers,”Appl. Opt.,vol. 16, no. 8, p. 2180, Aug. 19 77.

1691 K. Kobayashi and

M. Seki

“Micro-optic grating multiplexers or

fiber-optic communications,” in Tech. Dig. Topical Meet. Optical

[

701 R. Watanabe and K. Nosu, “Optical demultiplexer using concave

Fiber Communication, p. 54, Mar. 1979.

grating in 0.7 -0. 9 pm wavelength region ” Electron. Lett., vol.

16, no. 3, p. 10 6, Jan. 1980.

[ 711

Y.

Fujii et al., “Opticaldemultiplexerusingasiliconechelette

grating,” IEEE J . QuantumElectron., vol.QE-16, p. 16 5, Feb.

1980.

1721 R. Watanabe et al., “Optical grating mult iplexer n he1.1-1.5

pmwavelengthregion,” Electron. Lett., vol.16,no.3, p. 108,

Jan. 1980.

Nonlinearity in Fiber Transmission

ROGERS

H.

STOLEN

Abstract-Procedures

are

presented for estimating critical powers for

nonlinear optical processes in single-mode

fiber

transmission systems.

Crosstalk due to

Raman gain in

multiplexedsystems

can

appear at

powers of a few mW. The effects of self-phase modulation and stimu-

lated Brillouin scattering can appeararound

100 mW

while ypical

stimulated Raman

hreshold

powers are a few watts.

I .

INTRODUCT ION

PTICAL fibers can exhibit frequency conversion due to

nonlinear processes leading to loss, pulse spreading,

crosstalk, and in some cases, physical damage, The prob-

lems can be particularly severe in very long, low-loss fibers

because thecentralfeature of fibernonlinearities is theex-

change of length for optical intensity

[

11

- [ 3 ] .

Even

so,

critical or threshold powers have

so

far been high enough to

fy neglecting optical nonlinearities in transmission system

design. Recent interest n single-mode submarine-cable sys-

tems may, however, change this situation. Extremely low-loss

fibers

[ 4 ] ,

higher powers

[ 5 ]

and narrow-band lasers

all

move

toward the regime where nonlinear effects become important.

Other developments such as small fiber cores to increase the

minimum dispersion wavelength [6] and the use of germania

glass [71 would also enhance nonlinear effects.

This paper att empts to establish critical powers for various

onlinear processes and to describe their effects n ransmis-

sion systems. Worst case limitsare relatively straightforward

to calculate but, as will be seen, thesepowers can often be

safely exceeded. The nonlinear processes of interest are

timulated Raman cattering which serves to llustra te he

asic concepts nd could lead to crosstalk

in

wavelength-

ultiplexedystems, stimulated Brillouin scattering which

s the most ikelynonlinear loss mechanism, and self-phase

odulation which leads to pulse broadening. Another process,

arametric four-photon r three-wave mixing, could also

cause crosstalk but will only be discussed briefly since hese

ffects would appear at higher powers.

Manuscript received June 3 1980.

The

author is with Bell Laboratories, Holmdel, NJ 0 77 33 .

11.

STIMULATEDAMAN SCATTERING

A .

RamanAmplification

The Raman interact ion makes a fiber an optically pumped

optical amplifier. The process can be viewed as a three-

wave parametricamplifier in which pump and signal (called

the Stokes wave) are optical waves and the idler is a highly

dampedvibrational wave. The equivalent quantummechan-

ical interpretation is one of stimulated scatteringby the Stokes

photons. Raman amplificationepends onhentensity,

which is the pump power

P

divided by the spot size

A ,

the

interaction length L and the Raman gain coefficient g.

Amplification xp

Yl.

In low-loss fibers the ength can be more than a kilometer

which,rom ( l ) , reduces the powernd gain coefficient

necessary for significant Stokes amplification.

It is not necessary to injecta signal to see the effects of

Raman gain since there is always some light present from

spontaneous Raman scattering. If the powernd length

are large enough, this weak signal can be amplified to he

level of pumpdepletion. Because of theexponential ampli-

fication there is onlya small difference n pump powerbe-

tween negligible conversion and near total conversion of

pum p o signal and it is possible to define a threshold or

critical power

[

81.

16A

gL

p

E .

B.

Fiber Parameters

1 ) Raman Gain Coeffic ient: The Raman gain coefficient for

fused silica

as

derived from the spontaneous Raman spectrum

[9] is shown in Fig. 1. The gain sgiven for a pump wave-

length

of

1.0 pm.

To

convert t o a different pump wavelength

the gain coefficient scales by l / X At 1.Opm a shift of

500

cm-’ corresponds approximatelyto

5 nm.

In

general, dopants such as boron, germanium, or phos-

0018-9219/80/1000-1232 00.75 O

1980 IEEE



STOLEN:NONLINEARITY IN FIBER TRANSMISSION

1 ''''''''''1

0

200 400

600

800 IO00

1200 1400

FREOUENCY

SHIFT

(ern- )

Fig. 1 . Raman gain versus frequency hift or used ilica at apump

wavelength of

1.0

pm. The gain coeffic ient cales inversely with

pump wavelength and it is assumed that polarization is maintained.

phorusdonot appreciablymodify the silica gain spectrum

[ 101 particularly in the small percentages used in extremely

low-loss fibers. One should keep in mind, however, that there

are glasses such as pure GeOz which hasa gain coefficient

9.2X that

of

silica [ 11 There are also many silicate glasses

with

g in

curves which look qui te different from fused silica

[ 121 although the peak gains are not very different.

2 Length: Amplification will decrease inongiber

because of pumpabsorptiondue to fiber loss. This is han-

dled by replacing L in (1) by an effective length

[

13

-

€ I

L e f f=

where

I

is theactual engthand

Q

is the linear absorption

coefficient. In the present discussion we are usually inte r-

ested in he limitwhere Leff = l / ~ . orexample, l /a ina

dB/km fiber is 4.34 km.

3 Power:

The power

P

in (1) is the value at the fiber input

but for pulses there is always a question of whether to use

peak or average power in the gain calculation. We will use th e

average power, since n fiber transmission the fibers are long

and the optical pulses are closely spaced so group-velocity dis-

persion will cause relative slippage of pump and Stokes pulses

which averages the gain.

4

Area: We will use the fiber core area

as

the spot size for

single and multimode fiberswith step or graded boundaries.

More carefulalculationsequire computation

of

overlap

integrals

[

11, [21, 9]but he core-size approximation is

adequate within a factor of two. The approximation is very

good in single-mode fibers at cutoff of the second mode and

forstep-indexmultimode fiberswhereenergy

is

uniformly

distributed through the modes by either the initial excitation

or by mode mixing

[

141. In single-mode fibers below cutoff

the effective area is larger than the core size and in graded ndex

fibers the effective area can be smaller than the core size.

5 Polarization: The Raman gain coefficientn Fig. 1

assumes the ideal case in which the fiber maintains polariza-

tion. By maintaining polarization one usually refers to linear

polarization but hisstatement applies to circular and ellip-

tical polarizations as well. In typical single-mode fibers,

which do not preserve polarization, the polarizations of pump

1233

and stokes waves get out

of

stepand he average Raman

gain s reduced by a factor

of

two

[

151

.

While polarization-

maintaining ibers can be made [ 161, t is likely that such

properties would be sacrificed toobtain minimum loss in

fiber ransmission

so

wewill reduce the Raman gain by a

factor

of

two in all estimates of Raman amplification.

C.

Examples

of

Raman Amplif icat ion

Above thetimulated Ramanhresholdower, light is

effectivelyost by downshifting

to

theStokes Raman fre-

quency. An expression for he thresholdpower as defined

by (2) is given in Table I. Here it is assumed that polarization

is not preserved, thespot size can be approximated by the

corearea, and hat he fiber is sufficiently ong to replace

le^

by I/ in (3). As an example, we choose a single-mode

fiber

of

8j tm core diameter, 0.3-dB/km loss at a wavelength

of 1.5 pm, and a length much greater than 1/a. The threshold

power

is

then 1.7 W which is probably well above practical

power levels for communication applications.

Theeffects of Ramanamplification can appear atmuch

lowerowersn wavelength multiplexedystems where

energy can be transferred

to

a onger wavelength channel.

We choose as a criterion for a critical power that the ampli-

fication ofone wavelength by another should be ess than

1 percent which corresponds to 20-dBcrosstalk. This means

that exp g P , L / A) = 1.01 s gP,L/A

=

0.01 rather than 16 as

for hestimulated hreshold. n he worstcase,one is un-

lucky enough to choose a frequency separation corresponding

to t he maximum Raman gain. This case is used for the expres-

sion in Table

I .

For the same example of an 8 j tm core fiber

at h = 1.5 pm the critical power is then 1.1 mW.

This

shows

tha t it will be essential to keep multiplexed wavelengths sep-

arated by more than 500 cm-'

As a f iial example, we consider reducing the Raman thresh-

old power with feedback supplied by reflection from poorly

matched ends. The condit ion for oscillation is that the round-

tripStokes amplificationequals thecombined transmission

and eflection oss. Note hat backwards Raman gain is the

same

as

forward gain. To simplify the calculation weuse

the case of a

3

km,

10

pm core fiber at h =

1

Opm and assume

the loss at both hepump andStokes wavelengths is 1.0

dB/km. The fiber length is less than the absorption length so

(3)

is

used to calculate an effective ength of 2.17 km. If a

reflectivity of 4 percent is arbitrarilychosen for each end,

the round trip transmission

is

4.0

X lo4

so gain equals loss

at exp 2 g P t L / A)= 2500. Threshold power is now 2.9 W

as

comparedo an 11.8-W single-pass threshold in the same

3-km iber and a hreshold of 5.9 W ina similar fiber of

length much greater than

l/a.

111. S T I M U L A T E DRILLOUIN CATTERING

Stimulated Brillouin scattering can have a peak gain 300

times that for stimulated Raman scattering so threshold can

occur at much lower powers

[

171. The interaction is between

light and acoustic waves rather than with high frequency op-

tical phonons and because of wave vector matching considera-

tions [ I ] [31 Brillouin amplificationoccurs only in the

backward direct ion. The frequency shift is given by

2nV

v

=

(4)

where n is the refractive index and V is the velocity

of

lon-



1234

PROCEEDINGS OF THEIEEE,

VOL.

68, NO. 10, OCTOBER

1980

rABLE I

T h r e s h o l an ar i t i c a lo w e r s or st imulated Raman

s c a t t e r i n g , Raman a m p l i f i c a t i o no ra v e l e n g t h

s e p a r a t i o no r r e s p o n d i n go maximum daman g a i n ,

s t i m u l a t e dB r i l l o u i n c a t t e r i n g , a n d e l f - p h a s e modu-

l a t i o n . a v e l e n g t h A a n do r e i a me t e r d a r e n rm

f i b e r

loss

i s i n dB/km and aserbandwidth

b v

i s i n

GHz. D i s t h e u t ya c t o r .T h e s e u mb e r s r e a l c u -

l a t e dnh e limit w h e r eh ei b e r i s much longer

t h a nh eb s o r p t i o ne n g t h ,h ep p r o x i m a t i o nh a t

t h ef f e c t i v er e a

i s

t h eo r er e a ,n ds s u m i n g

t h a to l a r i z a t i o n

i s

n o ta i n t a i n e d .

As

e x a mp l e s ,

p o w e r s r e i v e n o r n 9 rm c o r ed i a m e t e r i b e rw i t h

a loss of .3 dB/kn a t a wave length o 1 . 5

J m

For

t h e r i l l o u i n h r e s h o l d A w was 1.0 GHz andhe uty

f a c t o r was 0 . 5 o r h ec r i t i c a l p o we r r o m e l f- p ha s e

mo d u l a t i o n .

Stimulated Raman

P t

=

5 . 9 ~ 1 0 - ~ 6 ~ 1 5 a t t s 1.1 w

Raman Amplif ication

P C

=

3 . 7 ~ 1 0 - ~ d ’ 1 5a t t s

1.1

mu

S t i m u l a t e da r i l l o u i n

P t =

4.4X10 3d2A25Av

w a t t s 1 9 0

S e l f - 3 h a s eE o d u l a t i o n

P C = 2 . 2 ~ 1 0 - ~ d ~ A S Dw a t t s 2 mu

gitudinal sound waves. For = 1.O pm in fused silica, us =

17.2 GHz.

Brillouin gain can be quite different depending on whether

the pump linewidth

A u , )

is much less or much greater than

the Brillouin linewidth A v B ) . Brillouin linewidths are typi-

cally around 100 MHz so communications systems will operate

in the limit where Au, >>A u B . In this imit the Brillouin

gain coefficient for silica-based glasses becomes

1.73 X

lo-’’

h 2 A

p

=

cm/W

where h is the wavelength in micrometers, Au, i s the pump

bandwidth (FWHM) in GHz and the constant contains factors

such

as

the elasto-optic coefficient, the density and the sound

velocity. In the opposite limit where Av , <<A u B , A u , in (5)

s replaced by A u B . As for Raman amplification, he gain is

a factor of two higher when polarization is maintained and

(5) assumes this ideal situation.

Threshold power for backward stimulated scattering s

[

81

21A

g L

P =

Equations 5 ) and 6 ) are combined to produce the relation

for threshold power in Table I. It is assumed in Table I that

polarization

is

not maintained which reduces the gain by a

factor of two from the value in (5) [ 151. As an example we

again choose a

0.3

dB/km, 8.O-pm core diameter fiber, a wave-

length of 1.5 pm, and Au, = 1.0 GHz. Brillouin threshold

power is then 190 mW.

Use of the pproximation

A U ,> > A y e

conceals several

complications associated with s timulated Brillouin scattering.

For example, the Brillouin linewidth is not a constantbut

varies

[

181 as approximately

u ; ;

the value corresponding to

h

=

1

.O

pm is 38.4 MHz. Doping modifies AuB in two ways.

First is the direct effect on the intrinsic linewidth of the doped

glass which has not been well studied. Second is the indirec t

effect through the change in the sound velocity which from

(4) leads to a change in the Brillouin frequency shift between

core and cladding because of dopingdifferences [ 151 This

shows up as an inhomogeneousbandwidth which couldbe

as

large as 1 GHz. In weakly doped fibers, however, the effects

should be much smaller and need to be included only if the

pump linewidth is small enough to be comparable to he

Brillouin linewidth.

IV.

SELF-PHASE

MODULATION

There is an intensitydependent broadening of the signal

spectrum romhe nonlinear process known

as

self-phase

modulation. This greaterrequency widthhen increases

pulse spreading through group-velocitydispersion. Thepro-

cess can be understood as adifferential phase shift of the

optical carrierbetween thecenter and the tails of a pulse

due to he ntensitydependent refractive index.The ndex

change is extremely small but the corresponding phase modu-

lation can be significant when added up in a long fiber

[

191.

The effect of the combinat ion of self-phase modulation and

group velocity dispersion can be calculatedn straight-

forward way for simple smooth pulses in single-mode fibers.



STOLEN:NONLINEARITY IN FIBER TRANSMISSION

The problem of establishing a critical power in a transmission

system where the pulses are not simple is less straightforward

and we are forced t o make some educated guesses.

For simple Gaussian pulses the ntensity dependent broad-

ening 6w can be related to the spectral width by

6w

=

0.86

AwAGmw (7)

where A -,= is the maximum phase shift in radians and is pro-

portional to the nonlinea r index, peak power, effective length

and inverse area. Real pulses usually have a frequency width

considerablygreater than a imple transform of the pulse

envelope.Thiscould be due to any combinationof phase

or amplitude modulat ion. We will assume that the frequency

width is entirely due to amplitudemodulation so the pulse

containsmicrostructureon a time scale of l/Aw where Aw

is the pectralwidth.For he mplitudemodulated pulse

thebroadening will be proportional

to

the subpulses

so 7)

should still applywithAw again referring to he spectral

width. There is a questionabout what to use for he peak

power. Even if i t were known, using the height of the strongest

spikewould certainly overestimate the broadening. We will

use the height of the pulse envelope as given by the average

power divided by the duty factor.

To calculate the actual pulse spreading the procedure is to

split the fiber into short sections and alternately calculate the

effects of self-phase modulation and group velocity dispersion

[20].

A

simpler approximateapproach would be to assume

that all the broadening takes place in a length l / a and all the

pulse spreading occurs in the remainder of the fiber. Neither

of these approaches is amenable to a quick estimate.

A simpler estimate of a critical power [191 can be obtained

by assuming tha t the system has been designed for maximum

capacity

as

set by linear pulse dispersion.

A

factor

of

two

additional broadening is then a serious problem. This occurs

at = 2.0 which established ritical eak ower of

XA

L

P C = 1 . 2 x

lo-2-w

(8)

where, and A are uni ts of pm and pm2 L is in km, linear

polarization is preserved, and heconstantcontains actors

such as the nonlinear index. In contrast to Raman gain, the

broadening from self-phase modulat ion is only decreased 17

percentwheninearpolarization

is

not maintained in the

fiber [ 191.

The criticalpower for self-phase modulation as defined in

(8) is written n erms of average power in Table I. Here D

is the duty fac tor and polarization is scrambled. The critical

power for the

8

pm core, 0.3 dB/km fiber at 1.5 pm is then

32 mW.

We have not reated such details as chup whichcould in-

crease or decrease the broadening, or the difference between

the positive and negative side of the minimumdispersion

wavelength. On the negative side of the minimum disper-

sion wavelength there hould be intensi tydependent pulse

narrowing

[

21 1 The pulse should first narrow in time and

then broaden again. This will decrease thenet broadening

from group-velocity dispersion.This ffect might be used

to advantage. Forexample, ata wavelength

of

1.6pm, in

a iber for which the minimum dispersion

is

1.3 pm, he

powers required t o cause narrowing of 100-ps pulses are

on the order f a few milliwatts.

1235

V .

FOUR-PHOTON

MIX ING

Four-photon mixing also called three-wave mixing) is a

parametric interaction which can generate new frequencies

oneitherorboth sides of the pump frequency [ 1 - [ 31

.

Forexample,stimulated Raman

or

Brillouin light can mix

with pump light to produce light on the high frequency side

of thepump. These effects equire phase matching which

can be provided by the differing phase velocities of the wave-

guide modes in a multimode fiber [2 2] . In single-mode fibers

the process appears only at small frequency shifts (-1 cm-’)

where coherence lengths are

of

the orderof kilometers [23].

It appears at present tha t this process will not be a serious

limit n iber ransmissionsystems because of the need for

phase matching n single-mode fibers nd the high powers

requiredn multimode fibers. It should be kept in mind,

however, that experiments with high optical powers and multi-

mode ibersalmost always produce spectra which combine

several nonlinear processes and are made even morecompli-

cated by the presence of four-photon mixing.

VI.

CONCLUSION

The foregoing treatment shows that nonlinear effects can

appear n single-mode transmission ystems at powers of a

few milliwatts. The nonlinear processes most ikely to cause

problems are Raman gain inmultiplexedystems, pulse

spreading due o self-phase modulation, and stimulated Bril-

louin scatteringwith very narrow line sources.

Threshold estimates were based on work done in the visible

and it is clear that here is aneed for measurements n the

spectral range beyond 1pm-particularlywith ources that

approximatehepectral and temporal characteristics of

real systems.

We have concentratedon he deleteriousaspects of fiber

nonlinearities but here are also some significantadvantages.

Forexample, fiber Raman lasers are a new-type of laser-

pumped tunable lasers which utilize both the long interaction

lengths

of

low-lossfibers and he broad Raman bandwidths

of glasses [2] , [3 ]. These lasers sharemanyproperties with

dye and F-center lasers and may someday prove to be compact

and inexpensive sources of tunable radiation in the ear infrared .

Raman lasers in their simple single-pass form have been used to

generate a band

of

optical pulses which have found application

in

fiber loss and pulse dispersion measurements 1241

There may even be direct system applications of fiber non-

linearities. For example, the nonlinear pulse narrowing in the

negative dispersion regime could lead to multiplexed systems

where both the minimum dispersion wavelength at 1.3 pm and

the low-loss band around 1.6 pm were used at high capacity.

ACKNOWLEDGMENT

The author would like to thank Tingye Li for helpful com-

ments and suggestions.

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Fiber Transmission osses in High Radiation

GEORGE H. SIGEL, JR .

Abstmct-A

summary is provided on the effects of ionizing radiation

[ 1 ]

[

21. The introduction of index modifying dopants into

on Present Fe n t i o n ptd fiben In

P d C “ h

the P W r focuses

SiOz glass may stronglymodify bondingstrengthsandcon-

on the magnituh and the spectr l and temporpl

characteristics Of

the

figurations and can thereforestrongly mpacton hedefect

radiation-inducedtransmission lossesn rradiated fiben The

most

=diption resistant

d-

of fi rsre identified and possible p p p r o ~ e s production and charge trapping processes [ 3

1

- 5 1 . In addi-

forurthermprovementsn

fiber

performance are suggested. tion, preexisting flaws andtrains are commonn glasses,

so

that they are ypically much more susceptible

to

damage

I.

I NTRODUCTI ON

from ionizingadiation thanheir crystalline counterparts

of

the same composition . Defects in the glasses are normally

bands in

the

near

uv,

visible and

infrared

spectral

result in the formation of lower energy light absorption and

regions. These

so

called “color centers” are particularly serious

in the case of optical fibers since the long optical pathlength

can make the optical link vulnerable to rather low radiation

doses.

The optical damage which is created within the g l a s s matrix

by ionizing radiation is usually not permanent so that recovery

Manuscript received May 22, 1980.

Theauthor

is

wit h he Naval ResearchLaboratory,Washington, DC and annealing

Of

absorption are

Observed

in

most

fibers

at

20375. roomemperature.he recovery process is strongly influenced

N UNDERSTANDING of he tYPiC‘ radiationeffects associated with energy levels within he forbidden gap

observed inoptica l fibers requires some knowledge of

the structure and bonding of the opt ical materials used

in waveguide fabrication.Since optical fibers for hemost

part are manufacturedrom high-purity glasses based on

SiOz, the discussion of radiation damage will focus on these

materials. onizing radiat ionproducesbrokenand dangling

bonds which serve as charge trapping sites

in

the

glass

matrix

U.S.

Government work not protected by

U.S.

copyright

Nonlinearity in Fiber Transmission

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