INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 8, NO. 5, DECEMBER 2010 107
© 2010 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
THE SOLITON TRANSMISSIONS IN OPTICAL FIBERS
Leoš BOHÁČ.1
1 Depa men o elecommunica ion enginee ing, Czech echnical uni e si y in P ague, P ague, Czech epublic Technicka
2, P ague 6, 16627
[email p o ec ed]
Abs ac . The objec i e o his pape is o amilia ize
eade s wi h he basic analy ical p opaga ion model o
sho op ical pulses in op ical ibe . Based on his model
simula ion o p opaga ion o he special ype o pulse,
called a soli on, will be ca ied ou . A soli on ansmission
is especially a ac i e in he ibe op ic elecommunica ion
sys ems as i does no change a pulses shape du ing
p opaga ing igh -down he ibe link o he ecei e . The
model o e y sho pulse p opaga ion is based on he
nume ical solu ion o he nonlinea Sch oedinge equa ion
(NLSE), al hough in some speci ic cases i is possible o
sol e i analy ically.
Keywo ds
Soli on, ul a sho pulses, NLSE, dispe sion,
Gaussian beam.
1. In oduc ion
Soli a y wa es, some imes simply called soli on, ha e
been a opic o heo e ical and expe imen al s udy o many
yea s. Though his pape is ela ed o ib e op ics, in eali y
soli a y wa es exis also in o he ield o science like
hyd odynamics, biology and plasma physics. His o ically
he one who i s obse ed a soli on wa e was James Sco
Russel in 1834 when he acciden ally no iced in he na ow
wa e canal a smoo hly shaped wa e heap ha o his
su p ise was able o p opaga e in he canal wi hou a
appa en change in i s shape a ew kilome es along. The
essence o p opaga ion o his soli a y wa e was no a long
ime unde s ood un il app op ia e ma hema ical model was
concei ed in he 1960’s oge he wi h a way o sol ing
nonlinea equa ion wi h he help o in e se sca e ing
me hod.
Now le us go back o he ield o op ics. Gene ally
speaking, he e exis wo o ms o soli a y wa es,
depending on whe he he ligh is being con ined in space
o ime. I he i s is he case wa e is e e ed o as spa ial
soli on o in he second case as empo al soli on. Soli on
o ming phenomenon s ems om nonlinea p ope ies o
medium whe e a pa icula wa e is p opaga ing. Namely,
in a ield o op ics i is Ke e ec ha is esponsible o
op ical nonlinea i ies. In he case o spa ial soli on he
na u al p ope y o ligh o dispe se in space is being
p oac i ely compensa ed by he nonlinea i y o he medium
in such a way ha highe in ensi y pa o an op ical beam
( ypically in he cen e o Gaussian beam) inc ease a alue
o e ac i e index o medium o ming de ac o a co e o
wa eguide ha is esponsible o con ine in e e se a
dispe sed ligh o he middle o he beam i sel . I can be
easily in ui i ely unde s ood ha i he sel induced
nonlinea i y is oo high he beam will ge ocused and on
he o he hand i i is e y small o none, beam will
dispe se in space – a p e ailing si ua ion in many cases
whe e a beam does no ha e enough powe densi y o
induce nonlinea i y in a medium.
2. Ma hema ical modeling o he
soli a y wa e
The p opaga ion o ligh can be p ecisely desc ibed
ma hema ically wi h Maxwell equa ions. When equa ions
o magne ic and elec ic ields a e combined oge he one
ge [1][2]:
2
2
2
0
2
2
2
211
P
c
E
c
E









(1)
whe e c is he speed o ligh in he acuum and 0 is he
acuum pe mi i i y. The induced pola iza ion P consis s
o wo pa s such ha [1][2]:
),(),(),( P P P NLL






 (2)
whe e he linea pa L
P

and nonlinea pa NL
P

a e ela ed
o he elec ic iled by he gene al ela ions [1][2] [3][4]:
108 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 8, NO. 5, DECEMBER 2010
© 2010 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
')',().'(),( )1(
0d E PL




 


(3)

  ),,(),( 321
)3(
0 P
NL


 (4)
321321 ),(),(),( d d d E E E 






whe e (1) and (3) a e he i s - and hi d- o de
suscep ibili y enso s.
3. P opaga ion o soli on pulse in
op ical ibe s
To be e unde s and a soli on pulse p opaga ion in
op ical ibe i necessa y o se up ou modelling on he
ma hema ical exp ession (1). We will suppose, ha a
solu ion o elec ic iled E ha e a o m [1]:
)exp(),(),(),( 0ZiYXF ZA E

 (5)
whe e F(X,Y) is ans e se ield dis ibu ion ha
co esponds o he undamen al mode o single mode ib e.
A(Z, ) is along p opaga ion axis Z and on ime dependen
ampli ude o he mode. A e some ma h manipula ions
one can come o he equa ion ha go e ns pulse
p opaga ion in op ical ib es [1]:
AAi
A
i
A
Z
A2
2
2
2
12












(6)
The pa ame e s 1 a 2 include he e ec o dispe sion
o i s and second o de s, espec i ely. Physically,
1=1/ g, whe e g is g oup eloci y associa ed wi h he
pulse and 2 akes in o accoun he dispe sion o g oup
eloci y. Fo his eason, 2 is called he g oup eloci y
dispe sion (GVD) pa ame e .
Pa ame e  is nonlinea pa ame e ha akes in o
accoun he nonlinea p ope ies o a ibe medium.
Pa ame e 1 is in eal case always posi i e bu on he
o he hand pa ame e s ZD and  can be in some speci ic
case ei he posi i e o nega i e. The pa ame e 1 is closely
associa ed in p ac ice wi h be e known pa ame e called
dispe sion pa ame e – D (ps/nm/km). The ela ion
be ween hem is in he o m [1]:
2
2
21




c
d
d
D
g









 (7)
As we know, dispe sion pa ame e D is a mono onically
inc easing unc ion o wa eleng h, c ossing a ze o poin a
wa eleng h ZD, which is called a ze o ch oma ic dispe sion
wa eleng h. I a sys em ope a es wi h wa eleng hs abo e
ZD, whe e D is posi i e, 2 mus be nega i e and a ibe is
said o wo k in anomalous dispe sion mode. I a ibe is
ope a ed below ZD, he D is nega i e and 2 mus be
posi i e. In his case a ibe is said o ope a e in no mal
dispe sion mode. As ega ds he nonlinea pa ame e  i
can be gene ally ei he posi i e o nega i e, depending on
he ma e ial o he wa e guide. Fo silica ibe is pa ame e
 posi i e bu o some o he ma e ials i can be nega i e.
Mo e speci ically, equa ion (6) has only wo solu ion, in
he o m o ei he da k o b igh soli on. The b igh soli on
co esponds o he ligh pulse bu da k soli on is a he a
pulse shaped dip in CW ligh “backg ound”. In o he
wo ds, he da k soli on is in a ac nega ion o he b igh
soli on. Whe e he e is maximum o ligh in he b igh
soli on, he e is minimum o he ligh in he da k soli on
and ice e sa.
The b igh soli on can p opaga e in only such a
wa eguide whe e he e is ei he he posi i e nonlinea i y
pa ame e  and anomalous dispe sion o he nega i e
nonlinea pa ame e bu no mal dispe sion. Fo a classical
silica ibe he i s is he case.
4. The soli on pulse and he simula ion
o i s p opaga ion in he op ical ibe
Equa ion (6) can be no malized in he o m:
0
2
2
2
2




uu
us
z
u
i

(8)
using his ans o ms:
O
TZ /)( 1




, D
LZz /

, ALu D

 (9)
INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 8, NO. 5, DECEMBER 2010 109
© 2010 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
whe e T0 is mo ing ime window wid h ( e y o en se o
he pulse wid h) and 2
2
0/

TLD is dispe sion leng h.
Using in e se sca e ing me hod e eals ha solu ion o
abo e men ioned equa ion has a o m:
2/2/ )(sech
2
),( iziz eNe
ee
Nzu
 


 (10)
I N is in ege , i ep esen he o de o he soli on pulse.
Ve y in e es ing si ua ion comes when N=1. In his case o
i s o de soli on, he pulse does no change i s shape a all
as i p opaga es in op ical ibe . In con as when N is
highe hen one, pulse shape is no s able and change
pe iodically wi h soli on pe iod D
LZ 2
0

. A he end
o e e y pe iod Z0 he soli on esembles i s ini ial simple
pulse shape. I is e iden ha o elecommunica ion
pu poses is he soli on o i s o de mos sui able, because
in his applica ion is necessa y o keep a pulse shape s able.
Pa ame e N, which de ines he soli on o de can be
u he exp essed by:
2
0


P
TN O
 (11)
whe e T0 [s] co esponds o inpu pulse wid h, P0[W] is
pulse peak powe 2

[ s2/m] akes in o accoun g oup
eloci y dispe sion and [(Wm)-1] is nonlinea pa ame e o
he ibe ma e ial.
I ha e s udied a p opaga ion o soli on pulses wi hin he
op ical ibe using a simula ion ool Op sim o m ARTIS
In his case a equa ion (6) is sol ed nume ically using a
spli -s ep-Fou ie me hod. The scheme used is shown on
Fig.1. As can be seen, I ha e used a soli on gene a o ( he
mos le side icon) ha nume ically gene a es sequence o
sech( ) pulses. The pulse wid h was se o 10 ps wi h pe iod
o 400 ps. I ha e used a s anda d single mode ibe model
wi h a ibe leng h o 100 km. I ha e se up wo peak
powe s o he pulse, namely 100 mW and 166 mW. As
pulses sa is y a condi ion o he soli on shape, i was only
necessa y o adjus app op ia e pulse wid h and peak
powe . Acco ding o he
Fig.1 – Simula ion o soli on in ibe
o mula (11) and o he case o i s o de soli on, we
need:


2
0
2
0
1T
PN  (12)
when eal alues a e subs i u ed o he abo e equa ion, in
pa icula T0=10 ps, 2= -20 ps2/km and =1,2 W-1km-1 he
adequa e powe o each i s o de soli on egime is
app oxima ely 166 mW. In o he wo ds i a sech( ) pulse
ha e his peak powe one should a leas acco ding o
heo y ha e a i s o de soli on ha is ansmi ed h ough
a ibe unchanged in shape. To e i y his I ha e used in
abo e model a ibe wi hou a loss, i is possible o cou se
only in simula ions no in he p ac ice. In he eali y i
would be necessa y o deploy op ical ampli ie s (like
EDFA o Raman, o bo h in combina ion) o o e come he
loss in he eal ibe link and keep he powe o he soli on
in equi ed limi s, o he wise he pulse will s a o sp ead
again, he e ec o nonlinea i y will be no so s ong o
e ec i ely supp ess unwelcome in luence o he ibe
ch oma ic dispe sion.
To illus a e c ea ion o he soli on I ha e used he scheme
on Fig.1. The pulse shape can be seen on Fig. 2. The igh
couple o o e lapping (highe one abo e o he ) pulses
co esponds o he inpu . These pulses we e in sequence
sen o 100 km sec ion o he single mode ibe link.
110 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 8, NO. 5, DECEMBER 2010
© 2010 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
Fig.2 – Resul s o simula ion
The one wi h he lowe powe (100 mW) ela es o he
lowe ou pu pulse ha is shown a he le side o he
pic u e. I is e iden ha in his case he pulse unde goes
sp eading caused by ch oma ic dispe sion and peak powe
is no ye high enough o compensa e dispe sion. On he
o he hand, i one inc ease op ical powe i is possible o
ge o he poin , whe e he nonlinea i y en i ely compensa e
dispe sion and he pulse p opaga es in he ibe link
wi hou any change in he shape, excep a small inc ease in
i s ampli ude. I is wo h o men ion igh now, ha his
inc ease in ampli ude is empo al phenomenon because i
one would ha e s udy he pulse p opaga ion mo e ca e ully
i would be e ealed ha i s peak oscilla es un il eaching a
poin o s eady ampli ude. This happens mos ly in cases
when he inpu pulse does no esemble he exac soli on
o m. The beau y o a soli on is among o he hings in i s
abili y o eassemble o iginal shape despi e o some
dis u bing ac o s ac ing upon i .
5. Conclusion
In his pape a e y basic analysis o soli ion
ansmission and i s dynamics in he op ical ibe was
pe o med. I has been shown by simula ion ha b igh
soli on can o m in classical op ical single mode ibe and
in he case o ze o loss i can p opaga e wi hou a change
o i s shape. As esul i is possible o o e come issue wi h
in luence o he dispe sion on pulse sp eading and achie e
a much longe ansmission dis ances and also inc ease line
capaci y. Howe e , he applica ion o soli on ansmission
is no en i ely wi hou p oblems. Soli on pulses should be
apa in ime conside ably o a oid excessi e o e lapping.
I soli on’s o e lap is no su icien ly supp essed soli ons
will ha e an e ec on he ansmission o hemsel es in a
way o cause indi idual soli on’s g oup eloci y o a y
along he line and esul ing in inc eased sys em ji e .
6. Acknowledgmen s
The wo k was suppo ed by g an The Resea ch in he
A ea o he P ospec i e In o ma ion and Communica ion
Technologies unde p ojec MSM6840770014.
Re e ences
[1] KIVSHAR, Yu i S., AGRAWAL, Go ind P. Op ical Soli ons :
F om Fibe s o Pho onic C ys als. San Diego : Academic P ess,
2003. 540 s. ISBN 0-12-410590-4.
[2] PORSEZIAN, K., KURIAKOSE, V.C. Op ical Soli ons :
The o ical and Expe imen ak challenges. 1s edi ion. Be lin :
Sp inge , 2003. 406 s. ISBN 3-540-00155-7.
[3] AGRAWAL, Go ind P. Applica ions on Nonline a Fibe Op ics.
San Diego: Academic P ess, 2001. 458 s. ISBN 0-12-045144-1.
[4] GAGLIARDI, Robe M., KARP, Sha man. Op ical
Communica ions. 2nd edi ion. New Yo k : Wiley, 1995. 347 s.
ISBN 0-471-54287-3.
Abou Au ho s ...
Leoš BOHÁČ ecei ed he M.S. and
Ph.D. deg ees in elec ical enginee ing
om he Czech echnical uni e si y in
P ague in 1992 and 2001, espec i ely.
F om 1992 he has been eaching Op ical
communica ion sys ems and Da a
ne wo ks a he same uni e si y. His
esea ch in e es is an applica ion o he
high speed op ical ansmission sys ems in a da a ne wo k.
He has also pa icipa ed in op ical esea ch p ojec s in
CESNET, he academic da a ne wo k p o ide , o help
implemen long haul high speed op ical esea ch ne wo k.