Depa amen o De Física Aplicada
In ege and F ac ional Talbo E ec :
Theo e ical S udy and
P ac ical Conside a ions
In ege and F ac ional Talbo E ec :
Theo e ical S udy and
P ac ical Conside a ions
A ea de Óp ica
Lau a Chan ada San odomingo
San iago de Compos ela, Junio 2006
UNIVERSIDADE DE
SANTIAGO DE COMPOSTELA
UNIVERSIDADE DE
SANTIAGO DE COMPOSTELA
D. Ca los G´omez-Reino Ca no a y D. Ca los Rod ´ıguez Fe n´andez-
Pousa, Ca ed ´a ico del ´
A ea de ´
Op ica del Depa amen o de F´ısica Aplicada
de la Uni e sidad de San iago de Compos ela y Con a ado Doc o del
´
A ea de Teo ´ıa de la Se˜nal y Comunicaciones del Depa amen o de F´ısica
y A qui ec u a de Compu ado es de la Uni e sidad Miguel He n´andez de
Elche,
CERTIFICAN
que la p esen e memo ia, i ulada “In ege and F ac ional Tempo al Talbo
E ec : Theo e ical S udy and P ac ical Conside a ions” ha sido ealizada
po D˜na. Lau a Chan ada San odomingo bajo su di ecci´on y
cons i uye la Tesis que p esen a pa a op a al G ado de Doc o en Ciencias
F´ısicas.
San iago de Compos ela, 8 de Junio de 2006.
V◦. B◦. Di ec o de la esis V◦. B◦. Di ec o de la esis
Ca los G´omez-Reino Ca no a Ca los Rod ´ıguez Fe n´andez-Pousa
La doc o anda
Lau a Chan ada San odomingo
Es e abajo ha sido inanciado po los p oyec os TIC99-0489 y
TIC2003-03041 del minis e io de Educac´ıon y Ciencia y el p oyec o
PGIDT04PXIC20605PN de la Xun a de Galicia
G acias al minis e io de Educaci´on y Ciencia po la ayuda econ´omica
ecibida a a ´es de la beca FPU.
Resumen
Las ´ecnicas y disposi i os que pe mi en la manipulaci´on de las p opiedades
de las ondas guiadas, cons i uyen la base ecnol´ogica de la espec oscop´ıa de
se˜nales moduladas, del p ocesado o ´onico, y de los m´e odos pa a el an´alisis
empo al de uen es pulsadas. Pa a implemen a odas es as ´ecnicas
se pueden u iliza an o p ocesos lineales como no lineales. Den o del
con ex o de la ´op ica lineal, la llamada analog´ıa espacio- empo al se ha
con e ido en los ´ul imos a˜nos en la una gu´ıa muy uc ´ı e a pa a el dise˜no
de nue os disposi i os. La dualidad espacio- empo al pe mi e aslada
concep os e ideas bien conocidas en p ocesado de se˜nales espaciales al
dominio empo al, p opo cionando de es e modo nue as o mas pa a el
p ocesado, manipulaci´on y con ol de se˜nales empo ales. Dicha analog´ıa
es ´a basada en la equi alencia ma em´a ica en e la ecuaci´on que desc ibe
la di acci´on pa axial de haces y la ecuaci´on de dispe si´on de p ime o den
que gobie na la e oluci´on de pulsos ´op icos cuando ´es os se p opagan en un
medio diel´ec ico o en una gu´ıa de onda. En ambos casos se a a de una
ecuaci´on di e encial pa ab´olica con coe icien es imagina ios, y po an o
admi e soluciones ondula o ias. Las ecuaciones que desc iben la e oluci´on
de en ol en es ´op icas len amen e a iables, co espondien es a ondas
planas o modos (en el caso de gu´ıas de onda) modulados empo almen e,
son o malmen e id´en icas a las ecuaciones pa axiales que desc iben la
p opagaci´on de ondas monoc om´a icas de ex ensi´on ini a. Exis e po an o
una co espondencia en e la a iable empo al de los p oblemas dispe si os
y la a iable espacial ans e sal de los p oblemas di ac i os. Asimismo,
en el dominio de Fou ie es a co espondencia se da en e los espec os
de ambos p oblemas. Es a dualidad ep esen a el ma co concep ual de
la comp esi´on lineal de pulsos, as´ı como de muchos o os disposi i os que
han sido p opues os en los ´ul imos a˜nos. En e ellos podemos menciona ,
sis emas de o maci´on de imagenes empo ales, il ado empo al po
medio de len es empo ales, educi´on de ji e , la ans o mada de Fou ie
empo al, la e si´on empo al del eo ema de Van Ci e -Ze nike, len es
iii
i Resumen
empo ales de g adien e de ´ındice, ´ecnicas de as o maci´on iempo-
ecuencia y ecuencia- iempo y el e ec o Talbo empo al en e o os.
El e ec o Talbo ue descubie o en 1836 po H.F. Talbo , cuando
iluminaba una ed de di acci´on ec angula de aguje os min´usculos con una
uen e de luz blanca de peque˜no ama˜no. A cie as dis ancias a lo la go de
la di ecci´on de p opagaci´on, pod´ıan obse a se ´eplicas de las es uc u as
pe i´odicas que hab´ıa iluminado. Es e e ec o no ue sa is ac o iamen e
explicado has a que Lo d Rayleigh, en 1881, lo a ibuy´o a la in e e encia
en e haces di ac ados po las es uc u as pe i´odicas. Mos ´o que, en el
caso de una ed lineal iluminada con un en e de onda plano, las im´agenes
de la ed se epe ´ıan a lo la go de la di ecci´on de iluminaci´on, localizadas a
dis ancias zγ=γd2/λ, donde γes un n´ume o en e o, del pe ´ıodo de la ed
yλla longi ud de onda de la uen e. Sin emba go hubo que espe a la gos
a˜nos has a que Cowley y Moodie (1957-1960), en sus es udios pione os sob e
las p opiedades de la di acci´on de F esnel de obje os pe i´odicos, die on una
explicaci´on del e ec o Talbo en el con ex o de la eo ´ıa di accional de la
o maci´on de im´agenes. A˜nos m´as a de, Mon gome y (1967) en un abajo
undamen al, desa oll´o las condiciones gene ales que debe sa is ace un
obje o pa a ob ene ´eplicas de su dis ibuci´on de ampli ud compleja a lo
la go de la di ecci´on de iluminaci´on. Fue en onces cuando Mon gome y
in odujo el ´e mino au oimagen que a pa i de ese momen o se us´o
conjun amen e con el ´e mino e ec o Talbo en los abajos que sob e es e
ema se han publicado has a la ac ualidad. Adem´as de la econs ucci´on
de imagenes, se pueden obse a pa ones con es uc u as mucho m´as icas.
Po ejemplo, en cie os planos a lo la go de la di ecc´ıon de p opagaci´on,
se poducen pa ones cuya celda unidad es ´a o mada po la supe posici´on
cohe en e de las celdas o iginales desplazadas y con una di e encia de ase
en e ellas. De es e modo, si las celdas es ´an su icien emen e con inadas
como pa a e i a la supe posici´on, se ob ienen ´eplicas del pa ´on o iginal
pe o con una pe iodicidad que es un cie o de n´ume o de eces meno
que la de la ed ilumnada. Es e e ec o es conocido como e ec o Talbo
acciona io.
La con apa ida empo al del e ec o de au oimagen o e ec o Talbo
iene luga cuando una se˜nal empo al pe i´odica, como po ejemplo un en
de pulsos ´op icos, se p opaga a a ´es de un medio dispe si o lineal bajo
condiciones de p ime o den de dispe si´on. Cuando la dispe si´on acumulada
es un m´ul iplo de una cie a escala b´asica, el en de pulsos se ep oduce a s´ı
mismo as la p opagaci´on, en lo que se conoce como e ec o Talbo en e o.
Po o a pa e, cuando la dispe si´on acumulada es una acci´on de dicha

Resumen
escala b´asica, a la salida del medio dispe si o se ob iene un en de pulsos
con una ecuencia de epe ici´on que es un n´ume o en e o de eces la del en
o iginal, siemp e y cuando los pulsos sean su icien emen e es echos como
pa a e i a la supe posici´on. Es e e ec o es conocido como e ec o Talbo
aciona io. El e ec o Talbo empo al en e o ue desc i o po p ime a ez
po Jannson y Jannson quienes p opusie on su uso pa a la ans e encia
de in o maci´on con endia en una se˜nal pe i´odica a lo la go de una ib a
´op ica. Es os au o es es ablecie on po ez p ime a las longi udes de la
ib a ´op ica en las que en´ıa luga la econs ucci´on de la se˜nal pe i´odica.
Pos e io men e, And ekson apo ´o la p ime a p ueba expe imen al del
e ec o Talbo , al p opaga un en de pulsos gene ado po un l´ase de
sinc onizaci´on de modo (mode-locking) a a ´es de una ib a ´op ica. A
inales de los a˜nos no en a Shake y A ahi a p oba on expe imen almen e
el e ec o Talbo acciona io en ib as momomodo y pos e io men e Aza˜na
y Mu iel hicie on lo p opio en ib as de pe iodo p og esi o (LCFG -linea ly
chi ped ibe g a ing). El e ec o Talbo empo al con in´ua a ayendo el
in e ´es de los in es igado es, en especial el e ec o Talbo aciona io, ya que
puede u iliza se pa a el inc emen o de la ecuencia de epe ici´on de una
secuencia de pulsos pe i´odica como la gene ada po l´ase es de sinc onizaci´on
de modo. La gene aci´on de enes de pulsos ´op icos con ecuencia de
epe ici´on ul a ´apida es de i al impo ancia en campos como sis emas
de compu aci´on ´op ica, po cesado de da os ul a ´apido o sis emas de
comunciaciones ´op icas en e o as ´a eas cien ´ı icas. A pesa de que se
ha p og esado mucho en la gene aci´on di ec a de enes de pulsos con al as
ecuencias de epe ici´on a pa i de l´ase es de semiconduc o es o de ib a,
esul a en ajoso ene ´ecnicas sencillas que pe mi an inc emen a a´un m´as
la ecuencia de epe ici´on ue a de la ca idad del l´ase . Po o a pa e, el
e ec o Talbo es un en´omeno lineal y po an o pe mi e su in eg aci´on con
o as ´ecnicas de p ocesado lineal ales como comp esi´on o econs ucci´on
( eshaping) de pulsos. El e ec o Talbo ambi´en ha sido usado jun o con
en´omenos no lineales, as´ı como con sis emas de imagen empo al.
Aunque el an´alisis expe imen al de la o maci´on de ´eplicas Talbo
de enes de pulses se cen a ´ıpicamen e en el dominio empo al, se
puede ex ae mucha m´as in o maci´on median e el an´alisis espec al de
la in ensidad del en de pulsos. En los il os con encionales dise˜nados
pa a aumen a la ecuencia de epe ici´on median e il ado ´op ico, un il o
lineal sup ime los a m´onicos ´op icos del en de en ada que no pe enecen
al en de salida de al a ecuencia. Po lo an o, el uncionamien o de es os
si emas se puede analiza po medio de la unci´on de ans e encia ´op ica
i Resumen
del il o lineal. Po el con a io, y debido al hecho de que la dispe si´on es
un il o de ase pu o en el dominio ´op ico, el an´alisis espec al de il os
Talbo ha de ealiza se en ´e minos de in ensidades, lo que da luga a un
an´alisis espec al no lineal.
En es e abajo se p esen a un es udio del e ec o Talbo empo al
en e o y aciona io, eniendo en cuen a conside aciones p ´ac icas an o
a ni el del en de pulsos como del medio dispe si o u ilizado. La memo ia
cons a de es cap´ı ulos. En el p ime o de ellos, in oducimos de o ma
b e e la dualiad espacio empo al que adica en la equi alencia ma em´a ica
en e las ecuaciones que desc iben la di aci´on pa axial unidimensional
y la p opagaci´on de pulsos en un medio dispe si o de p ime o de con
a enuaci´on desp eciable. Den o del con ex o de es a analog´ıa p esen amos
la con apa ida empo al del e ec o Talbo . La mayo pa e del an´alisis
se ealiza en el dominio espec al y se basa en el c´alculo exac o de la
densidad espec al de po encia de un en de pulsos cohe en es as su
p opagaci´on en un medio dispe si o con coe icien es de dispe si´on de p ime
y segundo o de a bi a ios. Hemos conside ado que la se˜nal empo al
es ´a o mada po un en de pulsos gaussianos con modulaci´on lineal de
ase (linea chi p). Es os esul ados anal´ı icos pe mi en iden i ica las
componene es espec ales que ep esen an la dispe si´on indi idual de los
pulsos, as´ı como los ´e minos que co esponden la in e e encia en e pulsos.
Desde el pun o de is a espec al las lineas Talbo se pueden in e p e a
como un il o pasabanda m´ul iple, donde cada pasabanda co esponde
a la in e e encia en e pulsos sepa ados un cie o n´ume o de pe ´ıodos.
El il o elimina los a m´onicos de in ensidad del en de en ada que no
pe enecen a la salida. La sup esi´on de los a m´onicos s´olo es exac a pa a
il os Talbo semi-en e os, con coe icien es de dispe si´on de segundo o den
y mayo es desp eciables, y pulsos en el l´ımi e de ans o mada, ya que en
es e caso las dis ancias Talbo coinciden con las dis ancias de sup esi´on
de po ado a. Los il os Talbo no son es ´a icos sino que dependen de la
anchu a espec al de la se˜nal de en ada y de las ca ac e ´ısi icas dispe si as
de la l´ınea. En pa icula , hemos deducido las condiciones que ga an izan
que el il o Talbo p opo ciona un en cuyas luc uaciones en in ensiad son
desp eciables incluso cuando el medio de p opagaci´on p esen a coe icien e
de dispe si´on de segundo o den. Finalmen e, hemos analizado la es abilidad
de las l´ıneas Talbo en e a a iaciones de ecuencia o de la longi ud de la
ib a. Cuando la anchu a espec al del en es g ande, la ines abilidad de
la dispe si´on iende a ensancha los pulsos empeo ando el uncionamen o
del il o debido a la in e e encia en e pulsos. Po el con a io, cuando
Resumen ii
la anchu a espec al es mode ada, a iaciones en la dispe si´on debili an la
sup esi´on de a m´onicos. La es abilidad ´op ima se alcanza cuando la anchu a
espec al de el en, medida como FWHM ( ull wid h hal maximum), es
el doble de la ecuencia de epe ici´on del en de salida.
El obje i o gene al del segundo cap´ı ulo es la desc ipci´on espec al de las
p opiedades de au o egene aci´on del e ec o Talbo empo al. Conside amos
que los pulsos iene o ma Gaussiana y es ´an linealmen e modulados en ase.
Adem´as suponemos que el disposi i o Talbo es ideal, es deci , es un medio
dispe si o lineal de p ime o den sin a enuaci´on, y con un ancho de banda
supe io al del en de pulsos de en ada.
En la p ime a secci´on analizamos la densidad de po encia espec al de
un en cuyos pulsos es ´an alea o iamen e dis ibuidos an es y despu´es de
un il o Talbo . Hemos conside ado que las a iables que desc iben la
p esencia o ausencia de los pulsos en el en no es ´an co eladas. Como
consecuencia de la in e e encia cohe en e en e los pulsos dispe sados,
la en ol en e del uido de banda ancha del espec o de adio ecuencia
despu´es de la de ecci´on del en se ensancha po causa de la modulaci´on de
ase de los pulsos. La p incipal ca ac e ´ıs ica del espec o es la p esencia
de bandas pasan es que il an el uido de banda ancha, y cuyos cen os
apa ecen desplazados espec o a los a m´onicos debido a la modulaci´on
de ase. Po an o, midiendo la des iaci´on ela i a de es os cen os con
espec o a la anchu a de las bandas pasan es es posible de e mina el signo
y la magni ud de la modulaci´on de ase de los pulsos del en de en ada.
Las bandas pasan es son an o m´as es echas cuan o mayo es la dispe si´on
del disposi i o, meno es la anchu a empo al de los pulsos y mayo es la
modulaci´on de ase. El m´aximo alo de es as bandas pasan es depende de
la can idad ela i a de pulsos p esen es en el en. Pa a disposi i os de baja
dispe si´on y pulsos anchos las bandas pasan es apa ecen moduladas po una
unci´on coseno cuad ado. La anchu a de es a modulaci´on es una acci´on
del alo del a m´onico undamen al de la en ada, y es independien e de la
can idad ela i a de pulsos en el en y de sus p opiedades.
En la segunda secci´on se analiza la in luencia del iming ji e de un
en de pulsos gaussianos con modulaci´on de ase lineal en un disposi i o
Talbo . A pesa de que en nues o es udio no hemos enido en cuen a
posible luc uaciones de ampli ud (ampli ude ji e ) del en de pulses, el
analisis del iming ji e puede gene aliza se a p oblemas con ampli ude
ji e o una combinaci´on de ambos. El an´alisis se lle a a cabo an o
en el domino empo al como en el espec al. En el an´alisis empo al
conside amos que las a iables que desc iben el iming ji e de los pulsos
iii Resumen
no es ´an co eladas. El es udio p esen ado mues a como los alo es de la
a ianza del en se homogeneizan as su paso po el disposi i o Talbo ,
ya que an o los pulsos indi iduales como su a ianza se ensanchan debido
a la dispe si´on, ex endiendose po los in e alos empo ales de los pulsos
ecinos. P esen amos adem´as exp esiones anal´ı icas ap oximadas pa a
la a ianza de la en ada y de la salida, analizando la alidez de dichas
ap oximaciones. Pa a disposi i os Talbo en e os, el alo de la a ianza
es cons an e e independien e del alo de la dispe si´on. Sin emba go pa a
disposi i os Talbo acciona ios, la a ianza a la salida pod ´ıa p esen a
una modulaci´on, dependiendo del cocien e en e la anchu a empo al del
pulso y el pe ´ıodo del en o iginal. En cualquie caso, pa a los disposi i os
Talbo dise˜nados pa a aumen a la ecuencia de epe ici´on de un en de
pulsos, la ap oximaci´on de la a ianza a una cons an e con in´ua siendo
´alida.
El p oblema del iming ji e se analiza ambi´en desde el pun o de
is a espec al. Aho a, y a di e encia del an e io an´alisis en ´e minos
de a ianza, conside amos que las a iables alea o ias que desc iben el
ji e de cada pulso es ´an en gene al co eladas. La p incipal ca a e ´ıs ica
del espec o esul an e es, como ocu ´ıa en el caso de enes de pulsos
alea o iamen e dis ibuidos, la p esencia de un mecanismo de il ado del
uido de banda ancha, dando luga a una educci´on de la po encia de
uido in eg ado al ededo de los a m´onicos en el dominio de adio ecuencia.
Tambi´en se analiza la in luencia de la modulaci´on de ase y de la anchu a
de l´ınea ´op ica de los pulsos indi iduales. Es os esul ados se u ilizan
pa a analiza la educci´on de uido de las ´eplicas Talbo bajo di e en es
condiciones de p opagaci´on. La educci´on del uido causado po el iming
ji e depende p incipalmen e de la modulaci´on de ase de los pulsos del
en, de la co elaci´on en e pulsos adyacen es y de la dispe si´on acumulada
de la l´ınea, pe o sin emba go no depende del alo de la des iaci´on
es ´anda del iming ji e . La modulaci´on de ase es la p incipal causa
de de e io o de la educci´on de uido, ya que p o oca el desplazamien o
de las bandas pasan es con espec o a la posici´on de los a m´onicos. Sin
emba go, un dise˜no adecuado de la l´ınea dispe si a pe mi e de un modo
sencillo compensa dicha modulaci´on de ase y al mismo iempo educi la
anchu a empo al de los pulsos.
Debido a que la co elaci´on con ola el ancho de banda del uido an es
de que el en de pulsos en e en la l´ınea dispe si a y debido ambi´en a que
el sua izado del iming ji e es m´as p onunciado a escalas del o den de
la ecuencia de epe ici´on, el uido inicial ha de se de banda ancha pa a
In oduc ion 3
a e o 196 GHz ( ou imes he o iginal one) and 400 GHz (eigh imes
he o iginal one) espec i ely. In ege and ac ional Talbo e ec ha e
been also demons a ed in linea ly chi ped ibe g a ings (LCFG) [27, 28]
and mul imode ibe s [29]. The empo al Talbo e ec is s ill a ac ing
he in e es o esea ches [30–33], especially he ac ional Talbo e ec ,
due o he ac ha i can be applied o mul iplying he epe i ion a e
o pe iodic pulse sequences. This p ope y has also been demons a ed o
mul iwa eleng h op ical pulse ains [34,35]. The gene a ion o op ical pulse
ains wi h ul ahigh epe i ion a es is o i al impo ance in ields such as
ul a as da a p ocessing op ical compu ing sys ems o ul a-high-bi - a e
op ical communica ions among o he s. Despi e i has been much p og ess
on di ec gene a ion o high- epe i ion a e pulse ains om semiconduc o
o ibe lase s, i is always ad an ageous o ha e simple me hods o u he
inc easing he pulse a e [36–39]. To day he highes epe i ion a e eached
by means o Talbo de ices is 2.5 THz, which was achie ed by mul iplying
250×a 10 GHz ain o pulses which had been p e iously comp essed [40].
Talbo e ec has also been applied o demos a e new ypes o mode-
locked lase s [41, 42]. Mo eo e and since Talbo e ec is a linea
phenomenon, i allows in eg a ion wi h o he linea p ocessing echniques,
such as comp ession [43, 44] o eshaping [28, 45]. I s unabili y has also
been demons a ed by he use o unable dispe si e lines [46, 47], and by
swi ching o soli on egimen [48]. Talbo dispe si e lines ha e also been
used oge he wi h nonlinea phenomena [42,47,49] and empo al imaging
sys ems [32].
Al hough he expe imen al analysis o he Talbo -imagined ains
usually ocuses on he ime domain, much in o ma ion ca be in e ed om a
spec al analysis o he in ensi y o he ain [24,25]. In con en ional il e s
designed o inc ease he epe i ion a e based on op ical spec al il e ing, a
linea il e supp esses he op ical ha monics o he inpu ain ha do no
belong o he high- equency ou pu ain [47]. The e o e he pe o mance
o hese sys ems can be analyzed om he op ical ans e unc ion o he
linea il e . In con as , and because dispe sion is a phase-only il e in
he op ical domain, Talbo dispe sion il e ing is nonlinea and equi es a
spec al analysis in he in ensi y domain.
He e we p esen a heo e ical s udy o Talbo de ices, aking in o
accoun p ac ical conside a ions ega ding bo h he incoming sequence o
pulses and he dispe si e medium. The s udy is mos ly ca ied ou in
spec al domain by means o in ensi y spec um, which leads o p esen
Talbo lines as a nonlinea il e . The wo k is di ided in ee chap e s. In

4In oduc ion
he i s one he basis o space- ime duali y and he empo al Talbo e ec
a e b ie ly e iewed. Likewise we analyze in e ms o in ensi y spec um
how dispe si e lines wi h a bi a y i s and second o de dispe sion ac
on a sequence pulses. Subsequen ly, hese esul s a e pa icula ized o a
Talbo line, hus showing he mechanism o image o ma ion depending on
Talbo index. The de e io a ion due o second-o de dispe sion as well as
he s abili y and ole ance o Talbo de ices agains iming and equency
a ia ions a e also ackled.
In he second chap e we in es iga e he es o a ion capabili y o Talbo
e ec . This p ope y is well-known in di ac i e op ics and has been
exploi ed o smoo h impe ec ions o a g a ing. He e we s udy non-ideal
ains a e i s p opaga ion in a Talbo line. In pa icula we analyze i s
ains o on-o -keyed pulses and la e ains o pulses su e ing om iming
ji e . In bo h cases he impe ec ions o he pulse sequence a e andom
and he e o e he p esen ed analysis o Talbo lines is no de e minis ic bu
s ochas ic.
Chap e 3 deals wi h he in luence o he cohe ence o he wa e on
Talbo de ices. We i s s udy he beha io o an a bi a y signal in a
linea ime-in a ian sys em unde pa ially cohe ence condi ions. When
he linea sys em is a lowes -o de dispe si e line he e ec o he cohe ence
on he empo al signal can be exp essed as a low pass il e . These esul s
a e pa icula ized o Talbo de ices. A ma ix analysis o Talbo lines
is also epo ed, p esen ing he in o ma ion en opy o he sys em as a
measu ed o he o e all deg ee o cohe ence o he sys em.
Chap e 1
Tempo al Talbo e ec
1.1 In oduc ion
The Talbo e ec is widely known as a cohe en phenomenon, which akes
place in he pa axial di ac ion egion when a pe iodic objec is illumina ed
by a cohe en sou ce [21,22]. I consis s in he o ma ion o sel -images o
eplicas o he objec a ce ain dis ances along he p opaga ion di ec ion,
and has i s o igin in he in e e ence among di ac ion o de s o he
g a ing. A ce ain dis ances om he objec exac eplicas a e ound,
which a e called in ege Talbo images. Be ween wo in ege Talbo planes,
pa e ns whose uni cells a e o med by he cohe en supe posi ion be ween
shi ed cells. These pa e ns a e called ac ional Talbo images. I cells a e
con ined enough o a oid o e lapping be ween neighbo ing cells, eplicas o
he g a ing bu wi h a pe iodici y lowe han ha o he o iginal pa e n
a e achie ed, which is known as ac ional Talbo e ec . Bo h in ege and
ac ional Talbo e ec s ha e been p o ed and applied no only in ee
space bu in inhomogeneous [50, 51] and wa eguide media [52], whe e he
p opaga ing modes play he ole o he di ac ed o de s in ee space.
The same phenomenon is ound in he ime domain when a pe iodic
signal p opaga es in a dispe si e medium [19, 27, 28, 53]. This ac , is
a consequence o he well-known ma hema ical analogy be ween he
equa ions go e ning one-dimensional pa axial di ac ion and lowes -o de
dispe sion o na owband signals [1, 2]. In his con ex , he pulse
en elope is equi alen o he complex ampli ude dis ibu ion, dispe sion
is equi alen o di ac ion and he pe iodici y o he g a ing co esponds
o he pe iodici y o he empo al sequence. Thus, when he accumula ed
dispe sion o he medium is a mul iple o a basic scale eplicas o he inpu
5
61.2. Space-Time Duali y
ain o pulse a e achie ed [20]. Fu he mo e i accumula ed dispe sion
is a ac ion o he men ioned basic scale, a sequence o pulses wi h a
epe i ion a e highe han ha o he o iginal ain is ob ained, as long
as pulse a e na ow enough o a oid o e lapping [1]. This p ope y has
been widely exploi ed and nowadays Talbo de ices cons i u e a well a
well-es ablished all-op ical echnique o inc ease he epe i ion a e o a
ain o pulses [20,25–28,54–56].
This chap e deals wi h he analysis o he empo al Talbo e ec ,
and is o ganized as ollows. In sec ion 1.2, we p esen he basis o he
space- ime duali y. The in ege and ac ional Talbo condi ions in bo h
space and ime domain a e es ablished in sec ion 1.3. The analy ical s udy
o he in ensi y o an a bi a y ain o pulses a e in ege and ac ional
Talbo de ice is also epo ed. Sec ion 1.4 is de o ed o he analysis o
a ain o linea ly chi ped Gaussian pulses which p opaga es h ough an
a bi a y second-o de dispe sion medium. This analysis is ca ied ou in
he Fou ie domain by means o he in ensi y spec um. These esul s a e
pa icula ized o a dispe si e line sa is ying Talbo condi ion in sec ion
1.5. Some examples a e p esen ed o show how he econs uc ion o he
ain akes place, as well as he dele e ious o he esul ing ain due o he
p esence o second o de dispe sion. Sec ions 1.6 and 1.7 a e espec i ely
de o ed o he analysis o he ejec ion capabili y and s abili y o Talbo
lines unde ibe leng h and iming a ia ions. Finally we end he chap e
p esen ing ou conclusions.
1.2 Space-Time Duali y
The so-called space- ime duali y in ol es he equi alen beha io o space-
limi ed beams p opaga ing in ee space and na ow band signals in a
i s -o de dispe sion medium. This analogy is based on he ma hema ical
equi alence be ween he equa ions desc ibing one-dimensional pa axial
di ac ion and na ow-band dispe sion [1]. Bo h equa ions a e de i ed
om he wa e equa ion by assuming wo s aigh o wa d app oxima ions.
They a e: monoch oma ic wa es and pa axial ays o he spa ial case; and
na owband ields o he empo al case [2].
In a linea non-magne ic medium wi h dielec ic cons an ǫ he space-
ime e olu ion o he elec ic ield, e, is go e ned by he wa e equa ion,
∇2e=µ0ǫ∂2e
∂ 2,(1.1)
1. Tempo al Talbo e ec 7
∇2and µ0being he Laplacian ope a o and he magne ic pe meabili y,
espec i ely. Since he ield componen s in Eq. (1.1) a e uncoupled,
he wa e equa ion can be exp essed wi hou loss o gene ali y in scala
o m. F om he wa e equa ion we will de i e he pa axial di ac ion
and i s -o de dispe sion equa ions by se ing he abo e men ioned
app oxima ions.
1.2.1 Pa axial di ac ion equa ion
In o de o achie e he pa axial equa ion, he elec ic ield is assumed o be a
monoch oma ic wa e wi h equency ω0,e(x, y, z, ) = e0(x, y, z) exp(iω0 ).
The ime de i a ions a e now mul iplica i e ac o s (∂/∂ →iω0),
con e ing he wa e equa ion (1.1) in o he Helmhol z equa ion,
(∇2+k2)e0= 0,(1.2)
whe e k=µ0ǫω02is he wa e numbe . We a e in e es ed in he s udy o
pa axial ays, i. e., ays con ined mos ly along he p opaga ion di ec ion z.
The e o e, he mos apid spa ial phase a ia ions ake place in zdi ec ion,
and he elec ic ield can be ew i en as
e0(x, y, z) = ψ(x, y, z) exp(−ikz).(1.3)
ψ(x, y, z) is he complex ampli ude dis ibu ion and a ies slowly compa ed
wi h he wa enumbe k. The pa axial app oxima ion implies ha he
cu a u e o he elec ic ield in p opaga ion di ec ion is much less han
he cu a u e o he ans e sal p o ile,
∂2/∂z2≪∂2/∂x2,∂2/∂y2and 2k|∂/∂z|.(1.4)
Unde his condi ions he Helmhol z equa ion esul s in
∂ψ
∂z +i
2k∇2
ψ= 0,(1.5)
∇2
being he ans e sal Laplacian ope a o . Eq. (1.5) is known as
pa abolic o pa axial equa ion and go e ns he e olu ion o he elec ic
ield o a monoch oma ic beam p opaga ing down z-axis.
81.2. Space-Time Duali y
1.2.2 Dispe sion Equa ion
The empo al analogy o he pa axial equa ion appea s by conside ing
na owband wa es p opaga ing h ough a i s -o de dispe si e medium.
In o de o de i e he na owband dispe sion equa ion we suppose he
elec ic ield o be na owband cen e ed a ω0. Then, i can be con enien ly
ew i en as
e(x, y, z; ) = e0(x, y, z; ) exp(iω0 ) (1.6)
By using exp ession (1.6), he wa e equa ion, Eq. (1.1), in scala way is
exp essed in he Fou ie domain as
∇2E0(x, y, z;ω−ω0) + µ0ǫω2E0(x, y, z;ω−ω0) = 0,(1.7)
whe e E0(x, y, z;ω−ω0) is he Fou ie ans o m o e0(x, y, z; ) e alua ed
a ω−ω0, ha is, he equency measu ed wi h espec o he cen al
equency ω0. Thus, E0(x, y, z;ω−ω0) ep esen s he elec ic ield a
baseband. Equa ion (1.7) can be sol ed by using he me hod o sepa a ion
o a iables [5]. I we assume a solu ion o he o m
E0(x, y, z;ω−ω0) = F(x, y)A(z, ω −ω0) exp(−iβ0z),(1.8)
whe e A(z, ω −ω0) is a slowly a ying unc ion o zand β0is he wa e
numbe a he ca ie equency o be de e mined la e . Equa ion (1.7)
leads o he ollowing wo equa ions o F(x, y) and A(z, ω −ω0):
∂2F(x, y)
∂x2+∂2F(x, y)
∂y2+µ0ǫω2−β(ω)2F(x, y) = 0,(1.9)
2iβ0
∂A(z, ω −ω0)
∂z −β(ω)2−β2
0A(z, ω −ω0) = 0.(1.10)
In ob aining Eq. (1.10), he second de i a i e ∂2A(z, ω −ω0)/∂z2was
neglec ed since A(z, ω −ω0) is assumed o be a slowly a ying unc ion
o z. The wa e numbe β(ω) as well as he ans e sal p o ile o he
elec ic ield, F(x, y), a e de e mined by sol ing he eigen alue equa ion
(1.9). F(x, y) ep esen s he he modal dis ibu ion, and o single-mode
ibe s, co esponds o he modal dis ibu ion o he undamen al ibe mode
gi en by equa ions (1.8), and (1.9). We a e in e es ed in he second equa ion
(1.10) which go e ns he ime p o ile e olu ion o he elec ic ield. I can
be w i en as

1. Tempo al Talbo e ec 9
∂A(z, ω −ω0)
∂z =−i[β(ω)−β0]A(z, ω −ω0) (1.11)
whe e we ha e app oxima ed β(ω)2−β2
0by 2β0(β(ω)−β0). This equa ion
ep esen s a linea ime-in a ian sys em wi h ans e unc ion [3]
H(z, ω) = exp[−i(β(ω)−β0)z].(1.12)
The in e p e a ion o equa ion (1.11) is clea . Each spec al componen
wi hin he pulse en elope acqui es, as i p opaga es, a phase shi whose
magni ude is equency dependen . Assuming a na owband signal cen e ed
a ω0, he p opaga ion cons an , β(ω), can be expanded in Taylo se ies
a ound he cen al equency ω0, as ollows
β(ω)∼
=β0+β1(ω−ω0) + β2
2(ω−ω0)2+β3
6(ω−ω0)3,(1.13)
whe e β0=β(ω0) is he p opaga ion cons an a he cen al equency,
β1=dβ(ω)/dω|ω=ω0is he in e se o he g oup eloci y and β2=
d2β(ω)/dω2|ω=ω0and β3=d3β(ω)/dω3|ω=ω0a e he i s and second-o de
dispe sion coe icien espec i ely. By using Eq. (1.13), Eq.(1.11) is gi en,
in he empo al domain, by
∂
∂z +β1
∂
∂ −iβ2
2
∂2
∂ 2−β3
6
∂3
∂ 3a(z, ) = 0.(1.14)
whe e a(z, ) is he op ical slowly a ying pulse en elope. I co esponds o
he Fou ie ans o m o A(z, ω −ω0), so ha he op ical en elope a(z, ) is
exp essed in baseband. By in oducing a change o a iables o a eling-
wa e coo dina e sys em, τ= −β1z, Eq. (1.14) esul s in
∂
∂z −iβ2
2
∂2
∂ 2−β3
6
∂3
∂ 3a(z, ) = 0.(1.15)
This is he na owband dispe sion equa ion exp essed in he p ope
e e ence ame. I he p opaga ion medium is pu e i s -o de dispe sion
(β3= 0), Eq. (1.15) is w i en:
∂a(z, τ)
∂z =iβ2
2
∂a(z, τ)
∂τ2.(1.16)
I is appa en now he ma hema ical equi alence be ween i s -o de
dispe sion equa ion (1.16) and he pa axial di ac ion equa ion (1.5) when
only one ans e sal dimension is ele an . They a e pa abolic equa ions
10 1.3. Tempo al Sel -Imaging E ec
Space domain Time domain
z z
x τ
k−1/β2
ψ(x, z)a(τ, z)
Table 1.1: Con e sion able be ween space and ime domain
which desc ibe he e olu ion o he en elope ield in space and ime domain
espec i ely. The compa ison be ween dispe sion and one dimension
pa axial equa ion leads o he con e sion able 1.1. The e o e, any medium
enable o p o ide la ampli ude and linea g oup delay o e he bandwid h
o he signal, is adequa e o ep oduce he ime-domain equi alen o
pa axial di ac ion e ec s. The mos common mediums used a e: single
mode ibe (SMF) and linea ly chi ped ibe g a ing (LCFG) [57]. SMF is a
good i s -o de dispe sion medium as long as he he equi ed ibe leng h
is no oo long and he bandwid h o he signal is na ow. O he wise he
second-o de dispe sion coe icien becomes ele an and mus be aken in o
accoun [25]. In hese cases, LCFG a e mo e adequa e since hey can be
speci ically designed o he sys em equi emen s in e ms o bandwid h
and g oup delay [58, 59]. Nowadays he ab ica ion echniques o B agg
g a ings a e well es ablished [57] and he esul ing de ice is compac (a
ew cen ime e s) compa ed wi h a single mode ibe (se e al kilome e s o
ibe ). Howe e , he B agg g a ings could also p esen some disad an ages
espec o SMF, such as de ia ions in he a e age dispe sion as well as high
equency ipples in he g oup delay esponse [60].
1.3 Tempo al Sel -Imaging E ec
The spacial Talbo e ec en ails he o ma ion o sel -images o a pe iodic
objec along he p opaga ion di ec ion [21, 22]. The sel -images o Talbo
images a e he esul o he cohe en in e e ence be ween all di ac ion
o de s. Fo uni o m illumina ion he Talbo planes a e hose sa is ying he
condi ion,
zγ/α =γ
α
d2
λ0
,(1.17)
1. Tempo al Talbo e ec 11
whe e dis he pe iod o he objec , λ0= 2π/k0 he wa eleng h o he
illumina ing sou ce and γand αa e cop ime and in ege numbe s. Fo
α= 1, exac images o he pe iodic objec a e ound shi ed by hal a pe iod
when γis odd. These planes a e called in ege Talbo planes. Fo α6= 1
pe iodic pa e ns wi h a iche s uc u e a e ound. In hese cases, each cell
in he pa e n is composed o he cohe en supe posi ion o α eplicas o
he o iginal one mu ually shi ed by d/α, wi h a phase di e ence be ween
hem and wi h powe educed by 1/α. These a e called ac ional Talbo
images. In case o he s uc u e o cells is su icien ly con ined o a oid
supe posi ion be ween neighbo ing cells, he pa e n esul s in a eplica o
he o iginal one bu wi h a pe iod α imes smalle , shi ed again by hal a
pe iod i αγ is odd.
Acco ding o he space- ime duali y he empo al analogy o he
sel -imaging e ec appea s when a cohe en pe iodic empo al signal is
p opaga ed h ough a dispe si e line, unde i s -o de condi ions. He e
we will cen e ed ou a en ion on he empo al Talbo e ec applied o he
inc easing o he epe i ion a e. We will desc ibe i in e ms o in ensi y
since i is in his domain whe e he mul iplica ion o he epe i ion a e
o he ain akes place. We conside an inpu signal wi h a epe i ion
pe iod 0. No ice ha his pe iodic signal is he empo al equi alen o he
pe iodic complex ampli ude dis ibu ion, in such a way ha he epe i ion
ime 0co esponds o he spa ial pe iod d. Ma hema ically, he op ical
en elope o he inpu pe iodic signal can be exp essed as
a( , 0) = ( )⊗
+∞
X
m=−∞
δ( −m 0) =
+∞
X
m=−∞
amexp(i2πm / 0),(1.18)
whe e ⊗deno es he con olu ion ope a ion. ( ) ep esen s an a bi a y
pulse shape con ined wi hin | |< 0/2, so ha he incoming op ical in ensi y
is I( , 0) = | ( )|2⊗Pkδ( −k 0). The se o a iables am ep esen s he
ha monics o he signal, am 0=F(ω=m2π/ 0), F(ω) being he Fou ie
ans o m o he pulse p o ile ( ).
As has been shown in p e ious sec ion a linea i s -o de dispe sion
medium is a linea ime-in a ian sys em, so ha i is comple ely
cha ac e ized by i s ans e unc ion, H(ω, z) [3]
H(ω, L) = exp (−iβ2zω2/2),(1.19)
o equi alen ly by i s impulse esponse, h( , z) which is he in e se Fou ie
ans o m o he ans e unc ion
12 1.3. Tempo al Sel -Imaging E ec
h( , z) = FT−1[H(ω, z)] = (i2πβ2z)−1/2exp(i 2/2β2z).(1.20)
ep esen s he p ope ime, ha is, he ime exp essed in he e e ence
ame mo ing wi h he pulse, = phy −β1z, phy being he physic ime.
A e a eling a dis ance zin a dispe si e medium desc ibed by Eq.
(1.19)-(1.20), he esul ing op ical en elope, a( , z) can be exp essed as he
con olu ion o he complex en elope o he inpu pulse wi h he impulse
esponse, a( , z) = a( , 0)⊗h( , z). Acco ding wi h he con olu ion heo em,
he op ical signal can be equi alen ly compu ed in he Fou ie domain by
di ec ly mul iplica ion o he spec um o he incoming signal, A(ω, 0) wi h
he ans e unc ion, A(ω, z) = H(ω, z)A(ω, 0). Thus, he dispe sed op ical
en elope esul s in:
a( , z) =
+∞
X
m=−∞
amexp(i2πm / 0) exp(−i2π2m2β2z/ 2
0) (1.21)
We now p oceed o he analysis o he op ical in ensi y o dispe si e
lines sa is ying Talbo condi ion. We i s ly de i e he esul ing op ical
in ensi y unde in ege Talbo condi ions, and subsequen ly in subsec ion
1.3.2 ac ional Talbo lines a e analyzed.
1.3.1 In ege Talbo e ec
The in ege Talbo condi ion, in ime domain, is di ec ly de i ed by se ing
α= 1 in Eq. (1.17) and making use o he con e sion able 1.1:
|β2|zγ=γ 2
0
2π,(1.22)
whe e γis an in ege numbe . Unde his condi ion he op ical en elope,
Eq. (1.21) is exp essed as
a( , zγ) =
+∞
X
m=−∞
amei2πm / 0e−iπm2γ(1.23)
The second phase elemen in Eq. (1.23), e−iπm2γcan be eexp essed as a
linea phase in he index m,e−iπmγ. I means ha in ege Talbo lines
wi h odd alues o γin oduce a phase di e ence o πbe ween consecu i e
ha monics which is ansla ed in o a shi ing o hal a pe iod in he ime
domain. The ou pu op ical in ensi y, I( , zγ/α) = |a( , zγ/α)|2, esul s in
1. Tempo al Talbo e ec 19
on he igh , he co esponding in ensi y p o iles. The b oadband spec al
en elope |J(ω)|2/|ζ(ω)|is depic ed wi h a hin con inuous cu e. The hick
con inuous cu e ep esen s he p oduc |J(ω)Z(ω)|2/|ζ(ω)| ha con ols
he powe ca ied by each ha monic o he ou pu in ensi y ain. The
powe o he ha monics is he alue o his hick line a he mul iples o he
undamen al equency Ω0= 2π×10 GHz, which a e depic ed wi h do s.
On he igh hand side, he hin con inuous cu e ep esen s wo pe iods
o he in ensi y o he inpu ain, and he hick cu e wo pe iods o he
ou pu in ensi y. The in ensi y PSD has been no malized o 0 dB a he
ca ie , whe eas he empo al signals ha e been no malized o uni y in he
cen e o he pe iod o help in isualizing he pulse in e e ence pa e ns.
−150
−120
−90
−60
−30
0
β2L=0 ps2/ ad
β3L=0ps3/ ad2
0 20 40 60 80 100 120 140 160
−150
−120
−90
−60
−30
0
β2L=−90 ps2/ ad
β3L=0.557ps3/ ad2
F equency (GHz)
In ensi y PSD (dB/Hz)
0
0.2
0.4
0.6
0.8
1
−100 0 100
0
0.2
0.4
0.6
0.8
1
Time (ps)
No malized in ensi y
Figu e 1.1: In ensi y PSD(le ) and empo al in ensi y ( igh )o a 10-GHz
ain be o e( op) and a e (bo om) i s p opaga ion in a s anda d SMF o
leng h L= 4.29 km (comp ession leng h)
In Fig. 1.1 he ains a e p opaga ed h ough a SMF wi h wo leng hs:
L= 0 (ini ial ain, op) and L= 4.29 km (bo om). β3is negligible in
bo h examples (ǫ= 0 and ǫ= 0.0034 ad, espec i ely). The in ensi y
PSD o he inpu ain ( op) has wo lobes ha g oup he ha monics o he
in ensi y. They a e cen e ed a DC and 140 GHz, and a e associa ed o he

20 1.4. In ensi y Spec um in Dispe si e Lines
i s wo con ibu ions o he sum (1.41) in Eq. (1.43). Since hei spec al
o e lapping is negligible, he sum can be app oxima ed o |g0(ω)|2+|g1(ω)|2.
The i s lobe |g0(ω)|2, oge he wi h he o e all en elope, ep esen s he
in ensi y spec um o a single pulse, see Eq. (1.45). The second lobe
|g1(ω)|2cen e ed a 140 GHz is due o he sligh in e e ence be ween
neighbo ing Gaussian pulses in he inpu ain. In p ac ice no pulse-
o-pulse in e e ence is p esen in he inpu ain, so ha he in ensi y
ha monics should be g ouped in a single lobe cen e ed a DC. Addi ional
in e e ence lobes only appea as dispe sion causes pulse b oadening and
subsequen in e e ence [68].
The exis ence o in e e ence lobes is a limi a ion o he ep esen a ion
o he inpu ain as a cohe en sum o Gaussian pulses: hei in ini e
empo al ex ension causes pulse- o-pulse o e lapping and in e e ence e en
in he inpu signal. Howe e , he alue o he powe o he ha monics
desc ibing he neighbo ing-pulse in e e ence e m, mo e han 90 dB below
he ca ie , is su icien ly small o ely on he alidi y o he model.
−150
−125
−100
−75
−50
−25
0β2L=−210 ps2/ ad
β3L=1.3 ps3/ ad2
0 20 40 60 80 100 120 140 160
−150
−125
−100
−75
−50
−25
0β2L=−273 ps2/ ad
β3L=1.69 ps3/ ad2
F equency (GHz)
In ensi y PSD (dB/Hz)
0
0.2
0.4
0.6
0.8
1
−100 0 100
0
0.2
0.4
0.6
0.8
1
Time (ps)
No malized in ensi y
Figu e 1.2: In ensi y PSD(le ) and empo al in ensi y ( igh )o a 10-GHz
ain a e i s p opaga ion in a s anda d SMF o leng h L= 10 km ( op)
and L= 13 km (bo om)
1. Tempo al Talbo e ec 21
The second example in Fig. 1.1 (bo om) shows he signal a e he
comp ession leng h LC=−C/σ2
ωβ2= 4.29 km, whe e pulses each he
minimum empo al wid h, 1/σω= 6.7 ps. A his leng h he equi alen
dispe sion, Deq de ined in Eq. (1.37), anishes. This means ha |Z(ω)|2∼
=
|g0(ω)|2= 1, see (1.42) and he de ini ion o ωn, so ha he spec um
ollows he en elope |J(ω)|2/|ζ(ω)|. In o he wo ds, his en elope is he
single-pulse en elope o comp essed pulses. This ac can also be de i ed
by se ing Deq = 0 in Eq. (1.46). The e o e, he ac o g0(ω) in he single-
pulse spec um (1.46) accoun s o he di e ence be ween comp essed and
non-comp essed pulse spec a.
In igu e 1.2 he ibe leng h is inc eased beyond he comp ession
condi ion up o L= 10 km ( op) and L= 13 km (bo om). The
second o de dispe sion is s ill negligible (ǫ= 0.0080 ad and 0.0104 ad,
espec i ely). In i s example ( op igu e) he in e e ence be ween pulses
yields an inc ease in powe o he second lobe |g1(ω)|2. In empo al domain
pulses a e b oaden and he supe posi ion be ween neighbo ing pulse is
now appa en . In second example (bo om igu e) he powe spec um
is composed o ee lobes: |g0(ω)|2,|g1(ω)|2and |g2(ω)|2which co espond
o single pulse dispe sion, in e e ence be ween neighbo ing pulses, and
in e e ence be ween al e na e pulses, espec i ely. The lobe |g1(ω)|2has
aised in powe and i s peak is now cen e ed a 80 GHz leading o an
inc ease in powe o he ha monics a ound ha equency. I can be
obse ed in i s empo al coun e pa as an in e e ence pa e wi h eigh
maxima pe pe iod.
1.5 In ensi y Spec um in Talbo Lines
In sec ion 1.3 we ha e shown ha sel -imaging e ec can be achie ed when
a pe iodic ain o pulses, such as (1.34), is p opaga ed in a i s -o de
dispe sion medium, whose lowes -o de accumula ed dispe sion is a mul iple
o he basic scale 2
0/2π. This e ec can be also achie ed by ma ching
he second-o de accumula ed dispe sion o e en highe o de dispe sion
[19,69,70]. He e we a e conside ing a dispe si e line wi h i s and second
o de dispe sion. The Talbo condi ion will be de ined by ma ching he
accumula ed i s -o de dispe sion, whe eas he second-o de dispe sion will
be conside ed as a de ia ion om he ideal beha io . Then, acco ding wi h
Eq. (1.26), he sel -images a e achie ed when he ibe leng h Lsa is ies
he condi ion:
22 1.5. In ensi y Spec um in Talbo Lines
|β2|LΩ2
0= 2πγ
α.(1.47)
whe e γand αa e in ege and cop ime numbe s. The quo ien γ/α is called
Talbo index.
Due o he s uc u e o he quad a ic phases in (1.35) and (1.36),
ano he alue o o al dispe sion is o in e es :
DeqΩ2
0=|β2L+C/σ2
ω|Ω2
0= 2πγ
α.(1.48)
We will e e o (1.48) as he comp essing Talbo condi ion. To in e p e
his condi ion, le us i s conside ha he leng h o he dispe si e medium
is adjus ed o e i y (1.48), he ou pu signal is a Talbo image (in ege
o ac ional, depending on α) o a ain o comp essed pulses. Le us
u he suppose ha phase modula ion and dispe sion ha e opposi e sign,
C·β2<0, so ha he leng h o he medium can be spli in wo posi i e
con ibu ions, L=LC+LT, he i s being he comp ession leng h and he
second he Talbo dis ance. This i s pa o he line is used o comp ess
indi idually he pulses in he ain [5], and he emaining leng h o he
ibe , LT, p oduces he Talbo image o he ain o comp essed pulses.
I bo h phase modula ion and dispe sion ha e he same sign, C·β2>0,
he comp ession condi ion (1.48) p oduces he same o e all esul because
he inal s uc u e o he spec al phases is equal o he p e ious case,
C·β2<0. The only di e ence is ha he dis ance o comp essing Talbo
condi ion when C·β2>0 is sho e han he dis ance when C·β2<0.
Be ween Talbo and comp essing Talbo condi ions (1.47) and (1.48), a
con inuum o pa ial comp ession is ob ained.
Now we analyze he in ensi y spec um o he empo al Talbo e ec
by means o some ep esen a i e examples. Le us i s conside in Fig. 1.3
an in ege sel -imaging dispe si e line wi h index γ/α = 1. Pulses ha e a
wid h o 15 ps and a chi p C= 2 ad, as in Figs. 1.1 and 1.2. The dispe si e
medium is a SMF a λ= 1550 nm, and comp essing condi ions (L= 80.1
km) a e shown in he igu es a he op, while non-comp essing condi ions
(L= 75.8 km) a e depic ed a he bo om. In bo h examples second-o de
dispe sion is negligible (ǫ= 0.064 ad and 0.061 ad, espec i ely). The
hick con inuous cu e ep esen s again he unc ion |J(ω)·Z(ω)|2/|ζ(ω)|
ha accoun s o he powe ca ied by he ou pu ha monics. A he ou pu
o he ibe we ge a ain o comp essed pulses in he i s example and
a ain o non-comp essed pulses in he second one, bo h o hem shi ed
by hal a pe iod because γis odd (γ= 1). The in ensi y PSD ollows wo
1. Tempo al Talbo e ec 23
−150
−120
−90
−60
−30
0β2L=−1.68x103 ps2/ ad
β3L=10.4ps3/ ad2
0 20 40 60 80 100 120 140 160
−150
−120
−90
−60
−30
0β2L=−1.59x103 ps2/ ad
β3L=9.85ps3/ ad2
F equency (GHz)
In ensi y PSD (dB/Hz)
0
0.2
0.4
0.6
0.8
1
−100 0 100
0
0.2
0.4
0.6
0.8
1
Time (ps)
No malized in ensi y
Figu e 1.3: In ensi y PSD(le ) and empo al in ensi y ( igh )o a 10-GHz
ain a e i s p opaga ion in a s anda d SMF o leng h L= 80.1km ( op)
and L= 75.8km (bo om), which co espond o he comp essing Talbo and
s anda d Talbo condi ions wi h index γ/α = 1, espec i ely.
di e en cu es. The i s one ( hin con inuous cu e) co esponds o he
in ensi y spec um o a comp essed pulse, while he second one (dashed
cu e) ollows he spec um o non-comp essed pulses. F om Fig. 1.3 we
can in e p e how Talbo e ec c ea es he sel -images o he ain. Mul iple
pulse- o-pulse in e e ence is e lec ed in he s uc u e o dips and bumps
o he unc ion |J(ω)·Z(ω)|2/|ζ(ω)|. This s uc u e is due o he sum o e
gn(ω) in Z(ω), see Eq. (1.41). In he absence o second-o de dispe sion
he spec al sepa a ion be ween consecu i e unc ions gn(ω) and gn+1(ω) is
0/Deq, while hei spec al wid h goes as 1/σωDeq. Then, o ibe leng hs
o he o de o magni ude o he Talbo dis ances, Deq ∼1/Ω2
0, he spec al
o e lapping be ween unc ions gn(ω) can be neglec ed i σω≫Ω0, i.e., i he
pulse wid h is smalle han he pe iod o he inpu ain. Then, |Z(ω)|2can
be app oxima ed by Pn|gn(ω)|2and he in ensi y PSD can be ew i en
as:
24 1.5. In ensi y Spec um in Talbo Lines
S(ω, L)∼
=Ω0|J(ω)|2|ζ(ω)|−1X
s
Gs(ω)X
k
δ(ω−kΩ0) (1.49)
whe e Gn(ω) = |gn(ω)|2. Dispe sion hus ac s as a mul iple bandpass il e ,
he posi ions and wid h o he passbands being de e mined by Gn(ω). Thei
cen e s a e loca ed a :
ωn=n 0/Deq,(1.50)
and hei 30-dB decay hal wid h is, i second-o de dispe sion is negligible:
∆ω= 3.72/σωDeq.(1.51)
Then, om Fig. 1.3 ( op), i can be obse ed ha he ou pu ha monics
o he comp essing Talbo eplica o he o iginal ain lie a he cen e o
he n- h passband ha desc ibes he in e e ence be ween pulses sepa a ed
n ime slo s. In o he wo ds, he n- h ha monic o he in ensi y ain
is c ea ed by in e e ence o pulses sepa a ed n ime slo s. Unde non-
comp essing Talbo condi ion he beha io o he in ensi y PSD is simila ,
bu he ou pu ha monics do no lie on he passband’s cen e s, see Fig. 1.3
(bo om). Howe e , he o e all pic u e is simila .
This poin o iew can be u he explo ed by conside ing he examples
p esen ed in Fig. 1.4. The same ain is p opaga ed h ough a SMF o he
comp essing Talbo leng hs wi h indices 1/2 and 3/2. The ou pu ain is
simila , since in bo h examples he ou pu is a 20-GHz ain o ans o m-
limi ed pulses. Again, second-o de dispe sion is negligible (ǫ= 0.034
ad and 0.094 ad, espec i ely). In bo h examples, he inc ease o he
epe i ion a e is mani es ed in he supp ession o he odd ha monics o
10 GHz. Howe e , he in e e ence s uc u es a e di e en . In he i s
example, abo e, he h- h ou pu ha monic is c ea ed by he in e e ence o
pulses sepa a ed by h ime slo s, while in he second example, below, i is
c ea ed by he in e e ence o pulses o iginally sepa a ed by 3h ime slo s.
In gene al, he passbands associa ed o comp essing Talbo lines a e
cen e ed a ωn=nαΩ0/γ,nbeing he index de e mining he in e e ence
be ween pulses sepa a ed by n ime slo s. When nis a mul iple o γ,n=hγ
o ce ain in ege h, he passband is cen e ed a he h- h ou pu ha monic
hαΩ0. Then, i he comp essing Talbo line is in ege (α= 1) he h- h
ha monic lies in he cen e o he passband wi h index n=h, so ha all
he ha monics su i e and be ween wo consecu i e ha monics he e a e
γ−1 passbands. Con e sely, i he comp essing dispe si e line is ac ional
(α6= 1), only he equencies mul iples o he α- h inpu ha monic lie o e

1. Tempo al Talbo e ec 25
−150
−120
−90
−60
−30
0β2L=−886 ps2/ ad
β3L=5.48ps3/ ad2
0 20 40 60 80 100 120 140 160
−150
−120
−90
−60
−30
0β2L=−2.48x103 ps2/ ad
β3L=15.3ps3/ ad2
F equency (GHz)
In ensi y PSD (dB/Hz)
0
0.2
0.4
0.6
0.8
1
−100 0 100
0
0.2
0.4
0.6
0.8
1
Time (ps)
No malized in ensi y
Figu e 1.4: In ensi y PSD(le ) and empo al in ensi y ( igh )o a 10-GHz
ain a e i s p opaga ion in a s anda d SMF o leng h L= 42.2km ( op)
and L= 118 km (bo om), which co espond o he comp essing Talbo
condi ions wi h index γ/α = 1 and 3/2, espec i ely.
he cen e o a passband. These su i ing ha monics o igina e he spec al
con en o he ou pu ain, he emaining ha monics being il e ed.
The nex example in Fig. 1.5 explo es he supp ession o ha monics in
Talbo de ices wi h index 1/3 and i s dependence on he op ical linewid h.
A 10-GHz ain wi h pulses o wid h 15 ps is launched in o a SMF
comp essing dispe si e line wi h index 1/3. Abo e, he chi p pa ame e
o he pulses is C= 2 ad, so ha he wid h o he comp essed pulses is
6.7 ps. The 1/3 comp essing Talbo line has a leng h L= 29.5 km. Below,
he chi p pa ame e o he pulses is C= 1 ad, he comp essed pulses
a e 10.6 ps wide, and he co esponding leng h is 30.6 km. Second-o de
dispe sion is s ill negligible (ǫ= 0.0237 ad and 0.0062 ad, espec i ely).
We obse e ha he supp ession o ha monics ha do no belong o he
ou pu 30-GHz ain is no exac . These ha monics will be e e ed o as
esidual ha monics. The pa ial supp ession o esidual ha monics is due
o he ini e wid h o he bandpass Gn(ω), which a e, acco ding o (1.51),
in e sely p opo ional o he spec al wid h.
26 1.5. In ensi y Spec um in Talbo Lines
−150
−120
−90
−60
−30
0β2L=−621 ps2/ ad
β3L=3.84ps3/ ad2
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120 140 160
−150
−120
−90
−60
−30
0β2L=−643 ps2/ ad
β3L=3.84ps3/ ad2
F equency (GHz)
In ensi y PSD (dB/Hz)
−100 0 100
0
0.2
0.4
0.6
0.8
1
Time (ps)
No malized in ensi y
Figu e 1.5: In ensi y PSD(le ) and empo al in ensi y ( igh )o a 10-GHz
ain a e i s p opaga ion in a s anda d SMF o leng h L= 29.5km ( op)
and L= 30.6km (bo om), which co espond o he comp essing Talbo
condi ions wi h index γ/α = 1/3. Pulses ha e a wid h o 15 ps and chi p
C=2 ad ( op) and C=1 ad (bo om) and 3/2.
In Fig. 1.5, below, we obse e ha a educ ion o he spec al wid h
implies, om he empo al poin o iew, ha pulses o e lap in he ou pu
ain causing pulse pedes al and in ensi y luc ua ion. As a esul , he
empo al p o ile shows a pe iodic s uc u e wi h equency Ω0, e lec ing
he absence o exac epe i ion- a e mul iplica ion. F om he spec al poin
o iew we no ice, on one hand, he educ ion o he spec al wid h, and, on
he o he , he inc ease o he powe ca ied by esidual ha monics, due o
he b oadening o he passbands Gn(ω). The ine icien il e ing o esidual
ha monics is clea ly shown a 10 and 20 GHz.
I is impo an o unde line ha he ac ional Talbo lines wi h indices
γ/α = (2k+ 1)/2 di e o m any o he ac ional Talbo de ice. In Fig.
1.4 he odd esidual ha monics a e o ally supp essed, independen ly o he
alue o he op ical spec al wid h, bu his ac is no gene al [25]. This
exac cancela ion occu s i he inpu pulses a e ans o m-limi ed, i. e., i
he inpu en elope is a eal unc ion so ha i s Fou ie coe icien s in (1.35)
1. Tempo al Talbo e ec 27
−150
−120
−90
−60
−30
0
β2L=−143 ps2/ ad
β3L=20 ps3/ ad2
0 40 80 120 160 200 240 280
−150
−120
−90
−60
−30
0
β2L=−275 ps2/ ad
β3L=38.5 ps3/ ad2
F equency (GHz)
In ensi y PSD (dB/Hz)
0
0.2
0.4
0.6
0.8
1
−50 0 50
0
0.2
0.4
0.6
0.8
1
Time (ps)
No malized in ensi y
Figu e 1.6: In ensi y PSD(le ) and empo al in ensi y ( igh )o a 20-GHz
ain a e i s p opaga ion in a s anda d DSF o leng h L= 285 km ( op)
and L= 551 km (bo om), which co espond o he comp essing Talbo
condi ions wi h index γ/α = 1/3and 2/3, espec i ely.
e i y ha ϕm=−ϕ−m. This, oge he wi h he spec al phases caused by
dispe sion, p oduces a ela i e phase shi o ±πbe ween op ical spec al
lines wi h indices ±m, which implies he o al supp ession o he esidual
ha monics. This obse a ion is simila o he well-known ca ie supp ession
e ec in analog communica ions [71]. In ac , comp essing Talbo dis ances
wi h indices γ/α = (2k+ 1)/2, and only hese dis ances, coincide wi h he
dis ances o ca ie supp ession.
To conclude his sec ion we conside he las example in Fig. 1.6
whe e second-o de dispe sion is no negligible. A 20-GHz ain o pulses
wi h wid h 5-ps and chi p C= 2 ad is p opaga ed h ough a dispe sion
shi ed ibe (DSF; β2= 0.5 ps2/km · ad and β3= 0.07 ps3/km · ad2
a λ= 1550 nm), whose leng h is adjus ed o e i y he comp essing
Talbo condi ion γ/α = 1/3 ( op) and 2/3 (bo om). The hin con inuous
and dashed cu es ep esen he en elope o he in ensi y PSD ha
would co espond o he comp essing Talbo and non-comp essing Talbo
condi ion, espec i ely, i he second-o de dispe sion coe icien was ze o.
28 1.6. Rejec ion P ope ies o Talbo Fil e s
The hick con inuous and he hin do ed cu es a e he en elope | (ω)·
J(ω)|2/|ζ(ω)|. Finally, he do s ep esen he ha monic con en o he
ou pu ain.
The op igu e 1.6 (ǫ= 3.32 ad) shows a sligh dec ease o o e all
bandwid h, as compa ed wi h he case β3= 0, and a p og essi e b oadening
o he passbands’ wid h along he RF spec um. These wid hs can be
app oxima ed by
∆nω∼
=3.72|ζ(ωn)|1/2
σωDeq
.(1.52)
The wid h enla gemen is due o he ac o |ζ(ω)|1/2, and weakens he
supp ession o il e ing o esidual ha monics. The e o e, he ou pu signal
con ains high equencies ha a e mani es ed in he ime domain as he
oscilla o y s uc u e in he pulse’s ailing edge. Fo u u e use, we compu e
he e he maximum alue o he 30-dB passband hal wid h in he signi ican
pa o he spec um, ω∼3.72σω:
∆ωmax ∼
=3.72(1 + ǫ2)1/4/σωDeq.(1.53)
whe e Eq. (1.44) has been used. No ice ha , despi e he p esence o
signi ican β3dispe sion, he ain o Fig. 1.6 ( op) has no in ensi y
luc ua ions, b oadening o pedes al, he only impai men being he
dis o ion induced by second-o de dispe sion. I he in luence o β3is
su icien ly la ge he passbands o e lap, causing he ading o he mul iple
passband s uc u e. The app oxima ion assumed in Eq. (1.49) is no longe
alid and i is equi ed o eso o (1.43) o he co ec desc ip ion o he
in ensi y PSD. This is shown a he bo om Fig. 1.6 whe e β3in luence
is now la ge (ǫ= 6.41 ad). No ice ha in his egime he ou pu signal
shows pe iodic a ia ions in pulse in ensi y in addi ion o pulse dis o ion.
Pulse in ensi y luc ua ions a e pe iodic, and hey a e due o he unde lying
non- il e ed inpu ha monics.
1.6 Rejec ion P ope ies o Talbo Fil e s
As has been illus a ed in he p e ious sec ion, he dispe si e mechanism o
Talbo e ec is desc ibed in he spec al domain by he passbands Gn(ω).
This il e ing is exac only when he dispe sion line is adjus ed o hal -
in ege comp essing Talbo indices γ/α = (2k+ 1)/2 and highe -o de
dispe sion is negligible. In any o he si ua ion, Talbo il e ing is no ideal.
1. Tempo al Talbo e ec 35
2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
10−2
10−1
No malized spec al wid h W
Rela i e leng h de ia ion δL/L
γ/α=1
γ/α=1/4
γ/α=1/2
Eq. (1.64)
Eq. (1.62)
Figu e 1.8: Rela i e leng h de ia ion allowed by he bounds (1.62) and
(1.64) wi h espec o he no malized spec al wid h W o di e en Talbo
il e s wi h negligible second-o de dispe sion.
dispe si e il e . Using he de ini ion o W, and assuming negligible second-
o de dispe sion, op imal ole ance condi ion (1.65) can be eph ased in
e ms o he spec al wid h o he inpu ain. When i is measu ed as he
FWHM, op imal condi ions equi e ha he FWHM is wice he epe i ion-
a e equency o he ou pu ain, 2αΩ0. F om he empo al poin o iew,
and assuming ans o m-limi ed pulses, his condi ion s a es ha 99% o
he ene gy o a pulse is con ained in hal a pe iod o he mul iplied ain,
0/2α. The op imal alue o ole ance is hen:
|δL|
L<1
K(4αγ + 1) ∼
=0.10
αγ(1 + ǫ2)1/4.(1.66)
No ice ha i is in e sely p opo ional o he p oduc γα and, he e o e, o
he complexi y o he il e . When we conside he usual ac ional Talbo
il e s o he se ies γ/α = 1/α , Eq. (1.66) s a es ha he expec ed op imal
ole ance in leng h is abou 10% di ided by he epe i ion- a e ac o α. Fo
ins ance, in he examples in Fig. 1.8, W= 4.4, and he ole ance in leng h
gi en by (1.66) is 5%. No ice inally om Fig. 1.8 ha op imal ole ance

36 1.8. Conclusions
(1.66) is loca ed a he edge o a pla eau. Nea ly op imal ole ances can be
ob ained in i s neighbo hood.
1.8 Conclusions
In his chap e we ha e b ie ly e iewed he well-known space- ime
duali y which is based on he ma hema ical equi alence be ween equa ions
desc ibing one-dimensional pa axial di ac ion and pulse p opaga ion in
a linea i s -o de dispe si e medium wi h negligible a enua ion. Unde
his analogy, he pulse en elope is equi alen o he complex ampli ude
dis ibu ion o ligh in di ac i e op ics. In his con ex , we ha e p esen ed
he empo al coun e pa o he sel -imaging o Talbo e ec . We ha e
analyzed, in e ms o in ensi y, an a bi a y ain o pulses a e p opaga ion
h ough a dispe si e medium unde Talbo condi ion. Fo in ege Talbo
de ices he ou pu in ensi y is a eplica o he inpu ain, whe eas
o ac ional Talbo lines an in ensi y ain wi h highe epe i ion a e
is ob ained p o ided ha inpu pulses a e su icien ly na ow o a oid
o e lapping. In bo h cases he ou pu in ensi y is shi ed by hal a pe iod
depending on he pa i y o Talbo index.
Talbo de ices a e also analyzed om he spec al poin o iew. This
s udy is based on he compu a ion o he exac in ensi y PSD o a cohe en
ain o pulses a e i s p opaga ion in a dispe si e medium wi h a bi a y
i s and second o de dispe sion and negligible a enua ion. We ha e
conside ed ains composed wi h linea ly chi ped Gaussian pulses. These
analy ical esul s enable he iden i ica ion o he spec al componen s ha
ep esen he dispe sion o indi idual pulses, as well as he e ms whe e
he in e e ence be ween pulses is enclosed. In his way we ha e made
connec ion wi h he exis ing esul s in he li e a u e [63, 65–67]. F om
he spec al poin o iew, Talbo dispe si e lines can be in e p e ed
as a mul iple bandpass il e whe e each passband co esponds o he
in e e ence be ween pulses o iginally sepa a ed a numbe o imes slo s.
The il e essen ially ejec s he in ensi y ha monics o he inpu ain ha
do no belong o he ou pu . The ejec ion is exac only in semi-in ege
Talbo il e s, wi h negligible highe -o de dispe sion, and ans o m-
limi ed pulses, since in his case he il e is adjus ed o he dispe sion o
ca ie supp ession [71]. The il e is no s a ic, bu depends on he spec al
wid h o he inpu ain and he dispe si e cha ac e is ics o he line. In
pa icula , condi ions ha e been de i ed ha gua an ee ha he Talbo
il e p o ides ou pu in ensi y ains wi h negligible in ensi y luc ua ions,
1. Tempo al Talbo e ec 37
e en in he p esence o highe -o de dispe sion. In hese cases, howe e ,
he pulses may p esen dis o ion. The de i ed condi ion, Eq. (1.59),
essen ially s a es ha he pulses o he ou pu ain do no o e lap. Finally,
he s abili y unde iming and leng h a ia ions o he Talbo line has been
analyzed. We ha e shown he p ope ies o he wo egimes o ope a ion o
Talbo il e s, desc ibed by wo bounds, Eqs. (1.62) and (1.64). When
he spec al wid h o he ain is la ge, dispe sion ins abili ies end o
b oaden he pulses and he il e ’s pe o mance wo sens by pulse- o-pulse
in e e ence. In con as , when he spec al wid h is mode a e, dispe sion
a ia ions dec ease he ejec ion abili y o he il e . Op imal s abili y is
eached when he spec al wid h o he inpu ain, measu ed as FWHM,
equals wice he epe i ion- a e equency o he ou pu ain. The expec ed
leng h ole ances a e abou 10% di ided by he epe i ion- a e ac o .
Chap e 2
Non-ideal cohe en
Tempo al Talbo e ec
2.1 In oduc ion
As has been shown in he p e ious chap e , he Talbo e ec is caused by
he in e e ence be ween all di ac ed o de s o a g a ing in space domain
and by he mul iple in e e ence among all pulses o a ain in i s empo al
coun e pa . I means ha all di ac ion o de s o all dispe sed pulses
ge in ol ed in he econs uc ion o each uni cell o pulse o he esul ing
Talbo image. In his sense, i is a collec i e phenomenon, since he pa e n
is a consequence o he whole pe iodic s uc u e o he g a ing o o he ain
o pulses. I is expec ed om a collec i e phenomenon like his some kind
o s abili y unde small a ia ion o he pa ame e s de ining he cells o
he pe iodic s uc u e. In di ac i e op ics, se e al wo ks ha e epo ed on
Talbo e ec as a me hod o smoo h impe ec ions o a g a ing, such as he
absence o a uni cell, o localized modi ica ions o he basic pe iod [21,73].
The deg ada ion o he e ec due o he ini e o de s om he op ical axis
is also known [22]. This walk-o e ec p og essi ely educes he pa axial
egion whe e Talbo e ec akes place. In he empo al domain he s udy
o he deg ada ion by he ini e wid h o pulse ains has epo ed in e [74]
The aim o his chap e is o desc ibe he ole ance o Talbo de ices
agains ain impe ec ions. Unlike he wo ks p e iously ci ed, ou
app oach o he p oblem is no de e minis ic bu s ochas ic. Al hough
he p esen ed analysis is alid in bo h di ac i e and dispe si e domain,
we will desc ibe i om he empo al poin o iew. Among all possible
de ia ion o he ideal pe iodic signal, we will ea i s he andomly on-
39
40 2.1. In oduc ion
o keyed pulses in he ain and la e he ji e y pulses in he ain. In
bo h cases Talbo de ices a e shown o ha e il e ing p ope ies a ound
equencies o he o de o he epe i ion a e o he ain. The use o his
p ope y is hen limi ed o b oadband noise educ ion.
This chap e is spli in o wo la ge sec ions. Sec ion 2.2 is de o ed o
he analysis o a andomly dis ibu ed pulses a e a Talbo de ice. This
s udy is ca ied ou in he Fou ie domain by means o he powe spec um
densi y, in he same way as he analysis o exac Talbo e ec in he p e ious
chap e . By assuming linea ly chi ped Gaussian pulses oge he wi h he
independence o he a iables which de e mine he p esence o absence
o a pulse in he ain, an exac s ochas ic esul o he RF-spec um
is achie ed. The noise spec al a e Talbo de ice consis s o a se o
passbands, each one being cosine-squa ed modula ed. The cen e s o he
passbands a e shi ed depending on he alue o he chi p. This ac , allows
an indi ec cha ac e iza ion o he pulse in he ain h ough he analysis o
he spec um. In pa icula , measu emen s o he posi ions and he wid h
o he bandpass esul s in he de e mina ion o he chi p o he ini ial pulses.
Mo eo e , he esul ing noise spec al a ies con inuously wi h espec o
he pa ame e s ha de ine he model. I means ha , in o de o obse e
he spec al ea u es o noise, no ine uning o he ini ial epe i ion a e
o he ain is necessa y o ma ch he Talbo condi ions, in con as wi h
he empo al domain whe e he ace o he Talbo e ec is he inc ease o
he epe i ion a e o he ain. Howe e , o cla i y, all esul s a e de i ed
unde exac Talbo condi ions.
Sec ion 2.3 deals wi h he beha io o Talbo de ices when hey a e ed
wi h a ain su e ing om iming ji e . Despi e he ac we ha e cen e ed
ou s udy in iming ji e , he o malism we p esen can be easily expanded
o ampli ude ji e o a combina ion o hem [78]. Fi s ly, we app oach he
p oblem in ime domain by means o he a iance, in subsec ion 2.3.1.
Analy ical exp essions o he inpu and ou pu a iance a e epo ed,
showing ha he a iance o he o iginal ain is la ed a e passing a
Talbo de ice. The p esence o pedes al in he ou pu ain is also s udied.
In subsec ion 2.3.2 we analyzed he in ensi y spec um o he ji e y ains
be o e and a e he p opaga ion o he ain h ough he Talbo il e .
This spec al analysis is mo e gene al han he p e ious one in e ms
o a iance, since co ela ions be ween a iables desc ibing iming ji e
a e now allowed. The esul s p esen ed he e a e no exac bu ely on a
small-signal app oxima ion o ji e sou ce. This echnique has been widely
applied in he analysis o he RF in ensi y spec um o iming ji e noise

2. Non-ideal cohe en Tempo al Talbo e ec 41
in mode-locked lase s [75–78]. The esul s a e exempli ied o wo models
o ji e pulse co ela ion, one desc ibing he noise o undamen ally mode-
locked lase s and he o he one desc ibing ha monically mode-locked lase s.
In subsec ion 2.3.3 we nume ically in es iga e he condi ions o a dec ease
in he noise powe a ound he ha monics. Finally, we end he chap e
p esen ing ou conclusions.
2.2 Talbo imaging o On-O -Keyed ains
In his sec ion we analyze he beha io o Talbo il e s when hey a e
ed wi h a cohe en ain composed by andomly dis ibu ed pulses. The
op ical en elope which will be launched in he Talbo line can be exp essed
as,
a( , 0) = X
k
bk ( −k 0) (2.1)
whe e ( ) deno es he pulse o m, 0is he uni in e al o he ain,
and he symbols bka e a se o andom a iables which will be assumed
unco ela ed. These a iables desc ibe he p esence (bk= 1) o absence
(bk= 0) o pulses in he ain wi h p obabili y pand 1 −p espec i ely.
Unde ideal condi ions all pulses a e p esen (p= 1) so ha he ain is
0-pe iodic. The a e age o e all ealiza ions will be deno ed by b acke s
h···i, so hbki=pand i s s anda d de ia ion hb2
ki=σ2
b=p(1 −p).
Unless o he wise s a ed all sums and in eg als uns om −∞ o +∞. Fo
simplici y he pulse o m is assumed o be linea ly chi ped Gaussian:
( ) = exp[− 2(1 −iC)/2 2
p] (2.2)
The pa ame e pis he ms wid h o e e y pulse and Cis he linea chi p o
phase modula ion. In o de o a oid o e lapping be ween incoming pulses
and also o allow epe i ion a e mul iplica ion a he ou pu o he Talbo
de ice, we assume ha he pulse wid h is smalle han he uni in e al
p< 0
I he andomly ain o pulses, Eq. (2.1), is p opaga ed along a is -
o de guided dispe si e medium wi h negligible a enua ion, Eq. (1.16),
he op ical en elope esul s in a cohe en supe posi ion o dispe sed pulses.
a( , ξ) = " 2
p
ρ(1 −iC)#1/2X
k
bkexp −( −k 0)2(1 −iC)
2ρ.(2.3)
42 2.2. Talbo imaging o On-O -Keyed ains
whe e
ρ= 1/σ2
ω+iξ +iC/σ2
ω,(2.4)
σω= (1 + C2)1/2/ pis he linewid h o he incoming pulses and he
pa ame e ξaccoun s o he accumula ed dispe sion in he medium. This
is he ele an pa ame e o he p opaga ion medium. I he guided medium
is chosen a ibe , ξ=β2L,Lbeing he ibe leng h; whe eas o a linea ly
chi ped ibe g a ing ξ=¨
Φ, ¨
Φ being he slope o he LCFG esponse in
e lec ion mode a ound he ca ie equency ω0. This slope is designed
o be cons an wi hin he bandwid h o he inpu signal. In his way,
he s udy will be p esen ed he e, is mani es ly independen on he high
dispe si e medium used as Talbo de ice.
The dispe sed ain, Eq. (2.3), can equi alen ly be exp essed in he
Fou ie domain, as ollows
A(ω, ξ) = p2π
1−iC 1/2
exp(−ρω2/2) X
k
bkexp(−iωk 0),(2.5)
−1000 −800 −600 −400 −200 0 200 400 600 800 1000
0
2
4
6
8
Ampli ude (a.u.)
−1000 −800 −600 −400 −200 0 200 400 600 800 1000
0
1
2
3
4
5
Ampli ude (a.u.)
Time (ps)
Figu e 2.1: Simula ion o he o m o he pulse en elope o 20-uni in e als
o a 10-GHz, p=0.5 andom ain o unchi ped, 12-ps ms Gaussian pulses
be o e (abo e) and a e (below) i s pass h ough a Talbo de ice wi h γ/α =
1.
2. Non-ideal cohe en Tempo al Talbo e ec 43
In Fig. 2.1, we p esen a simula ion o he dispe sion o a po ion o a
10-GHz ain ( 0= 100 ps) andomly on-o keyed, a e i s pass h ough
he i s in ege Talbo de ice, γ/α = 1. The o al dispe sion o he de ice
is ξ= +1592 ps2/ ad. The p obabili y o appea ance o pulses in he ain
is p= 0.5. Pulses a e unchi ped and ha e a ms wid h o 12 ps. The ideal
Talbo would ep oduce he incoming ain shi ed by hal a uni in e al.
The empo al sp eading o pulses due o dispe sion c ea es Talbo images
o med by in e e ence o pulses. This phenomenon is local, depending on
he numbe o neighbo ing pulses a ound a gi en ime slo . This leads o
an unsa is ac o y empo al signal, since he empo al cha ac e is ics o
he ou pu ain depend on he local beha io o he inpu . Ne e heless,
whe e e wo pulses in e e e in Fig. 2.1, hey end o o m he desi ed
ideal ou pu ain. Then, a spec al analysis o he signal can un eil he
abili y o he Talbo de ice o econs uc ou pu ha monics in a non-ideal
si ua ion.
2.2.1 In ensi y Spec um
As has been shown in p e ious chap e , dispe si e lines lea e he powe
spec um o he op ical en elope, |A(ω, ξ)|2, in a ian . Howe e , since
de ec ion is a non linea ans o ma ion o he s ochas ic a iables, a( , ξ)
o A(ω, ξ), he pulse sp eading caused by he dispe sion-induced phases,
can be obse ed in he de ec ed ain.
The powe spec um o he de ec ed ain i( , ξ) = |a( , ξ)|2, is compu ed
by Fou ie ans o ming he mean co ela o , which is he empo al a e age
o he co ela o [79]:
S(ω, ξ) = Z+∞
−∞
dτ exp(−iωτ)lim
T→∞
1
2TZT
−T
d hi( +τ, ξ)i( , ξ)i,(2.6)
whe e he pho ocu en co ela o is
hi( , ξ)i( ′, ξ)i=Z+∞
−∞ Z+∞
−∞
dω1
2π
dω2
2πexp[i(ω1 −ω2 ′)]
×hI(ω1, ξ)I(ω2, ξ)∗i.
(2.7)
44 2.2. Talbo imaging o On-O -Keyed ains
The Fou ie o m o he pho ocu en I(ω, ξ), is ela ed o he Fou ie o m
o he op ical en elope A(ω, ξ) by an au oco ela ion,
I(ω, ξ) = Z+∞
−∞
d exp(−iω )i( , ξ) =
Z+∞
−∞
dω′
2πA(ω+ω′, ξ)A(ω′, ξ)∗.
(2.8)
Then, using Eq. (2.5), he Fou ie o m o he pho ocu en can be
exp essed as
I(ω, ξ) = J(ω)X
k
Bk(ω) exp(−iωk 0),(2.9)
whe e
J(ω) = p√πexp(−ω2/4σ2
ω),(2.10)
Bk(ω) = bkX
s
bk+sgs(ω) exp(−iωs 0/2) (2.11)
and
gs(ω) = exp −σ2
ω
4ξω +Cω/σ2
ω−s 02.(2.12)
To compu e he powe spec al densi y, Eq. (2.6), le us suppose ha he
expec ed alue a di e en equencies o he co ela o o unc ions Bk(ω)
is o he o m
hBk(ω1)Bm(ω2)∗i=hBk(ω1)ihBm(ω2)∗i
+δkmΩ(ω1, ω2) + ∆k−m(ω1, ω2),(2.13)
whe e δkm is he K onecke del a, he as e isk deno es complex conjuga ion,
and Ω(ω1, ω2) and ∆k−m(ω1, ω2) a e adimensional unc ions o be e alua ed
a pos e io i. The i s e m in Eq. (2.13), hBk(ω1)ihBm(ω2)∗igi es ise
o he in ensi y spec um o he signal which is composed o a se o
ha monics mul iples o he undamen al equency Ω0. The beha io o he
signal powe spec um has al eady been analyzed in de ail in he p e ious
chap e , so he e we will ocus in he noise spec al densi y. The second and
hi d e ms in Eq. (2.13) o igina e he noise con ibu ion o he mic owa e
spec um, esul ing in
2. Non-ideal cohe en Tempo al Talbo e ec 51
0 10 20 30 40 50 60 70 80 90 100
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Powe spec um (dB/Hz)
F equency (GHz)
Figu e 2.3: The same as in Fig. 2.2, bu wi h chi ped pulses wi h C= 1
ad.
∆Wω=4.3
σωξ+C/σ2
ω−1∼
=4.3 Ω0
1
0σω
α
γ,(2.24)
whe e, in he las app oxima ion, we assume again ha |ξ| ≫ σ2
ω. Thus
chi p educes he wid h o he passbands i espec i ely o i s sign, h ough
he ac o σω. Finally, he peak noise le el in he op o he passbands o
a ain o chi ped pulses can be aised o lowe ed depending on he alue
o p, as in he unchi ped case. Bu his peak noise le el also su e s he
bandwid h enla gemen o he wideband ou pu noise en elope, |J(ω)|2,
which always aises he peak noise le el o e he ini ial b oadband noise.
In Fig. 2.3, we p esen a simula ion o a ain composed o chi ped
pulses. The chi p pa ame e is chosen as C= 1 ad. The emaining
pa ame e s a e as in Fig. 2.2. The ou pu noise en elope is b oade han
he inpu le el owing o he exis ence o chi p. The passbands’ wid h is
∆Wω= 2π×1.32 GHz, and he ela i e edshi o he cen e s is 8.85×10−3.
Fo ins ance, he ou h ou pu ha monic a 80 GHz is edshi ed 708 MHz.
We see a modula ion o high-o de ha monics, as in Fig. 2.2, and he
lowe ing o he noise peak le el o passbands, co esponding o nonou pu

52 2.2. Talbo imaging o On-O -Keyed ains
ha monics in he le pa o he plo , owing o nonmaximal modula ion.
F om ela ions (2.23) and (2.24), i is s aigh o wa d o ealize ha
he magni ude and chi p o he indi idual pulses can be ex ac ed om he
alues o he passband wid h and he passband shi . To be p ecise,
C=−2.9 sign(ξ)Ω0
s
∆(s)
peakω
(∆Wω)2.(2.25)
This measu emen can be pe o med o any passband, no necessa ily one
o hose su ounding he ha monics (smul iple o γ). No ice also ha i is
independen o he conc e e Talbo de ice used, since i does no depend on
he alue o he Talbo index. This exp ession is also alid when modula ion
o he passband is p esen , al hough in hose cases a i ing o he whole
passband can be necessa y o ob ain p ecise alues o wid h and shi .
2.2.3 Analysis o he Modula ion
The passband unc ions Gi(ω) a e modula ed by a cosine-squa ed e m bo h
in he chi ped and in he unchi ped cases. Gi en a passband index s, he
maximum alue o he modula ion ac o Ms(ω) = 4p2cos2(ωs 0/2) + σ2
b
occu s o
ω=mΩ0/s, (2.26)
wi h many in ege . I is easy o show ha in ains composed o
unchi ped pulses, he modula ion and passband maxima coincide o he
ou pu ha monics, and only in hese cases. The e o e nonou pu ha monic
passbands (s6=hγ) can show ips on op o he passbands, o lowe ing o
he peak noise le el i hey a e su icien ly na ow. When he pulses a e
chi ped, he modula ion maxima s ill emain o e he ha monics, bu he
passbands a e shi ed.
The spec al modula ion ull wid h in he passband o index sis
∆Mω= Ω0/s. (2.27)
In pa icula , he ou pu ha monics co espond o indices s=hγ so ha
∆Mω= Ω0/hγ. This wid h becomes smalle as he ha monic o passband
index becomes la ge , bu i does no depend on he ain cha ac e is ics,
such as he empo al wid h, p, he alue o he chi p, C, o he alue o
p. Then, modula ion o he spassband appea s when ∆Mω < ∆Wω, o
equi alen ly when
2. Non-ideal cohe en Tempo al Talbo e ec 53
4.3 p>γ
α
0
s.(2.28)
Then, modula ion is p esen when he passband index sis high, he
dispe sion γ/α is small, o he ela i e wid h o he pulses p/ 0is high.
The limi desc ibed in ela ion (2.28) is exempli ied in Fig. 2.4, whe e
we simula e a andom 10-GHz ain composed o 400 pulses wi h p= 10 ps,
γ/α = 1, ξ= 1592 ps2/ ad, and p= 0.9, so ha he ou pu ain also has
10 GHz. The 10-dB passband ull wid h is 4.3 GHz, while he modula ion
wid h in ou pu ha monics is 10/h GHz, so ha modula ion is expec ed
o be obse able a e he hi d ou pu ha monic. The con en ions in his
plo a e as in he p e ious igu es. The peak noise le el is aised 5.2 dB
owing o he simula ion o a nea ly ull ain wi h p= 0.9. When
0 10 20 30 40 50 60 70 80 90 100
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Powe spec um (dB/Hz)
F equency (GHz)
Figu e 2.4: Powe spec um o a de ec ed 10-GHz, p=0.9 andom ain
o unchi ped pulses wi h 10-ps ms wid h a e i s pass h ough a Talbo
il e wid h γ/α = 1. The uppe pa o he plo shows (con inuous cu e)
he powe spec um o he de ec ed ou pu ain, (dashed cu e) he powe
spec um o he de ec ed inpu ain, and (do ed cu e) he mean signal
powe spec um o bo h inpu and ou pu ains. The lowe pa shows
analy ical app oxima ions o he noise powe spec um: (con inuous cu e)
ou pu noise; (dashed cu e) inpu noise.
54 2.2. Talbo imaging o On-O -Keyed ains
0 10 20 30 40 50 60 70 80 90 100
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Powe spec um (dB/Hz)
F equency (GHz)
Figu e 2.5: Same as in Fig. 2.4, bu wi h chi ped pulses wi h C=−1
modula ion o a passband appea s, he decay a ound he peak alue has o
conside bo h ∆Wωand ∆Mω. Howe e , since he modula ion occu s o
high ha monics, small dispe sion, and b oad pulses, he decay due o he
s uc u e o he passband is usually negligible, since ∆Wωis la ge han
∆Mω. Using ela ion (2.22), i is s aigh o wa d o compu e he decay Γ
be ween noise le el in he ha monic and he i s modula ion ze o due only
o modula ion, Γ(dB) = 10 log(1 + 3p)/(1 −p).
In Fig. 2.4, he decay Γ is 16 dB. In he wo s case (p= 0.5), Γ a ains
a minimum alue o 7 dB. This is also he ypical alue o decay be ween
consecu i e maxima and minima o modula ion in he same passband.
When he app oxima ion ails because ∆Wωand ∆Mωa e o he same
o de , o when he cen e o he passband is shi ed owing o he chi p
o he basic pulses, he alue o he decay Γ has o be ee alua ed om
ela ion (2.21).
To comple e ou examples, in Fig. 2.5 we plo ou inal simula ion whe e
chi p is aken in o accoun . We choose he same pa ame e s o simula ion
as in Fig. 2.4, excep he alue o he chi p, whose alue is C=−1 ad. The
chi ped ain blueshi s he cen e o he passbands, while he modula ion
does no change he posi ion o i s maxima. This esul s in asymme ic
2. Non-ideal cohe en Tempo al Talbo e ec 55
pa e ns o modula ion. The o he ea u es o he spec um a e as in Fig.
2.4.
2.3 Talbo imaging o mis imed pulses
This sec ion deals wi h he s udy o a Talbo de ice, when i is ed wi h
a ji e y ain o pulses. The analysis will be ca ied ou in he empo al
domain by means o he a iance o he esul ing ain as well as in he
Fou ie domain by means o he powe spec um o he de ec ed ain.
Le us conside a ain o pulses su e ing om iming ji e , which can
be ep esen ed by
a( , 0) = X
k
0( −k 0−ak).(2.29)
The unc ion 0( ) deno es he pulse o m and 0is he uni ime o he ain.
The cen e o pulses in he ain occu s a k=k 0+ak(k=−∞, . . . , +∞).
The se o a iables akis an in ini e collec ion o andom numbe s wi h
dimensions o ime ep esen ing he iming ji e o each pulse. I he
andom numbe s a e se o ze o, he ain o pulses has a pe iod o 0. No ice
ha we ha e conside ed ha ji e shi s pulses om i s ideal posi ion
bu does no a ec o hei shape. The andom a iables a e assumed o
ha e a Gaussian p obabili y densi y unc ion wi h ze o mean and s anda d
de ia ion σj:
p(ak) = 2πσ2
j−1/2exp −a2
k/2σ2
j.(2.30)
The a e age o e all he ealiza ions o each one o ji e pa ame e s will
be deno ed by b acke s, h···i. The e o e, haki= 0 and ha2
ki=σ2
j o any
pulse index k. As in he p e ious sec ion, we will conside linea ly chi ped
Gaussian pulses wi h ms wid h equal o p:
0( ) = exp − 2(1 −iC)/2 2
p,(2.31)
Cbeing he phase-modula ion pa ame e o linea chi p. In o de o a oid
o e lapping be ween incoming pulses we will assume ha he pulse wid hs
a e smalle han he exac pe iod, p< 0, and he s anda d de ia ion o
pulse cen e is smalle han pulse wid h σj< p.
I his ji e y ain, Eq. (2.29) is p opaga ed h ough a dispe si e line,
he esul ing op ical en elope can be exp essed as a cohe en in e e ence
56 2.3. Talbo imaging o mis imed pulses
o dispe sed pulses:
a( , ξ) = X
k
ξ( −k 0−ak)
=X
kZ+∞
−∞
h( − ′, ξ) 0( ′−k 0−ak)d ′,
(2.32)
whe e ξis he accumula ed dispe sion o he sys em, and h( , ξ) he impulse
esponse o p opaga ion ke nel o a lowes -o de dispe sion medium wi h
negligible a enua ion. Using Eqs (1.20) and (2.31) he dispe sed signal
(2.32) esul s
a( , ξ) = " 2
p
ρ(1 −iC)#1/2X
k
exp −( −k 0−ak)2
2ρ,(2.33)
wi h
ρ= 2
p
(1 −iC)+iξ. (2.34)
We also compu e o u u e use he dispe sed signal in he Fou ie domain,
which can be w i en as
A(ω, ξ) = p2π
1−iC 1/2
exp(−ρω2/2) X
k
exp(iωk 0+iωak).(2.35)
A e aged ields can be compu ed in he ime domain as ollows. The
mean alue o he op ical en elope is a sum o a e aged indi idual pulses,
o he esul o he p opaga ion h ough he linea sys em o he ini ial
a e aged ield:
ha( , ξ)i=X
kh ξ( −k 0−ak)i=Z+∞
−∞
h( − ′, ξ)ha( ′,0)id ′(2.36)
A simple compu a ion o he dispe sed ield yields o:
ha( , ξ)i=X
kZ+∞
−∞
ξ( −k 0−ak)p(ak)dak
=" 2
p
˜ρ(1 −iC)#1/2X
k
exp −( −k 0−ak)2
2˜ρ,
(2.37)

2. Non-ideal cohe en Tempo al Talbo e ec 57
whe e we ha e in oduced he no a ion ˜ρ=ρ+σ2
j. On he o he hand, he
ini ial a e aged ield is eco e ed by se ing ξ= 0 in he alue o ˜ρin Eq.
(2.37):
ha( , 0)i=" 2
p
2
p+σ2
j(1 + iC)#1/2
×X
k
exp "−( −k 0)21 + iC
2( 2
p+σ2
j(1 + iC))#.
(2.38)
Then, he pulse wid h o he a e age inpu pulses can be exp essed as:
h 0i= 2
p1 + ǫ21 + ǫ2(1 + C2)−C2
1 + ǫ2(1 + C2),(2.39)
whe e he pa ame e ǫis he a io be ween he s anda d de ia ion o
he iming ji e and he o iginal pulse wid h, ǫ=σj/ p. Thus, he
a e aged inpu ield is a ec ed by andom ji e only as change in he
pulse wid h. Fo unchi ped ains, he incoming pulses a e b oaden by he
ac o ( 2
p+σ2
j)1/2. Howe e , when inpu pulses a e phase modula ed, hey
may b oaden o ge na ow depending on chi p alue. F om Eq. (2.39) he
ansi ion occu s when
C=Climi ≡1 + ǫ2
1−ǫ21/2
.(2.40)
Then, i C < Climi pulses a e b oaden by iming ji e while C > Climi
yields o na owe pulses han he o iginal ones. The compu a ion and
in e p e a ion o he ou pu ield is simila o a de e minis ic signal, and
he de ails o he compu a ion a e widely known [20–22] and a e equi alen
o hose shown in chap e 1. In his sec ion we will simply quo e he
esul s.
2.3.1 Va iance
We a e in e es ed in he s a is ical a ia ions o a conc e e ealiza ion o
a ain o pulses a ec ed by iming andom ji e , a he han in mean
alues. To cha ac e ize hem, i is necessa y o calcula e he co esponding
a iance. This quan i y is de ined as he a e aged squa ed de ia ion o he
ield wi h espec o he mean o each empo al ins an :
58 2.3. Talbo imaging o mis imed pulses
V( , ξ) =h |a( , ξ)−ha( , ξ)i|2i
=ha( , ξ)a∗( , ξ)i−ha( , ξ)iha∗( , ξ)i,(2.41)
and i has he usual in e p e a ion as a measu e o he a ia ion o he
ou pu ield a each sampling ins an . Exp essed as a cohe en sum o
dispe sed pulses, Eq. (2.36), he a iance is
V( , ξ) = X
kX
mh ξ( −k 0−ak) ξ( −m 0−am)∗i
−X
kh ξ( −k 0−ak)i
2
.
(2.42)
He e we will assume ha andom a iables aka e mu ually independen ,
which implies ha hg(ak)h(ap)i=hg(ak)ihh(ap)iwhen k6=p o any
unc ions gand h. Taking his in o accoun , he a iance can be easily
cas as a sum o e dispe sed pulses:
V( , ξ) = X
kh| ξ( −k 0−ak)|2i−|h ξ( −k 0−ak)i|2,(2.43)
whe e he i s pa o he sum co esponds o he a e aged squa e o each
dispe sed pulse and he second o he squa e o he a e age. I is a simple
ask o compu e bo h con ibu ions. The esul is he di e ence o wo
ains o Gaussian unc ions:
V( , ξ) = p
√2η1(ξ)X
k
exp −( −k 0)2
2η1(ξ)2
− p
√2η2(ξ)
p
X
k
exp −( −k 0)2
2η2(ξ)2,
(2.44)
whe e we ha e in oduced he ollowing no a ions:
η1(ξ)2= 2
pb 2+ξ2(1 + C2)
2 2
p
,(2.45)
η2(ξ)2= 4+ξ(1 + C2) + C 2
p2
2(1 + C2) 2,(2.46)
2. Non-ideal cohe en Tempo al Talbo e ec 59
2= 2
p+σ2
j(1+C2) and b 2= 2
p+2σ2
j. Then he wid h η1(ξ) ep esen s he
ms wid hs o he a e aged squa ed dispe sed pulses, and η2(ξ) deno es he
ms wid h o he squa e o he a e age dispe sed pulse. The e o e, a iance
is composed o wo con ibu ions coming om indi idual dispe sed pulses,
which will be called o h| |2iand |h i|2 ype, espec i ely. Since V( , ξ) is
a pe iodic se ies, i can also be expanded in a Fou ie se ies o he o m
V( , ξ) =
+∞
X
k=−∞
gkexp(2πik / 0) = g0+ 2g±1cos(2π / 0) + ... (2.47)
wi h gk=g−kand
gk=√π p
0
exp(−2π2k2η1(ξ)2/ 2
0)
−√π 2
p
0 exp(−2π2k2η2(ξ)2/ 2
0).
(2.48)
Bo h o ms o he a iance, Eqs. (2.44) and (2.47), will be used in wha
ollows. F om Eq. (2.47) i is wo h no ing ha
1
0Z 0/2
− 0/2
V( , ξ)d =g0=√π p
01− p
,(2.49)
ha is, he mean alue is cons an and he e is no dependence on he
speci ic alue o he dispe sion pa ame e ξ.
The a iance o he inpu ield V( , 0) can be eco e ed by se ing ξ= 0
in Eq. (2.44) h ough he co esponding alues o he wid hs, Eqs. (2.45)
and (2.46). Since he ji e pa ame e is small compa ed wi h he wid h
o he o iginal pulses, σj< p, bo h wid hs become o he o de o he
o iginal pulses, η1(0), η2(0) ≈ p. Mo eo e , since he pulses do no o e lap,
only wo Gaussian unc ions con ibu e app eciably o V( , 0) in he sum
Eq. (2.44) o each uni in e al. Fo ins ance, o he cen al pe iod,
− 0/2< < 0/2, he ollowing app oxima ions applies:
V( , 0) ∼
=h| ( −a0)|2i−|h ( −a0)i|2(2.50)
= p
√2η1(0) X
k
exp − 2
2η1(0)2− p
√2η2(0)
p
X
k
exp − 2
2η2(0)2
60 2.3. Talbo imaging o mis imed pulses
On he o he hand, he a iance o he ou pu en elope canno be
app oxima ed in his way: when ξis di e en om ze o, he dispe sed
wid hs [Eqs. (2.45) and (2.46)] a e la ge han 0, and hen he Gaussian
unc ions in Eq. (2.44) o e lap be ween adjacen pe iods. Indeed, o a
gene al Talbo condi ion, ξ≈ 2
0, so ha he o de o he pe iod o he
pulses, η1(ξ), η2(ξ)≈ 0. In o de o ob ain a simple exp ession o he
a iance V( , ξ) a e he dispe si e de ice, i is necessa y o e u n o he
Fou ie expansion (2.47), whose coe icien s a e gi en in Eq. (2.48). Now,
since η1(ξ), η2(ξ)≈ 0, all ha monics in his expansion a e damped. The
leading-o de app oxima ion is jus he DC e m:
V( , ξ)∼
=g0.(2.51)
This, oge he wi h Eq. (2.49), means ha he ou pu a iance equals he
mean alue along he uni in e al, and i is insensi i e o ξ.
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1
0
0.2
0.4
0.6
0.8
1
No malized ime
Inpu con ibu ions
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
0.02
0.04
0.06
0.08
No malized ime
Ou pu con ibu ions
Figu e 2.6: Con ibu ions o ype h| |2i(solid cu e) and |h i|2(dashed
cu e) o he inpu ( op g aph) and ou pu (bo om g aph) a iances due o
a signle pulse
In ig. 2.6 we p esen he cen al pulse con ibu ions h| |2iand |h i|2
o inpu and ou pu a iances in an example. The pe iod o he ain has
been chosen as 0= 1 in sui able uni s; hen, he cen al uni in e al
is −0.5< < 0.5. Mo eo e , he γ/α = 1/4, p= 0.050, C= 0 and
2. Non-ideal cohe en Tempo al Talbo e ec 67
Wi h his con en ion we can compu e he damping coe icien s 2π2η2
1(ξ)/ 2
0
and 2π2η2
2(ξ)/ 2
0o he i s ha monic in he expansion (2.47):
2g±1=2√π p
0
exp(−2π2η1(ξ)2/ 2
0)
−2√π 2
p
0 exp(−2π2η2(ξ)2/ 2
0).
(2.53)
He e we will ocus only on he second one, which in his example is
smalle ; see Fig. 2.6. Ne e heless, bo h ha e in gene al he same o de
o magni ude. This damping coe icien can be bounded o γ/α = 1/N as
ollows:
2π2η2(ξ)2
2
0
=π2 2
(1 + C2) 2
0
+ 2
0(1 + C2)
4N 2+ 4
pC2π2
2
0 2(1 + C2)+πC 2
p
2
> 2
0
2
p
1 + C2
8N2+Cπ
2= (1 + C2) ln 10 + Cπ
2≥ln 10
(2.54)
whe e we ha e used he Talbo condi ion, Eq. (1.26), and no a ions (2.45)
and (2.46), and, in he las inequali y, ha σj< p. Then, he alue o
2π2η2(ξ)2/ 2
0dec eases as 1/N2, bu since he a io p/ 0is also educed by
a ac o N o a oid o e lapping [see Eq. (2.52)], his damping coe icien is
cons an o any N. Mo eo e his coe icien is highe o phase modula ed
signals. In he mos un a o able case, C= 0, he alue in Eq. (2.54)
co esponds o a conse a i e damping ac o o exp(−2π2η2(ξ)2/ 2
0)<0.10.
This esul s in a maximum modula ion o he cons an a iance o 20%. The
ac ual alue o modula ion is ypically lowe , especially i ji e is negligible.
Fo he pa ame e s p esen ed he e, wi h σjand po he same o de , he
modula ion is o 6%. Nume ical esul s in Fig. 2.12 illus a e hese alues.
Thus, we conclude ha when he sequence o pulses is su icien ly na ow o
a oid o e lapping in he ac ional Talbo se ies, he con ibu ion o highe -
o de ha monics o exp ession (2.51) is o a good app oxima ion negligible.
I o e lapping is allowed such ha he a io p/ 0is kep cons an , he
cons an app oxima ion ails as γ/α dec eases.
A new ea u e appea ing in his las simula ion is he inc ease o pulse
pedes al in ac ional Talbo de ices. This ac is obscu ed in Fig. 2.11
because o he o e lapping be ween adjacen pulses. In Fig. 2.13 we p esen
a new simula ion o he ac ional de ice co esponding o γ/α = 1/2, wi h
p= 0.040, C= 0 and σj= 0.007. The p esence o pedes al he e is clea . To

68 2.3. Talbo imaging o mis imed pulses
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
No malized ime
Ampli ude
Figu e 2.13: Supe posi ion o en sequences o ou pu pulses in a uni
in e al in he ac ional Talbo de ice cha ac e ized by he index γ/α =
1/2, leading o a 2× epe i ion a e. The pulses a e unchi ped and hei
ms wid h is p= 0.040. The s anda d de ia ion o andom iming ji e is
σj= 0.007.
analyze his ea u e, le us conside a gene al Talbo de ice cha ac e ized
by an index γ/α. The epe i ion- a e se ies co espond o he ac ional
Talbo se ies γ/α = 1/N. F om Eq. (2.41) he powe o he mean de ec ed
ain ha( , ξ)a( , ξ)∗iis composed o he cons an a iance V( , ξ) = g0plus
he squa ed mean ain ha( , ξ)iha( , ξ)∗i, which anishes be ween pulses.
In u n, he peak alue o he ou pu pulses in a gene al Talbo de ice
is damped [2,15] in ampli ude by a ac o o 1/√αwi h espec o he
o iginal ain, Eq. (2.38). Then, o Gaussian pulses he a io be ween he
peak ampli ude o he de ec ed ain (Apeak) o he alue be ween pulses
(σ2
ξ=g0) is
Apeak
σξ
=1
σξ" 2
p
α √2η2(0) +σ2
ξ#1/2
∼
= p
21/4p αη2(0)
∼
=√2
π1/4
1
√α p
01/2 0
σj(1 + C2)1/2,
(2.55)
2. Non-ideal cohe en Tempo al Talbo e ec 69
whe e in he las app oxima ion we ha e e ained he leading o de in he
a io σj/ p om Eq. (2.49). The e o e, o a gi en amoun o ji e he
pedes al inc eases as √α=√N. The chi p o he pulse inc eases he
pedes al equi alen ly o an inc ease o he iming ji e by he ac o (1 +
C2)1/2. Mo eo e , he a io dec eases when he inpu pulses a e wide .
Fo he simula ion in Fig. 2.13, his a io is 21.2, wi h Apeak = 0.70 and
σξ= 0.033.
2.3.2 In ensi y spec um
In his sec ion we app oach he p oblem o a mis iming ain om a spec al
poin o iew, by analyzing he powe spec al densi y o he de ec ed ji e y
ain be o e and a e a Talbo de ice. He e, and in con as o he p e ious
ime-domain s udy, in which independen pulse- o-pulse ji e a ia ions
we e assumed, now a bi a y pulse- o-pulse iming ji e co ela ion is
allowed. We will conside ha co ela ion be ween pulses a e s a iona y,
and desc ibed by he unc ion R, de ined as R(k) = R(−k) = hamam±ki.
As in he p e ious sec ion he ji e a iables a e assumed o ha e ze o
mean and s anda d de ia ion σj. Fu he mo e we will also assume ha
σjσω≪1. The app oxima e o mulas o he powe spec um ha we
de i e a e based on a second-o de expansion o σjσωo equi alen ly, on a
small-signal app oxima ion in he a iables ak.
Fo de i ing he exp ession o he powe spec al densi y o he de ec ed
ain, Eq. (2.6), we ollow he same s eps han in sec ion 2.2.1, whe e he
in ensi y spec um o a andom dis ibu ed pulse ain was calcula ed. Le
us assume ha he co ela o o he Fou ie ans o med de ec ed ain can
be ac o ized as:
hI(ω1, ξ)I(ω2, ξ)∗i=hI(ω1, ξ)ihI(ω2, ξ)∗i
+ 2πK(ω1, ω2)X
s
δ(ω1−ω2−2πs/ 0).(2.56)
whe e he unc ion Kwill be de i ed below. F om Eq.(2.56),
he powe spec al densi y can be spli in o wo con ibu ions:
S(ω, ξ) = S(S)(ω, ξ) + S(N)(ω, ξ). S(S)is he signal con ibu ion,
p e iously analyzed in chap e 1 and S(N)is he noise one. Using Eqs.
(2.6) and (2.7), we can s aigh o wa dly show ha he noise spec al
densi y is he diagonal pa o egula unc ion K: S(N)(ω, ξ) = K(ω, ω).
Thus he p oblem o compu ing S(N)(ω, ξ) educes o compu a ion o
he Fou ie -domain co ela o and he mean Fou ie in ensi y. Using
70 2.3. Talbo imaging o mis imed pulses
Eqs. (2.35) and (2.8) yields he pho ocu en by Gaussian in eg a ion.
In oducing he no a ion µm(ω) = (ωρ −im 0)σ2
ω/2 esul s in
I(ω, ξ) = p√πexp(−ρω2/2) X
pk
exp[−(ap−ak)2σ2
ω/4]
×exp(−iωk 0+µ2
p−k/σ2
ω−iµ∗
p−kak−iµp−kap).
(2.57)
Now we subs i u e Eq. (2.57) in o Eq. (2.56) and expand he exponen ials
in Eq. (2.57) o second o de in ji e a iables ak, pe o ming he
expec a ion alues acco ding o small-signal app oxima ion. The esul
is p opo ional o he co ela ions o iming ji e a iables:
hI(ω1, ξ)I(ω2, ξ)∗i∼
=hI(ω1, ξ)ihI(ω2, ξ)∗i+π 2
pexp(−ρω2
1/2−ρ∗ω2
2/2)
×X
k
exp[−i(ω1−ω2)k 0]X
sq
exp(−iω2q 0) exp[(µ2
+µ∗2
s)/σ2
ω]
×[µ∗
sµ R(q−s+ ) + µ∗
sµ∗
R(q−s) + µsµ∗
R(q) + µ µsR(q+ )]
(2.58)
whe e µ is e alua ed a ω1and µsa ω2. F om Eq. (2.58) he diagonal
pa o unc ion Kcan be di ec ly ead:
0S(N)(ω, ξ)∼
=π 2
pexp(−ω2/σ2
ω)X
sq
exp(−iωq 0) exp[(µ2
+µ∗2
s)/σ2
ω]
×[µ∗
sµ R(q−s+ ) + µ∗
sµ∗
R(q−s) + µsµ∗
R(q) + µ µsR(q+ )]
(2.59)
whe e he unc ions µ and µsa e now e alua ed a ω. No ice ha in
Eq. (2.59), |exp(µ2
/σ2
ω)|= exp(ω2/4σ2
ω)G (ω)1/2,Gnbeing he passbands
shown in p e ious sec ion:
Gn(ω) = exp[−(ωξ +ωC/σ2
ω−n 0)2σ2
ω/2].(2.60)
The passbands ha e Gaussian o m wi h peaks a ω≃ 0/|ξ|and ms
wid h 1/σ2
ω|ξ|. Fo Talbo lines, |ξ| ≈ 2
0≫ 2
p≈1/σ2
ω, so ha
o e lapping be ween passbands can be neglec ed. Unde his condi ions,
exp[µ2
(ω)/σ2
ω]∼
=exp(ω2/2σ2
ω)G (ω)δ s and subs i u ion in o Eq. (2.59)
yields
2. Non-ideal cohe en Tempo al Talbo e ec 71
0S(N)(ω, ξ)∼
=π 2
pexp(−ω2/2σω)X
n
Gn(ω) (2.61)
×X
q
exp(−iωq 0)[2|µn|2R(q) + µ∗2
nR(q−n) + µ2
nR(q+n)]
By eo de ing he sums o e q, using he de ini ion o µn he powe spec al
densi y can be exp essed in he ollowing mo e compac way:
0S(N)(ω)∼
=ω2J(ω)Φ(ω)X
n
Gn(ω)Mn(ω) (2.62)
whe e
ω2J(ω) = π 2
pω2exp(−ω2/2σ2
ω),(2.63)
Φ(ω) = X
k
R(k) exp(−iωk 0) (2.64)
and
Mn(ω) = cos(ωn 0/2) −(ξ+C/σ2
ω−n 0/ω)σ2
ωsin(ωn 0/2)2.(2.65)
The powe spec um is composed o a sum o ou ypes o con ibu ion.
The i s one is a smoo h b oadband unc ion, ω2J(ω), which ep esen s he
en elope unc ion o noise in he spec al domain ha has bandwid h o he
o de o σω. The second unc ion, Φ(ω), is he powe spec um o he iming
ji e , which is he Fou ie ans o m o he ji e co ela ion unc ion. This
unc ion is mani es ly Ω0pe iodic, and, in gene al, i is na owband abou
he ha monics [75, 78]. Howe e , in he nonco ela ed iming noise limi ,
R(k) = σ2
jδk,0, he ji e noise becomes whi e, i.e., Φ(ω) = σ2
j.
The e ms in he sum in o mula (2.62), howe e , a e dispe sion
dependen and ji e independen . Each e m in his sum is composed
o a na owband Gaussian unc ion Gn(ω), o which we e e as he n
passband, and an oscilla o y unc ion Mn(ω), o which we e e as he
modula ion unc ion o he npassband. The unc ions Gn(ω) accoun o
he po ions o noise a which he p e ious noise con ibu ions, ω2J(ω)
and Φ(ω), a e damped, hus p o iding he mechanism o noise il e ing by
mul iple in e e ence.
Fi s ly we analyze he noise powe spec um o he incoming ain by
72 2.3. Talbo imaging o mis imed pulses
se ing ξ= 0 in Eqs. (2.60) and (2.65). This is he RF noise o he in ensi y
o he ain be o e i en e s he dispe si e line, measu ed by di ec de ec ion
o he ain wi h a high bandwid h pho ode ec o . F om Eq. (2.60), he
passbands become cen ed a ω=n 0σ2
ω/|C|. Because 0>1/σωand he
bandwid h o he spec um o he indi idual pulses is o he o de o σω,
he peak con ibu ion o Gn(ω) when n6= 0 lies ou side he pulse spec um.
The e o e we neglec he con ibu ions o all he passband unc ions excep
ha wi h index n= 0. In addi ion, he co esponding modula ion unc ion
is uni y, M0(ω) = 1, and hus he inpu noise spec um educes o
0S(N)(ω, 0) ∼
=ω2|F(ω)|2Φ(ω),(2.66)
whe e |F(ω)|2=π 2
pexp(−ω2 2
p/2) is he spec um o he in ensi y o a
single pulse, | 0( )|2. I is well known in he con ex o cha ac e iza ion o
noise in undamen ally mode-locked lase s [75–78]. I co esponds o he
noise spec um o a de ec ed ji e y ain Pk| 0( −k 0−ak)|2, whe e we
ha e neglec ed pulse o e lap in he incoming ain. I shows again he ac
ha o neglec he con ibu ions o Gn unc ions is equi alen o neglec
o e lapping among pulses, since Gn ep esen s he in e e ence among n- h
neighbo ing pulse. No ice ha Eq. (2.66), is chi p-independen owing o
he ac ha di ec de ec ion o o iginal ain misses phase in o ma ion.
When a eplica o he ain is c ea ed a e a Talbo dispe si e line,
he basic ea u es o he noise spec um a e simila o hose ound in he
analysis o noise induced by on-o keying o an exac ain (see Eq. (2.21)
in sec ion 2.2). Fi s , he b oadband en elope o he spec um, ω2J(ω),
ex ends along he op ical linewid h o a single pulse o he ain such ha ,
i he pulses a e chi ped, he noise bandwid h is b oade han he bandwid h
o he inpu noise en elope ω2|F(ω)|2. Second, noise is il e ed a ound he
maxima o he passbands Gn(ω). These maxima a e loca ed a
ωn=n 0
|ξ+C/σ2
ω|∼
=αΩ0
n
γ1−1
2πsign(ξ)Cα
γ
Ω2
0
σ2
ω.(2.67)
Acco ding o o mula (2.67), when he ini ial pulses a e unchi ped and
passband unc ion index nis a mul iple o γ, hese passbands a e cen e ed
a he ha monics o he ou pu ain. When Talbo index γis no uni y,
be ween wo consecu i e ou pu ha monics he e a e γ−1 passbands
uni o mly sepa a ed. When he ini ial pulses a e chi ped he e exis s a
de ia ion o he maxima o he passband unc ions Gn(ω) ha depends on
he ela i e sign be ween dispe sion and chi p. When dispe sion is posi i e,
ξ > 0, he maxima [ o mula (2.67)] a e edshi ed (blueshi ed) i he chi p

2. Non-ideal cohe en Tempo al Talbo e ec 73
is posi i e (nega i e). The passband wi h index nwill he e o e exhibi
a maximum in he neighbo hood o a ha monic o he ou pu ain when
n=hγ, whe e his he ha monic index. The noise maximum can be aised
wi h espec o he noise alue be o e dispe sion. Ne e heless, he p ecise
alue o noise a he ha monic equencies hαΩ0does no change. The
ull wid h o he 10-dB decay o hese passbands is compu ed om Eq.
(2.60), p o iding a simple es ima e o he bandwid h o he esul an noise
spec um abou he ha monics:
∆Wω=2√2
√log e
1
|ξ+C/σ2
ω|∼
=4.3
σω|ξ|=4.3
0Q.(2.68)
This wid h is common o all passbands. In o mula (2.68) we ha e de ined
he pa ame e Q:
Q=γ
α
σω
Ω0
.(2.69)
Because he empo al wid h o a pulse a e a dispe si e line o high
dispe sion ξgoes as 0Q[5], Qis he empo al dispe sion-induced sp ead
o an indi idual pulse wi h espec o he pulse sepa a ion and ep esen s
he numbe o basic ime pe iods o he o iginal ain ha occupies a pulse
a e dispe sion.
Finally, each o hese passband unc ions, Gn(ω) is modula ed by he
unc ions Mn(ω). In his case he s uc u e o he modula ion is iche
han ha ound in he analysis o on-o -keyed ains a e Talbo de ices.
Fi s , no ice ha on he op o he passbands ha co espond o ou pu
ha monics (n=hγ) he alue o Mn(ω) is uni y, showing ipples as he
o se om he op inc eases. When he passband index does no co espond
o an ou pu ha monic, howe e , he modula ion unc ion does no each a
local maximum on op o he passband. Then hese passbands show ips on
op ha a e due o his nonmaximal modula ion. Mo eo e , modula ion
maxima a e in gene al bigge han uni y, so noise can ise abo e he
b oadband en elope [Eq. (2.63)]. We p esen an example o his p ope y
in nex subsec ion below. Noise anishes when modula ion eaches ze o,
Mn(ω) = 0; nea he npassband ha co esponds o he hha monic, he
equa ion Mn(ω) = 0 can be ep esen ed by means o a a iable x ha
accoun s o he o se om he alue o he ha monic, ω(x) = Ω0(hα +x).
Then he ze os o he modula ion a e he solu ions o
co (πhγx) = sign(ξ)C+ 2πQ2α
γ
x
x+hα.(2.70)
74 2.3. Talbo imaging o mis imed pulses
A g aphic analysis o his unc ion shows ha he e exis wo di e en se s
o solu ions, which co espond o equal numbe s o posi i e and nega i e
alues o x. The e exis s a ze o in each inc emen al o se o wid h Ω0/n
on he le and he igh sides o he ha monics ha co espond o he
b anches o he co angen unc ion. The loca ion o he ze os is no
exac ly symme ic wi h espec o he ha monic, e en in he absence o
chi p. The e o e, noise ski s a ound ha monics will always be asymme ic.
Because o he dec easing cha ac e o he b anches o he co angen
unc ion, when bo h chi p and dispe sion ha e di e en sign he ze os
o modula ion a e loca ed a lowe alues o he o se han hose wi h
unchi ped pulses. I chi p and dispe sion ha e he same signs, he o se
o he ze os inc eases. Fo la ge alues o Q he wo se s o le and igh
solu ions app oxima e he solu ions o an(πhγx) = 0. These p ope ies
a e also exempli ied and analyzed in he nex subsec ion.
To p esen examples o he powe spec um i is necessa y o in oduce
a conc e e model o he co ela ion be ween pulses. I is no ou objec i e
o e iew he possible cha ac e is ics o noise and i s o igin. They
ul ima ely depend on he conc e e sys ems ha gene a e he ains. He e
we show wo simple models de i ed om he noise heo y o mode-locked
lase s [78], which pe mi s a gene al illus a ion o he aces in he in ensi y
spec um by conside a ion o wo di e en ypes o pulse- o-pulse iming
ji e co ela ion.
2.3.2.1 Pulse-To-Pulse Pa ially Co ela ed Noise
The noise spec al densi y unc ions o undamen ally mode-locked lase s,
be o e dispe si e p opaga ion, p esen iden ical peaks a mul iples o he
pulse epe i ion equency [78]. This pa ially co ela ed noise can be
po ayed by a ecu si e ela ion be ween adjacen pulses. The de ia ion o
he cen e o he pulses is a ac ion η > 0 o he de ia ion o he p e ious
pulse, augmen ed by an addi ional e m ǫm:
am=ηam−1+ǫm.(2.71)
I is assumed ha bo h quan i ies ha e ze o mean, hami=hǫmi= 0, and
ha he addi ional e ms a e mu ually independen and hus unco ela ed,
hǫmǫni=T2δm,n. The e o e ηcon ols he co ela ion be ween pulses
and depends on he unde lying mechanism o pulse gene a ion. The ji e
a ia ions desc ibed by pa ame e ǫm, howe e , ep esen noise induced by
spon aneous emission o acuum luc ua ions.
2. Non-ideal cohe en Tempo al Talbo e ec 75
The ela ion o he wo pa ame e s, ηand T, o he ms alue o ji e
is T2=σ2
j(1 −η2). The co ela ion unc ion is R(k) = σ2
jη|k|. Thus
ji e noise is unco ela ed and he e o e whi e when η= 0, whe eas o ally
co ela ed noise co esponds o η= 1. The iming ji e s spec al densi y
is s aigh o wa dly compu ed om he co ela ion, esul ing in a Fab y-
Pe o - ype unc ion:
φ(ω) = σ2
j
1−η2
1−2ηcos(ω 0) + η2.(2.72)
No ice ha his unc ion is mani es ly pe iodic, wi h a pe iod equal o he
undamen al ha monic o he ain, and is peaked in he ha monics. Thus
he noise spec um’s densi y be o e i en e s he Talbo dispe si e line has
symme ic noise RF sidebands o ski s a ound each ha monic. Expansion
o Eq. (2.72) a ound a ha monic hΩ0 educes he o m o he ski s o a
Lo en zian shape [78].
In all he examples p esen ed he pulse- o-pulse- empo al sepa a ion
is 0= 100 ps, co esponding o ains wi h epe i ion a e Ω0= 2π×10
GHz. The empo al wid h o he pulses in he ain is p= 10 ps, and
he ji e ’s s anda d de ia ion is σj= 100 s. The simula ions p esen ed
he e we e gene a ed om a s ing o 1024 basic pe iods, each wi h 64
sample poin s. The powe spec um is es ima ed nume ically by use o
he Ba le algo i hm [80], a e aging 8 spec a, each ob ained om a
sequence o 128 ime pe iods. The equency sepa a ion o he poin s in
he spec um is he e o e 78 MHz (= 1/128 ×100 ps). Ji e is c ea ed
nume ically by use o s anda d pseudo andom numbe gene a o s and
ecu si e ela ion (2.71).
In Fig. 2.14 we ep esen he spec um o a ain o unchi ped pulses
(C= 0) ha co esponds o he in ensi y o he ji e y ain be o e and
a e he Talbo dispe si e line. Dispe sion is adjus ed o c ea e he i s
in ege Talbo image o he o iginal ain (γ/α = 1), so Q= 1.59. The
pa ame e ha desc ibes co ela ion is η= 0.7. In he lowe pa o
Fig. 2.14 we ep esen by a dashed cu e ou analy ical app oxima ion
o he noise spec al powe o he de ec ed ain be o e i en e s his
dispe si e de ice, as gi en by o mula (2.66). By a con inuous cu e we
plo analy ical o mula (2.62) o he noise spec al powe o same ain
de ec ed a e he Talbo de ice. In he uppe pa o he igu e we
depic he simula ion o he spec a be o e and a e he Talbo dispe si e
line, no malizing he DC componen o 0 dB. These spec a show peaked
con ibu ions ha co espond o he ha monics o a pe ec ain. To help
76 2.3. Talbo imaging o mis imed pulses
0 10 20 30 40 50 60 70 80 90 100
−260
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Powe spec um (dB/Hz)
F equency (GHz)
Figu e 2.14: In ensi y powe spec al densi y (PSD) o a ji e y ain o
pulses (σj= 100 s, η= 0.7) be o e and a e a empo al Talbo de ice
wi h index γ/α = 1. Pulse wid h, p= 10ps; chi p, C= 0; epe i ion a e,
10 GHz; Q=1.59. Abo e, nume ical powe spec al densi ies o he inpu
and ou pu ains (con inuous cu es) and nume ical powe spec al densi y
o he mean signal (do ed cu e). Below, analy ical noise powe spec al
densi ies o he inpu (dashed cu e) and ou pu (con inuous cu e) ains.
in isualizing he con ibu ions o noise we ep esen he spec um o he
mean signal by using a do ed cu e. The ag eemen be ween simula ions
and analy ical o mulas is excellen .
To app ecia e he di e en con ibu ions o he RF noise ski s
a ound he ha monics, we plo in Fig. 2.15 he con ibu ions o noise
in he igh -hand o se nea 50 GHz. The o se uns up o 5 GHz, he
midpoin be ween ha monics. A he le in Fig. 2.15, and no malizing
he powe spec um o 0 dB a ze o o se , we ep esen ji e powe
spec um Φ(ω) by using a dashed cu e, modula ion unc ion Mn=5(ω) by
using a con inuous cu e, and passband unc ion Gn=5(ω) wi h a do ed
cu e. The di e en con ibu ions o he o al powe spec um a e clea ly
obse ed. The ji e powe spec um has a 10-dB hal -wid h o 1.71 GHz;
he 10-dB decay hal -wid h o passband unc ion Gn=5(ω) is 2.15 GHz.
2. Non-ideal cohe en Tempo al Talbo e ec 83
0 10 20 30 40 50 60 70 80 90 100
−320
−300
−280
−260
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Powe spec um (dB/Hz)
F equency (GHz)
Figu e 2.21: The same as in Figu e 2.14.
0 10 20 30 40 50 60 70 80 90 100
−320
−300
−280
−260
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Powe spec um (dB/Hz)
F equency (GHz)
Figu e 2.22: The same as in Figu e 2.14 bu γ/α = 1/2which leads Q=
0.80.

84 2.3. Talbo imaging o mis imed pulses
consecu i e ha monics.
In Figu e 2.21 he Talbo index is chosen o be γ/α = 1. The passbands
unc ions a e cen e ed a ound he ha monics o he signal. Howe e he
supe mode noise peaks a e loca ed a midpoin be ween wo passbands,
so ha hey a e comple ely il e ed by he Talbo de ice. I is wo h o
compa e his igu e wi h igu e 2.14. The pa ame e s o bo h igu es a e he
same, excep he ji e spec al noise o he incoming ain. We can see he
appea ance o noise supe noise peaks due o lack o co ela ion be ween
he iming ji e o consecu i e pulses. Figu e 2.22 shows he esul s o
he noise spec um a e a ac ional Talbo de ice wi h index γ/α = 1/2.
The passband unc ions desc ibing dispe sion a e now cen e ed a he e en
ha monics, so ha he odd ha monics a e il e ed and he mul iplica ion
a e o he signal is doubled. In his case he supe mode noise peaks
loca ed be ween ha monics, a e damped bu no comple ely il e ed by
Talbo de ices. This ac is due o he b oadening o he passbands wi h
espec o he p e ious case (γ/α = 1) weakening he supe mode noise
il e ing. No ice ha he damping o hese peaks is mo e e icien o high
equencies, since he modula ion helps in na owing he passbands.
2.3.3 Ji e Smoo hing
The in eg a ion o he RF noise ski s in a symme ic domain o bandwid h
Ω0abou he h h ha monic is one o he s anda d ope a ional ools used o
ob ain expe imen ally he ms alue o he iming ji e o a ain om noise
analysis o he in ensi y be o e i passes h ough a dispe si e line [75]. Since
a e a ac ional Talbo de ice he spec um is composed o ha monics
mul iples o α, we will compu e he iming ji e a ound he hα-ha monic
o he o iginal ain, which enables o compa e he ji e be o e and a e
he dispe sion line. The p ocedu e ha pe mi s a simple de e mina ion o
ji e is o app oxima e he b oadband noise en elope in o mula (2.66) by
i s alue in he ha monic and o in eg a e along a passband o wid h Ω0
abou he ha monic. This ende s he o al in eg a ed noise p opo ional
bo h o squa ed ha monic index α2h2and o s anda d de ia ion σ2
jo he
2. Non-ideal cohe en Tempo al Talbo e ec 85
ms ji e :
Z(αh+1/2)Ω0
(αh−1/2)Ω0
dω S(N)(ω, 0) ∼
=(2.75)
∼
= −1
0Ω2
0h2α2|F(hαΩ0)|2Z(αh+1/2)Ω0
(αh−1/2)Ω0
dω Φ(ω)
= −1
0Ω3
0α2h2|F(αhΩ0)|2σ2
j.
Then, om a compa a i e measu emen o di e en in eg a ed RF noise
sidebands, he alue o ms ji e is expe imen ally de e mined.
As we illus a ed in p e ious sec ions, he RF noise ski s a e a
Talbo dispe si e line change as a esul o o he p esence o he passband
and he modula ion unc ions. Thus o al noise, measu ed as he esul
o in eg a ion o he noise ski spec um along a passband a ound he
ha monics, can ge smoo hed. No ice ha , because o he s uc u e o he
noise ski s a e he Talbo dispe si e line, he ou pu noise canno be
asc ibed pu ely o he iming ji e bu a combina ion o bo h ampli ude
and iming ji e s in he pulses o he ou pu ain. The in e p e a ion o
ou pu noise as a speci ic combina ion o bo h ampli ude and iming ji e s
in he pulses o he ou pu ain will no be add essed he e.
The in eg a ed noise along bandwid h Ω0abou he h h ha monic o
he ou pu signal (αhΩ0) a e he Talbo dispe si e line is exp essed as
Z(αh+1/2)Ω0
(αh−1/2)Ω0
dω S(N)(ω, ξ) (2.76)
∼
= −1
0Ω2
0α2h2J(αhΩ0)Z(αh+1/2)Ω0
(αh−1/2)Ω0
dωΦ(ω)Gn=hγ(ω)Mn=hγ(ω).
In o mula (2.76) we ha e also app oxima ed he b oadband en elope
by i s alue in he ha monic and ha e e ained in he sum only he
passband unc ion ha co esponds o he ha monic αhΩ0, ha is, n=hγ.
Acco ding o he examples p esen ed in p e ious sec ions, his is a good
app oxima ion when he dispe si e line in adjus ed o he i s Talbo
image, γ= 1, because no new passbands be ween ha monics appea in he
spec um. The ex ension o he in eg a ion o a bigge numbe o passband
unc ions is ne e heless s aigh o wa d.
Then he a io ρ(αh)o he in eg a ed RF noise powe a e dispe sion
[ o mula (2.76)] o ha be o e dispe sion [ o mula (2.75)] is exp essed, om
o mulas (2.62)-(2.66) and (2.60), as
86 2.3. Talbo imaging o mis imed pulses
ρ(αh)∼
=exp n2C2/2Q2Ln=hγ(η, Q, C),(2.77)
whe e he i s e m accoun s o he di e en noise en elopes and
Ln=hγ(η, Q, C) = Z1/2
−1/2
dx Φ(x)Gn=hγ(x)Mn=hγ(x).(2.78)
In Eq. (2.78) we pe o m he in eg a ion by again using he a iable
ω(x) = Ω0(αh +x) and wi h he no a ion Φ(x)≡Φ[ω(x)]/σ2
j,
Gn=hγ(x)≡Gn=hγ[ω(x)], and Mn=hγ(x)≡Mn=hγ[ω(x)].
F om he s uc u e o o mula (2.77) he ollowing consequences a e
de i ed: Fi s , he a io in o mula (2.77) depends on co ela ion, no on
he p ecise alue o σ2
j. Second, i pulses a e unchi ped, noise is essen ially
symme ic abou he ha monic, and he esul an a io is always less han
uni y because o he p esence o he passband unc ions. The educ ion
inc eases as he wid h o he passband na ows, so smoo hing inc eases
wi h Qand also as η→0, because he inpu noise is whi e in a bigge
band. Finally, when he pulses a e chi ped he alue o he a io is aised
h ough he exponen ial ac o in o mula (2.77) ha accoun s o he
di e en noise en elopes. This inc ease becomes bigge a high ha monics.
In Fig. 2.23 we plo he a io ρ(αh)o he i h ha monic (h= 5) o he
i s in ege Talbo dispe si e de ice (γ/α = 1) o co ela ion pa ame e
ηand o h ee alues o Q. The alues o he a io [ o mula (2.77)] we e
ob ained by nume ical in eg a ion o Eq. (2.78). Fi s , no ice ha , owing
o he quad a ic dependence o he in eg a ed noise powe o mula (2.75)]
on he ji e s s anda d de ia ion σj, a educ ion o 10 dB in he a io ρ(αh)
amoun s o a educ ion o σjby a ac o o 3, whe eas a 6-dB decay is
associa ed wi h a educ ion o σjby a ac o o 2. In Fig. 2.23(a) he a io
is compu ed o an ini ial ain composed o unchi ped pulses, whe eas
in Fig. 2.23(b) he pulses a e sligh ed chi ped (C= 0.5 ad). As he
co ela ion pa ame e con ols he whi e-noise bandwid h o he RF noise
ski s o he ha monic, his is one o he mos de e minan pa ame e s o
he smoo hing; see Fig. 2.23(a). Fo high alues o η, noise is co ela ed,
he noise ski s a e na ow, and consequen ly he noise educ ion induced
by he p esence o passbands is small. Howe e , when he co ela ion is
low, he p esence o he passbands induces a signi ican dec ease in he
RF noise ski s, which inc eases wi h pa ame e Q. Chi p is he mos
dele e ious ac o o his smoo hing. F om Figs. 2.17 and 2.18, when
pulses a e chi ped he noise ski s a e asymme ic and he peak alues a e
aised abo e he ini ial noise le el, leading o he inc ease in he a io in
2. Non-ideal cohe en Tempo al Talbo e ec 87
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−12
−9
−6
−3
0
3
6
(a) C=0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−12
−9
−6
−3
0
3
6
(b) C= 0.5
Co ela ion pa ame e η
Ra io ρ(h=5) (dB)
Q=2
Q=5
Q=7
Q=2
Q=5
Q=7
Figu e 2.23: Ra io ρ(h) o he i h ha monic h= 5 ( i h passband, n= 5)
and o he i s in ege Talbo eplica γ/α = 1 as a unc ion o co ela ion
pa ame e η o h ee alues o dispe sion pa ame e Q: con inuous cu es,
Q= 2; dashed cu es, Q= 5; do ed cu es Q= 7.
1 2 3 4 5 6 7 8 9 10
−9
−6
−3
0
3(a) C=0
1 2 3 4 5 6 7 8 9 10
−9
−6
−3
0
3(b) C= 0.5
Dispe sion pa ame e Q
Ra io ρ(h) (dB)
h=1
h=5
h=7
h=1
h=5
h=7
Figu e 2.24: Ra io ρ(h) o he i s in ege Talbo eplica γ/α = 1 and o
ji e y ains wi h co ela ion pa ame e η= 0.7as a unc ion o dispe sion
pa ame e Q o h ee ha monics h: con inuous cu es, h= 1; dashed
cu es, h= 5; do ed cu es h= 7.
88 2.3. Talbo imaging o mis imed pulses
o mula (2.77) illus a ed in Fig. 2.23(b). Howe e , e en in he p esence
o chi p, he combined e ec o high Qand low co ela ion can lead o a
signi ican dec ease in he a io ρ(αh).
To isualize i s dependence on he Q ac o , and also i s small a ia ions
wi hin ha monic index h, in Fig. 2.24 we ep esen again he a io o
he i s in ege Talbo eplica, γ/α = 1, o ji e y ains wi h co ela ion
pa ame e η= 0.7 as a unc ion o Q o di e en ha monics hand o
he same wo alues o chi p, C= 0 and C= 0.5 ad. Bo h he dec ease
in he a io ρ(αh)wi h Qand i s small dependence on he ha monic a e
obse ed. Again, small chi ps in Fig. 2.24(b) spoil he noise educ ion.
This e ec is mo e p onounced o high ha monics, because he passband
shi and he subsequen inc ease in he noise peak alue a e p opo ional
o he ha monic equency. Bu no ice ha his inc ease in a io ρ(αh) o
chi ped pulses can be compensa ed o by means o an inc ease in Q.
In Figs. 2.25 and 2.26 we ep esen he a io ρ(αh)in he second Talbo
de ice, γ/α = 2. The p esence o new passbands be ween ha monics has
been aken in o accoun in he nume al e alua ion o in eg al equa ion
(2.78). In Fig. 2.25 we plo a io ρ(αh) o he i h ou pu ha monic,
h= 5, wi h espec o he co ela ion pa ame e o di e en alues o C
and Q. When C= 0 ad, he cu es in Fig. 2.25(a) a e simila o hose in
Fig. 2.23(a), p o ing ha he a io depends basically on he alue o Q
h ough he wid h o he passbands loca ed in he ha monic. The p esence
o new passbands adds a subsidia y con ibu ion o a io ρ(αh). The e o e
he inc ease in Talbo o de p oduces an inc ease in he educ ion o
smoo hing. Howe e , when he chi p is nonze o, he subsequen change
in he passband posi ion always inc eases he alue o he a io. In Fig.
2.25(b) he a io becomes bigge han uni y o low alues o Q, bu , again,
inc easing he alue o Qcompensa es o he inc ease in he a io.
Finally, in Fig. 2.26 we depic he e olu ion o a io ρ(αh) o he
second in ege Talbo eplica, γ/α = 2, o ji e y ains wi h co ela ion
pa ame e η= 0.7, as a unc ion o Q o h ee ha monics h. In Fig.
2.26(a), unchi ped pulses a e conside ed. These cu es a e simila o
hose o Fig. 2.24(a) excep o he i s ha monic, h= 1, whose
alues o he educ ion a e wo se because o he p esence o he new
passband a 5 GHz; Fig. 2.19. In Fig. 2.26(b) we conside pulses wi h
chi p, C= 0.5 ad. Again, a io ρ(αh)inc eases owing o a shi in he
passband maximum and can be compensa ed o by use o high alues o Q.

2. Non-ideal cohe en Tempo al Talbo e ec 89
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−12
−9
−6
−3
0
3
6
(a) C=0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−12
−9
−6
−3
0
3
6
Co ela ion pa ame e η
Ra io ρ(h=5) (dB)
(b) C=0.5
Q=2
Q=5
Q=7
Q=2
Q=5
Q=7
Figu e 2.25: Ra io ρ(h) o he i h ha monic h= 5 ( en h passband, n= 5)
and o he i s in ege Talbo eplica γ/α = 2 as a unc ion o co ela ion
pa ame e η o h ee alues o dispe sion pa ame e Q: con inuous cu es,
Q= 2; dashed cu es, Q= 5; do ed cu es Q= 7.
1 2 3 4 5 6 7 8 9 10
−9
−6
−3
0
3
(a) C=0
1 2 3 4 5 6 7 8 9 10
−9
−6
−3
0
3
(b) C=0.5
Dispe sion pa ame e Q
Ra io ρ(h) (dB)
h=1
h=5
h=7
h=1
h=5
h=7
Figu e 2.26: Ra io ρ(h) o he i s in ege Talbo eplica γ/α = 2 and o
ji e y ains wi h co ela ion pa ame e η= 0.7as a unc ion o dispe sion
pa ame e Q o h ee ha monics h: con inuous cu es, h= 1; dashed
cu es, h= 5; do ed cu es h= 7.
90 2.4. Conclusions
2.4 Conclusions
In his chap e we ha e analyzed he beha io o Talbo lines when hey
a e ed wi h non-ideal cohe en ain o linea ly chi ped Gaussian pulses.
We ha e assumed an ideal Talbo il e , i. e., a pu e i s -o de dispe sion
medium wi hou a enua ion and non linea e ec s. Fu he mo e, we
conside ed he bandwid h o Talbo line is la ge enough o cope wi h he
op ical bandwid h o he incoming ain.
In sec ion 2.2 we ha e analyzed he powe spec al densi y o a
on-o -keyed ain a e passing a Talbo il e . We ha e assumed ha
he a iables desc ibing he p esence o absence o pulses in he ain a e
unco ela ed. As a consequence o he cohe en in e e ence o dispe sed
pulses, he mic owa e spec um a e he de ec ion o he ain has a
b oadband noise en elope ha ollows he linewid h spec um o an
indi idual pulse, i.e., he mic owa e en elope bandwid h is enhanced by
he alue o he chi p. The basic ea u e o he spec um is he p esence
o passbands il e ing his b oadband noise, whose cen e s a e shi ed
wi h espec o he ha monics due o he chi p o he inciden pulses.
Measu emen o he ela i e de ia ion o hese cen e s wi h espec o
passband wid h allows a de e mina ion o he chi ps sign and magni ude.
The wid h o he passbands is na owe as he dispe sion o he de ice
dec eases, he pulse wid h is sho e , and he chi p is la ge . The peak
alue o hese passbands depends on he ela i e amoun o pulses in
he ain. Fo small dispe sion de ices, high passbands, and wide pulses,
he passbands show a cosine-squa ed modula ion. The wid h o his
modula ion is a ac ion o he inpu ha monic alue, being independen o
he ela i e amoun o pulses in he ain o o he p ope ies o he pulses
in he ain.
In sho , ou analysis shows ha he pa ame e s desc ibing he pulse
ain ha e a nea in luence in he esul ing mic owa e spec um, allowing
he indi ec cha ac e iza ion o pulse ains h ough he andom Talbo
e ec . In expe imen al si ua ions, howe e , he andom Talbo e ec has
o coexis wi h o he ypes o b oadband and na owband noise due o
he sou ces o modula o s used, such as ampli ude noise o iming ji e .
Fo ins ance, as is well known, he spec um o de ec ed pulse ains om
mode-locked lase s show ypical noise ski s a ound ou pu ha monics,
whose wid h depends on he co ela ion p ope ies o noise [76–78].
Mo eo e , de ia ions o he ideal beha io o he Talbo il e will also
a ec he cha ac e is ics o he spec um. Al hough i seems unlikely
ha he andom Talbo e ec o pulse ains shows all he ine spec al
2. Non-ideal cohe en Tempo al Talbo e ec 91
de ails con ained in Eq. (2.18) o (2.21), he o e all ea u es should be
s ill p esen . Anyway he noise ea u es o Talbo de ices can be enla ged
when equi ed by conc e e expe imen al ci cums ances.
In he sec ion 2.3, we ha e s udied he in luence o andom ji e o a
linea ly chi ped Gaussian pulses in Talbo de ices. The analysis has been
pe o med in he bo h empo al and spec al domain. In he empo al
analysis we ha e assumed he ji e o be unco ela ed. We ha e shown
ha he a iance o he ain is la ened by use o dispe sion a e passing
a Talbo de ice. This is due o he dispe sion o he indi idual pulses ha
o m he ain, as hei a iance is sp ead on neighbo ing in e als. We
ha e also gi en app oxima e o mulas o he inpu and ou pu a iances
and discussed he alidi y o hese app oxima ions. Fo in ege Talbo
de ices, he alue o he a iance is cons an i espec i e o he alue o he
dispe sion. Fo ac ional Talbo de ices wi h γ/α < 1, a modula ion o
he cons an alue o he a iance may be p esen , depending on he a io
be ween he pulse wid h and he pe iod o he o iginal ain. Ne e heless,
o Talbo de ices leading o N× epe i ion a es, he a iance s ill is, o
a good app oxima ion, cons an . The p esence o pulse pedes al in hese
de ices has also been analyzed.
F om he spec al poin o iew we epo on a wide analysis. Now an
a bi a y co ela ion among ji e a iables is allowed. The basic ea u e
o he esul an spec um is, like in he case o on-o -keyed ains, he
p esence o a b oadband noise- il e ing mechanism by mul iple in e e ence
a he scales o a ac ion o he ains epe i ion a e, which p o ides a
educ ion o he in eg a ed noise powe a ound he ha monics as measu ed
in he RF egime. The in luence o chi p and op ical linewid h o he
indi idual pulses in he ain was also analyzed. We ha e used he esul s
de i ed o explo e he noise educ ion o di e en cohe en Talbo -imagined
ains unde di e en p opaga ion condi ions. The smoo hing o ji e
noise by Talbo in e e ence depends basically on he chi p o he pulses
o he ain, on he co ela ion be ween adjacen pulses in he ain, and
on he accumula ed dispe sion o he line bu no on he absolu e alue
o he iming ji e s s anda d de ia ion. Chi p has been shown o be he
mos dele e ious ac o because i induces a shi o he peak noise le el
wi h espec o he posi ion o he ha monic ha causes an inc ease in he
o al noise powe . Howe e , ca e ul design o he dispe si e line can easily
compensa e o i [43, 82] and comp ess he pulses simul aneously, as has
been shown in i s chap e .
As co ela ion con ols he ji e noise bandwid h be o e he ji e noise
92 2.4. Conclusions
en e s he dispe si e line and because ji e smoo hing is mo e p onounced
a scales o he o de o he epe i ion a e ( ypically less han 1 GHz
o he 10 GHz ains), he ini ial noise mus be b oadband o p oduce
signi ican ji e educ ion. This ac limi s he possible applica ion o his
p ocedu e o ains composed o sligh ly co ela ed pulses. Ul ima ely,
his applica ion depends on he mechanisms ha p oduce he ains. Fo
ins ance, he use o a Talbo dispe si e line canno be success ully applied
in undamen ally mode-locked lase s ha ha e noise o se s up o 100
MHz; see, e.g., Re . [77]. In his case he ac i e locking in he ca i y o
he lase p oduces highly co ela ed iming ji e . Howe e , e en in hese
cases he Talbo eplica o a mode-locked lase ou pu can in p inciple help
o educe o he ypes o non-na owband noise, such as he supe mode
noise [81] in ha monically mode locked lase s, o which bo h phase noise
and iming ji e a e unco ela ed be ween adjacen pulses. [77,78].
The noise educ ion depends on he accumula ed dispe sion o he line
h ough he o de o he Talbo eplica. Basically, high Talbo o de s
imply he appea ance o new noise windows be ween ha monics and he
na owing o he passband wid hs, which p o ides he bandwid h o he
smoo hed spec um abou he ha monics. Excep o he i s ha monic,
he p esence o new windows is a subsidia y e ec when one is compu ing
he dec ease in noise a io ρ(h). The na owing o he passbands is
con olled by dispe sion pa ame e Q, which accoun s o he wid h o
he dispe sed pulses ela i e o he epe i ion a e o he ain. High
Q alues imply ha he esul an pulses in a Talbo eplica a e o med
om con ibu ions o a high numbe o dispe sed pulses, hus p o iding
a cohe en ly a e aged ou pu . The same smoo hing mechanism has been
ound in he ime domain.
3. Pa ially cohe en Tempo al Talbo e ec 99
3.3 Pa ially Cohe en Ligh in Talbo Lines
In his sec ion we specialize he p e ious esul s o a Talbo line. We now
conside he pa ial ligh sou ce is modula ed by a pe iodic unc ion, so
ha he op ical en elope a he inpu o dispe si e medium, a( , 0) is gi en
by
a( , 0) = ǫ( )m( ) = ǫ( )" ( )⊗X
k
δ( −k 0)#.(3.21)
Then, using (3.12) and (1.32), we ob ain he ollowing exp ession o he
a e age in ensi y a he Talbo dis ances zγ/α, Eq. (1.32):
hI( , zγ/α)i=α
γ
1
2
0
Sα
γ
2
0⊗| ( )|2⊗1
αX
k
δ( −τ−k 0/α).(3.22)
This o mula can be educed o he esul s o [84] when dispe sion is
specialized o in ege Talbo dis ances (α= 1) and ( ) is eal. No ice
ha , acco ding o (3.12), pa ial cohe ence esul s in a smoo hing o he
a e age in ensi y, i espec i ely o he exac o app oxima ed cha ac e o
he (monoch oma ic) empo al Talbo e ec due o pulse o e lapping. I ,
howe e , he ini ial pulses a e su icien ly na ow o a oid o e lapping a he
Talbo dis ance unde conside a ion, he cohe en in ensi y ain is exac ly
pe iodic wi h pe iod 0/α see Eq. (1.32). Then, o mula (3.22) desc ibes
he smoo hing o he a e age in ensi y p o ile in each exac pe iod 0/α due
o pa ial cohe ence.
Fo mula (3.22) also desc ibes an example o he empo al Colle -Wol
heo em [13,14]. This esul s a es ha wo op ical modula ed ligh ields
wi h di e en cohe ence p ope ies may p opaga e in dispe si e media
wi h iden ical a e age in ensi y dis ibu ion. In ou case, i is appa en
om (3.22) ha , gi en a Talbo dis ance zγ/α, a pulse ain ca ied by
a monoch oma ic wa e o in ensi y I0, and wi h single pulse p o ile q( )
would p o ide he same a e age in ensi y o a ain o pulses ( ) ca ied
by a pa ially cohe en wa e i :
α
γ
1
2
0
Sα
γ
2
0⊗| ( )|2=I0|q( )|2.(3.23)
The dual beha io o single-pulse in ensi y and spec al con en becomes
clea h ough his con olu ion. Equi alen ly, pa ial cohe ence can be
in e p e ed as de ining he low pass il e (3.13). The equi alen bandwid h
(3.15) o he il e a Talbo dis ances zγ/α is:

100 3.3. Pa ially Cohe en Ligh in Talbo Lines
Beq =1
2
τc
2
0
α
γ,(3.24)
No ice ha he bandwid h inc eases wi h he in e se o he Talbo index
γ/α. This means ha , o he same ype o pulses, cohe ence equi emen s
a e mo e es ic i e o high alues o he Talbo index γ/α and ha , o a
gi en alue o he epe i ion a e α, he inc ease o γsmoo hes he a e age
in ensi y o he Talbo -imagined ain. In he ime domain, his obse a ion
can be in e p e ed as ollows. The p ocess o econs uc ion o pulses a
Talbo dis ances is mul iple pulse- o-pulse in e e ence due o dispe sion.
Pa ial cohe ence damps he in e e ence pa e n be ween neighbo ing
pulses, since he unde lying ca ie misses p og essi ely i s own phase. This
damping is mo e p onounced when he in e e ing pulses a e ini ially mo e
dis an . The e o e, high alues o he Talbo index γ/α, o , al e na i ely,
highly dispe si e de ices, which equi e cons uc i e in e e ence be ween
pulses ha we e ini ially mo e dis an , a e mo e sensi i e o he cohe ence
o he ca ie .
Fo mula (3.24) can also be used o de e mine cohe ence equi emen s
o he eco e y o he Talbo ain o in ensi ies. Since his ain is
composed o in ensi y ha monics o he undamen al equency 1/ 0, he
quan i y Beq 0 ep esen s he numbe o ha monics ha a e no il e ed by
pa ial cohe ence. The equi emen o ai h ul econs uc ion o he pulses
is he e o e Beq 0≫1. Fo ins ance, a ypical DFB lase has a spec al
line wi h FWHM o 50 MHz. I s cohe ence ime is he e o e τc∼20 ns. I ,
mo eo e , he epe i ion a e is 2.5 GHz ( 0= 400 ps), and γ/α = 1/4, he
esul ing ela i e bandwid h is Beq 0= 32.
On he o he hand, he quan i y Beq 0/α, which ep esen s he numbe
o in ensi y ha monics ha cons i u e he ou pu ain, does no depend on
he epe i ion- a e index α:Beq 0/α =τc/2 0. This is a consequence o he
in e se ela ionship be ween equi alen bandwid h and dispe sion: lowe
dispe sion -o , equi alen ly, highe epe i ion- a e index implies highe
bandwid h o a less se e e smoo hing o he single-pulse in ensi y, see (3.24).
Bu he inc ease in index αalso implies he inc ease in he sepa a ion α/ 0
o he in ensi y ha monics ha con o m he ou pu ain, so ha he o al
numbe o he ou pu in ensi y ha monics is independen o α. In ac ,
Beq 0/α is a measu e o he cohe ence ime ela i e o he pe iod o he
ain.
This conclusion can also be ob ained by di ec compu a ion. Le us
conside he ha monic expansion o he ini ial in ensi y ain:
3. Pa ially cohe en Tempo al Talbo e ec 101
| ( )|2⊗X
k
δ( −k 0) = X
k
Ckexp(i2πk / 0),(3.25)
whe e Cka e he ha monics o he pe iodic modula ion unc ion. Then,
he co esponding expansion o he cohe en ou pu ain wi h inc eased
epe i ion a e only in ol es he ha monics mul iples o α:
| ( )|2⊗1
αX
k
δ( −τ−k 0/α) = X
k
Ckα exp(i2πk( −τ)α/ 0).(3.26)
I we inse his expansion in (3.22) o ake in o accoun he pa ial
cohe ence o he ca ie wa e, we ob ain:
hI( , zγ/α)i=X
k
CkG(−k 0γ) exp(i2πk( −τ)α/ 0),(3.27)
which shows ha cohe ence limi s he numbe o ou pu in ensi y ha monics
independen ly o he epe i ion- a e index α.
Finally, we conclude he analysis o o mula (3.22) wi h he desc ip ion
o i s gene aliza ion o mul iple ca ie wa eleng hs [99, 101]. Le us
assume ha he ca ie is composed o Nindependen pa ially-cohe en
wa eleng hs. Le us also assume ha he de ec ion condi ions assu e ha
he co esponding modula ed wa es a e mu ually incohe en , so ha he
inal a e age in ensi y is a sum o he in ensi ies associa ed o di e en
wa eleng hs bu wi h he same modula ion. Then, he il e in (3.22)
gene alizes o a sum o e he di e en spec al lines:
hI( , zγ/α)i=α
γ
1
2
0
N
X
k=1
Skα
γ
2
0⊗| ( )|2⊗1
αX
m
δ( −τ−m 0/α).
(3.28)
and hus he spec al con en o he ca ie wa es de e mines he il e ’s
empo al esponse. In his sum, equency shi s be ween di e en ca ie s
ansla e in o ime delays. Fo ins ance, le us assume ha he modula ion
is ca ied by wo monoch oma ic wa es wi h di e en in ensi ies and
equencies ν0and ν0+ ∆ν0, espec i ely. Then, he sum in Eq. (3.28)
con ains wo e ms, associa ed o he powe spec al densi ies S1( ) =
I0δ( ) and S2=I′
0δ( −∆ν0). The esul ing il e is:
α
γ
1
2
0
2
X
k=1
Skα
γ
2
0=I0δ( ) + I′
0δ( −γ
α 2
0∆ν0),(3.29)
102 3.4. Ma ix Theo y and En opy
hus desc ibing a weigh ed supe posi ion o wo eplicas o he signal, wi h
a mu ual delay ha depends on he equency di e ence ∆ν0. A mapping
o his ype - om he spec al con en o he ca ie o a delay in he
modula ed wa e o ms- inds applica ion in he cons uc ion o mic owa e
ans e sal il e s [102]. In o mula (3.28), howe e , he il e ac s o e
pulses c ea ed as Talbo eplicas o he o iginal ain. The e o e, he
wa eleng h-dependen delay in oduces a ela i e delay be ween Talbo -
imagined ains, no be ween eplicas o a dispe sed ain, as in (3.12).
This obse a ion ep esen s he basis o he empo al Lau e ec .
3.4 Ma ix Theo y and En opy
The i s objec i e o his sec ion is o exp ess he ou pu in ensi y o a
Talbo de ice in a ma ix o m, sepa a ing he in luence o pa ial cohe ence
and he spec al con en o he incoming ain o pulses. To his end, we
obse e ha he ha monics Cαk o he cohe en ou pu in ensi y in eq.
(3.26) can be exp essed in e ms o he ampli ude α-ha monics aαk :
Cαk =αX
a αa∗
( −k)α.(3.30)
This can be de i ed om (3.26), since he absence o pulse o e lapping
leads o he ollowing equali y:
Iu( , zγ/α) = 
( )
√α⊗X
k
δ( −τ−k 0/α)
2
=√αX
s
aαs exp[i2παs( −τ)/ 0]
2
.
(3.31)
On he o he hand, in Eq. (3.27) we obse e ha he Talbo e ec in ol es
he sampling o he complex cohe ence unc ion wi h sampling pe iod s=
γ 0. This pe mi s he desc ip ion o he in luence o pa ial cohe ence in
a ma ix way. Fo his pu pose, we de ine he cohe ence ma ix Mwi h
elemen s:
M s =M∗
s =g(γ( −s) 0).(3.32)
The p ope ies o he complex deg ee o cohe ence imply ha he cohe ence
ma ix is a He mi ian Toelpli z ma ix [103,104]. We also de ine a ime-
3. Pa ially cohe en Tempo al Talbo e ec 103
dependen column ec o Awhose en ies a e gi en by:
A ( ) = a α exp(i2πα / 0),(3.33)
which co esponds o he α-ha monics o he incoming op ical en elope.
F om now on, we will assume ha N= 2n+1 is he numbe o α-ha monics
desc ibing he inpu pulse ain, so ha he indices in (3.32) and (3.33) un
om −n o n. I he spec al con en o he inpu pulses, ( ), is con ained
in a band [−Wp, Wp] a ound he op ical ca ie , hen:
n=⌊WpT/α⌋,(3.34)
and ⌊x⌋is he la ges in ege smalle o equal o x. Using hese de ini ions,
he a e age in ensi y (3.27) can be ew i en as:
hI( , zγ/α)i=I0αA+( −τ)·M·A( −τ),(3.35)
whe e A+is he He mi ian conjuga e o ec o A. Specializing his
exp ession o =τi ollows ha Mis a posi i e-de ini e ma ix, since
in his case Ais an a bi a y ime-independen ec o and he in ensi y is
clea ly a non-nega i e quan i y.
The cohe ence ma ix is a He mi ian ope a o and he e o e diagonal-
isable by means o a ans o ma ion ma ix B:
diag(λ1, λ2,···, λN) = 1
NB·M·B+,(3.36)
whe e λ ep esen i s eigen alues and he ac o 1/N has been in oduced
in o de o no malize he ace o he diagonal ma ix o uni y. Since he
cohe ence ma ix is posi i e semi-de ini e, i s eigen alues a e nonnega i e
[103]. Wi h his decomposi ion he a e age in ensi y is:
hI( , zγ/α)i=I0Nα
n
X
=−n
λ |b ( −τ)|2,(3.37)
whe e:
b ( ) =
n
X
q=−n
B qaαq exp(i2παq / 0).(3.38)
Eq. (3.37) ep esen s he a e age in ensi y a Talbo dis ances as a weigh ed
sum o in ensi y pa e ns, |b ( −τ)|2, he eigen alues λ being he weigh s o
he sum. Thus, he a e age in ensi y can be ein e p e ed as an incohe en
sum o ampli ude pa e ns b ( −τ) whose numbe and ela i e in ensi y is
104 3.4. Ma ix Theo y and En opy
de e mined by he eigen alues λ [85,86].
In he cohe en limi he cohe ence ma ix is uni y M s = 1, ∀ , s, and
he e is a single eigen alue di e en o m ze o, λK= 1 o ce ain index
K. The co esponding eigen ec o is BKq = 1/√N∀q, so ha he only
signi ican ampli ude bK( ) ep esen s he ampli ude o he ain o pulses
wi h inc eased epe i ion a e unde o al cohe ence. On he o he hand,
he incohe en limi is ep esen ed by a cohe ence ma ix M s =δ s. In his
case all eigen alues a e equal o 1/N, and he e o e all pa e ns in he sum
(3.37) a e equally weigh ed. The ma ix o eigen ec o s is B q =δ q, and
he in ensi y is cons an in he incohe en limi . The e o e, he o m o he
ampli ude pa e ns b ( ) is a linea combina ion o he α-ha monics o he
ain ha in e pola es be ween o al cohe ence and incohe ence condi ions.
On he o he hand, he numbe and ela i e magni ude o he
eigen alues enclose he in luence o pa ial cohe ence in he pe o mance
o he empo al Talbo e ec . In Appendix B we e iew se e al pa ame e s
ha ha e been in oduced as a measu e o he pa ial cohe ence in
simila ma ix decomposi ions o pa ially-cohe en op ical e ec s om
i s eigen alue decomposi ion [87, 88]. These can also be used o measu e
he o e all impac o pa ial cohe ence in he in ensi y pa e ns a ising
a Talbo dis ances. Among hem, we will ocus on he in o ma ional
en opy [85]:
H=−
n
X
k=−n
λkln λk.(3.39)
The en opy anges om i s minimum alue Hmin =Hcoh = 0 ha
ep esen s o al cohe ence; and i s maximum, Hmax =Hincoh = ln N,
which co esponds o he incohe en limi .
A somewha simila measu e o he cohe ence is he e ec i e numbe
Ne o unco ela ed andom a iables ha ep esen a signal [89]. In ou
con ex his numbe is an equi alen measu e o he numbe o signi ican
non-ze o eigen alues ha con ibu e o he expansion o he in ensi y (3.37).
Since his expansion can be in e p e ed as he incohe en sum o he
in ensi ies associa ed o Ndi e en sou ces [86], we e e o Ne as an
e ec i e numbe o incohe en sou ces.
Le us iden i y his expansion wi h an equi alen expansion associa ed
o he in ensi y eco ded a a ce ain Talbo dis ance wi h exac ly Ne ≤
Nunco ela ed, equally-weigh ed sou ces, so ha he en opy o his
equi alen sys em would be H= ln Ne . Then, gi en a Talbo de ice
ope a ing unde pa ial cohe ence wi h en opy H, we de ine he numbe

3. Pa ially cohe en Tempo al Talbo e ec 105
o e ec i e sou ces as:
Ne =⌈exp(H)⌉,(3.40)
whe e ⌈x⌉ ep esen s he minimum in ege la ge han x. The ela ion o
his quan i y wi h hose in oduced in [87–89] is also b ie ly e iewed in
Appendix B.
In o de o p esen examples o he use o he expansion in eigen alues
(3.37) we conside Talbo de ices ed by pa ially-cohe en ains wi h
a Lo en zian ca ie linewid h, whose complex deg ee o cohe ence is an
exponen ial unc ion [3]:
g(τ) = exp(−|τ|/τC),(3.41)
whe e τCis he cohe ence ime. Le us conside a modula ion signal wi h
pe iod 0= 1 ns, desc ibed by 201 ha monics, which eeds a Talbo de ice
wi h index γ/α = 1/2. F om Eqs. (3.27) and (3.30), only hose ha monics
mul iples o α= 2 ha e in luence o e he alue o he a e age ou pu
in ensi y. The e o e, he ou pu in ensi y is desc ibed by a cohe ence ma ix
wi h dimension N×N= 101 ×101. The o de ed eigen alues o his
sys em (λ1≥λ2≥...) a e p esen ed in igu e 3.1. On i s le hand side,
he la ges en eigen alues o he cohe ence ma ix a e depic ed o di e en
alues o he cohe ence ime. The eigen alues o a pa ially cohe en sou ce
wi h τC= 100 ns a e depic ed by squa es. Each pulse in he o iginal ain
is able o in e e e wi h he nea es 2τC/ 0= 200 ini ial pulses, so ha we
expec ha he ou pu ain is composed o ai h ul eplicas o he ini ial
pulses. The alue o he en opy has inc eased om i s alue in he cohe en
limi bu i is s ill a om i s maximum alue. I he empo al cohe ence
is educed up o 10 ns (ci cles) o 3 ns ( iangles-down), pulses in he
ini ial ain can in e e e only wi h he nea es 20 o 6 pulses, espec i ely.
This leads o a p og essi e equaliza ion o he magni ude o he eigen alues.
Co espondingly, he en opy is inc eased. Finally, o τC= 1 ns ( iangles-
up) we almos each he incohe en limi , since each pulse can in e e e wi h
only hei wo neighbou s. The en opy is close o i s maximum alue and
he eigen alues almos ha e he same magni ude.
On he igh hand side o igu e 3.1, we depic he e olu ion o he
wo la ges eigen alues (λ1, λ2), oge he wi h he en h (λ10) and lowes
(λ101) eigen alue, wi h espec o he cohe ence ime. Thei beha io
is equi alen o he p e ious plo : he ela i e impo ance o he main
eigen alue dec eases wi h he cohe ence ime. When all eigen alues a e
equally weigh ed wi h λ ∼1/N = 1/101 ∼0.01 we each he incohe en
limi . In his case, his happens o cohe ence imes o abou τC∼100 ps.
106 3.4. Ma ix Theo y and En opy
12345678910
10−3
10−2
10−1
100
Eigen alues’ index
Eigen alues
10−1 100101102
10−3
10−2
10−1
100
Cohe ence ime (ns)
Eigen alues
H=1.06
H=2.94
H=3.90
H=4.47
λ1
λ2
λ10
λ101
Figu e 3.1: Le hand side: he en highe eigen alues o cohe ence ma ix
associa ed o di e en cohe ence imes: τc= 100 ns (squa es), τc= 10 ns
(ci cles), τc= 3 ns ( iangles-down) and τc= 1 ns ( iangles-up). Righ
hand side: e olu ion o he eigen alues λ1,λ2,λ10 and λ101 as a unc ion
o he cohe ence ime. The pa ame e s o he Talbo de ice a e 0= 1 ns,
N=101, and γ/α = 1/2.
In igu e 3.2 we exempli y he in luence o pa ial cohe ence in he
ou pu a e age in ensi y. We also explo e he e ec i e numbe o sou ces
and hei abili y o econs uc he ou pu by unca ion o expansion
(3.37). In his plo we ep esen he ou pu a e age in ensi y ha esul s
a e aking in o accoun all eigen alues in (3.37) (con inuous line) and
he unca ion o Ne eigen alues (dashed line), o di e en alues o he
cohe ence ime. The ini ial ain is composed o Gaussian pulses o τP= 50
ps wid h (measu ed as he hal -wid h a 1/e decay in powe ) and pe iod
0= 1 ns. I is p opaga ed h ough a Talbo de ice wi h index γ/α = 1/2.
In spi e o he ac ha a Gaussian pulse ain is composed o an in ini e
collec ion o ha monics, we will assume ha he se o ha monics wi h
3. Pa ially cohe en Tempo al Talbo e ec 107
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
Time (ns)
In ensi y (a.u.)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
Time (ns)
In ensi y (a.u.)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
Time (ns)
In ensi y (a.u.)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
Time (ns)
In ensi y (a.u.)
H=0.17 → Ne =2 H=0.89 → Ne =3
H=2.07→ Ne =8
H=1.58 → Ne =5
Figu e 3.2: A e age in ensi y o a 1GHz ain o 50 ps Gaussian pulses
a e p opaga ion along a γ/α = 1/2-Talbo de ice. Con inuous cu e:
exac a e age in ensi y. Dashed cu e: app oxima e a e age in ensi y a e
unca ion o Ne sou ces.
powe abo e −30 dB he maximum desc ibes accep ably he ain. Unde
his c i e ion and aking in o accoun (3.34) he dimension o he ma ix
p oblem N= 2n+ 1 is de e mined by
n=⌊0.419
τP
0
α⌋,(3.42)
which in his example leads o N= 9. No e ha his unca ion o
he numbe o ha monics does no assu e ha he ini ial pulses a e ime-
limi ed o he in e al [0, 0/2], as equi ed o exac inc ease o he ain’s
epe i ion a e, bu will be su icien o desc ibe he e ec o pa ial
cohe ence.
In igu e 3.2(a) we p esen an example o a highly cohe en ca ie wi h
τC= 100 ns. The e ec i e numbe o sou ces is wo. The cu e depic ed by
108 3.5. En opy in he Limi o La ge N
conside ing wo sou ces is indis inguishable o m he exac in ensi y ou pu .
We ha e also checked ha he in ensi y associa ed o a cohe en ca ie is
indis inguishable om hose depic ed he e. In igu e 3.2(b) we conside
a ca ie wi h τC= 10 ns, which is he second example in ou p e ious
igu e 3.1. In he exac pa ially-cohe en in ensi y pa e n we obse e he
p og essi e smoo hing o he in ensi y pa e n, as expec ed. On he o he
hand, he e ec i e numbe o sou ces is now 3, and he esul ing a e age
in ensi y a e unca ing he expansion (3.37) is s ill in good ag eemen
wi h i s nume ical alue.
In igu es 3.2(c) and (d) he cohe ence ime o he ca ie a e 3 ns
and 1 ns, so ha he e ec i e numbe o sou ces inc eases o 5 and 8,
espec i ely. We obse e he p og essi e ading o he ou pu pulse ain,
which ends o a cons an alue in he incohe en limi . We also no ice
ha he accu acy o he unca ion by Ne e ec i e incohe en sou ces
dec eases when sou ces become less cohe en . This is due o he ac ha
he ela i e impo ance o he i s neglec ed eigen alue in he expansion,
wi h index k=Ne + 1, wi h espec o he p e ious one, wi h k=Ne ,
ends o uni y in he incohe en limi . The e o e, he unca ion o he
in ensi y expansion by he e ec i e numbe o sou ces p o ides a good
app oxima ion o he a e age in ensi y as long as he cohe ence o he sou ce
is no ex emely low. In gene al, i is necessa y o use sou ces wi h high
phase and ampli ude s abili y o achie e a cohe en phenomenon such as
he Talbo e ec , and he e o e he app oxima ion o expansion (3.37) by
unca ion o Ne e ms is alid in p ac ice.
3.5 En opy in he Limi o La ge N
The applica ion o he empo al Talbo e ec o mul iplying he epe i ion
a e o pulse ains equi es he absence o pulse o e lapping in he ou pu ,
o equi alen ly, na ow ini ial pulses. The desc ip ion o na ow pulses
equi es a la ge numbe N, which is he numbe o α-ha monics in he
ain and also he dimension o he cohe ence ma ix. Fo una ely, he
p ope ies o Toepli z ma ices in he limi o la ge Na e well unde s ood.
They a e based on Szeg¨o heo em on he dis ibu ion o eigen alues [104],
which s a es ha :
lim
N→∞
N
X
k=1
Fτ(N)
k=Z+π
−π
dΩ
2πF[P(Ω)].(3.43)
Conclusions
F om he spec al poin o iew, a Talbo line is a mul iple bandpass il e ,
whe e each passband ep esen s he in e e ence be ween dispe sed pulses
sepa a ed a ce ain numbe o ime slo s. Thus, ac ional Talbo lines,
ac il e ing he in ensi y ha monics o he inpu ain which do no belong
o he ou pu , leading o an inc ease o he epe i ion a e o he ain.
The ejec ion is exac only o ans o m-limi ed pulses and semi-in ege
Talbo de ices wi h negligible second-o de dispe sion. Talbo il e s a e
no s a ic bu depend on he dispe si e cha ac e is ics o he line as well as
he bandwid h o he inpu ain. In he pa icula case o linea ly chi ped
Gaussian pulse ain, we ha e de i ed he condi ions which gua an ee
ha he Talbo de ice p o ides an ou pu ain wi h negligible in ensi y
luc ua ions, e en in p esence o second o de dispe sion. Howe e , in his
case, he ou pu ain may su e om single pulse dis o ion.
The s abili y and he ole ance o Talbo il e s unde iming and leng h
a ia ions ha e been s udied. I inpu ains ha e la ge bandwid h, small
dispe sion a ia ions lead o no able b oadening o pulses, wo sening he
pe o mance o Talbo line by pulse- o-pulse in e e ence. Ne e heless,
when spec al wid h is mode a e, dispe sion ins abili ies de e io a e he
ejec ion capabili y o he il e . Inpu T ains wi h a bandwid h measu ed
as FWHM equals o wice he he epe i ion- a e o he ou pu ain yields
o he op imal s abili y. The ole ance leng h a e expec ed o be abou
10% di ided by he epe i ion- a e ac o .
We also ha e explo ed Talbo de ices as smoo he s o impe ec ions
o he inpu signal. In pa icula , we ha e i s s udied andomly on-o
keying o he pulses in he ain, and hen iming ji e y ains be o e
and a e passing h ough a Talbo line. We ha e assumed pulses o be
linea ly chi ped Gaussian. In bo h cases he basic ea u e o he spec um
is he p esence o passbands which il e he b oadband noise. I pulse
a e unchi ped he passbands a e cen e ed in he ha monics o he pe ec
ain, o he wise hei cen e s a e sligh ly shi ed. Measu emen o he
115

116 Conclusions
ela i e de ia ion o his cen e s wi h espec o he passband wid h allows
a de e mina ion o he chi p’s sign and magni ude. Passbands p esen a
modula ion, which inc eases in equency along he RF spec um.
The iming ji e y noise educ ion has been analyzed om he a iance
and by he in eg a ion o he RF noise ski abou he ha monics. When
ji e a iables a e unco ela ed, he a iance o he ain a e passing
Talbo de ice is la ened, becoming ime-independen . I leads o he
smoo hing o he iming ji e bu also o he appea ance o pulse pedes al.
The same esul is achie ed by means o RF noise in eg a ion. Howe e
when pulse ji e is co ela ed, he noise educ ion is less signi ica i e. I
is easy unde s ood om he RF spec um. In his case, ji e noise is
na owband a ound he ha monics and consequen ly he noise educ ion
induced by he p esence o passbands is smalle . Chi p has been shown o
be he mos dele e ious ac o because i induces a shi o he peak noise
le el wi h espec o he posi ion o he ha monic ha causes an inc ease
in he o al noise powe .
The in luence o he sou ce cohe ence in Talbo de ices has been
analyzed. The pa ial cohe en ca ie o he modula ed wa e a e
p opaga ion along a lowes -o de dispe sion medium is as a low pass il e
ac ing o e he a e age in ensi y associa ed o a monoch oma ic ca ie .
Unde Talbo condi ions, he cohe ence limi a ions a e mo e es ic i e o
high dispe si e lines, i.e, high Talbo index, since longe cohe ence ime is
need o p oduce ai h ull pulse- o-pulse in e e ence. Inc easing dispe sion
o dec easing he cohe ence ime esul s in a p og essi e dec ease o he
il e ’s bandwid h and he e o e, in loss o he low- ime scale de ails when
compa ed wi h he in ensi y ob ained wi h a monoch oma ic ca ie . We
also app oach he Talbo e ec unde pa ial cohe ence om a ma ix
poin o iew. The diagonaliza ion o he ma ix p oblem ep esen ing
he a e age in ensi y a e Talbo de ice yields o he in e p e a ion o he
a e age in ensi y as an incohe en sum o in ensi y pa e ns. In his con ex ,
we ha e compu ed he in o ma ional en opy and he numbe o e ec i e
sou ces as measu es o he o e all pe o mance o he Talbo de ice.
Appendix A
Le be de ine he unc ions 1( ) and 2( ), om he a bi a y unc ion ( )
as:
1( ) = ( ) exp(2πis / 0)⊗X
m
δ( −m 0/α) (A.1)
2( ) = ( ) exp(2πi(s−q) / 0)⊗X
m
δ( −m 0/α) (A.2)
I ( ) is con ined in o [− 0/2α, 0/2α], he p oduc 1( ) 2( )∗can be
exp essed as ollows
1( ) 2( )∗=| ( )|2exp(2πiq / 0)⊗X
m
δ( −m 0/α) (A.3)
which does no depend on he a iable s. By using he Fou ie expansion o
1( ) 2( )∗and ca ying ou he mul iplica ion among ha monics we ob ain
FT [ 1( ) 2( )∗](ω) = 2πα2X
p
δ(ω−2πpα/ 0)X
a α+sa∗
α+s−pα−q(A.4)
F om (A.3) and (A.4), i ollows ha he sum
X
a α+sa∗
α+s−pα−q(A.5)
does no depend on s, so he p oo is comple e.
117
Appendix B
Se e al pa ame e s ha e been in oduced o measu ing he o e all deg ee
o cohe ence o he ma ix ep esen a ion o a pa ially-cohe en spa ial
sys em. In he con ex o a s udy o he e ec o pa ial cohe ence in
he Wigne ep esen a ion o a pa axial op ical sys em [87,88], Bas iaans
in oduced a se o quan i ies µm(m > 1) as a measu e o pa ial cohe ence.
The pa ame e s µma e based on he Neeigen alues λko he ma ix
ep esen ing he impac o pa ial cohe ence in he sys em’s pe o mance:
µm= Ne
X
k=1
λm
k!1
m−1
.(B.1)
Pa ame e µ1is de ined as he limi µ1= limm→∞ µm. I can be shown
ha pa ame e s µma e bounded by 0 and 1:µm= 0 co esponds o he
o ally incohe en case whe eas µm= 1 ep esen s he cohe en limi . The
alue o pa ame e s µmis a nondec easing unc ion o he index m, and
he e o e µ1is he smalles o he se µmand hus he mos conse a i e
measu e o he deg ee o cohe ence.
Fo m→ ∞,µ∞ ep esen s he maximum eigen alue. The o iginal
de ini ion o he e ec i e numbe o unco ela ed andom a iables
ep esen ing he op ical signal is p ecisely i s in e se, 1/µ∞[89]. The special
case µ1can be exp essed as a unc ion o he en opy:
µ1= exp Ne
X
k=1
λkln λk!= exp(−H).(B.2)
Then, pa ame e µ1is he in e se o he e ec i e numbe o sou ces as
in oduced in he ex , Ne = 1/µ1, and ep esen s he la ges measu e o
he numbe o e ec i e sou ces when conside ing he comple e se ies µm.
119

Acknowledgmen s
Es a esis ep esen a la culminaci´on del abajo ealizado en los ´ul imos
cua o a˜nos. Du an e es os a˜nos han sido muchas las pe sonas con las que
me he c uzado y quienes, bien a ni el acad´emico, bien a ni el pe sonal han
con ibuido a que es e abajo llegase a buen pue o. A odos ellos quisie a
mos a aqu´ı mi m´as since o ag adecimien o. Espe o no deja me a nadie
en el in e o, pe o po si eso ocu ie a pido disculpas de an emano.
En p ime luga , y como no pod ´ıa se de o a mane a, quisie a
ag adecie a mis di ec o es de esis Ca los G´omez Reino y Ca los R.
Fe n´andez-Pousa, el habe me dado la opo unidad de inco po ame a su
g upo de in es igaci´on, su con inua a enci´on y seguimien o, as´ı como las
inume ables palab as de apoyo, a ec o y animo, que me han b indado a lo
la go de es os a˜nos. Sin su ayuda la p esen aci´on hoy de es a esis no hubie a
sido posible. Quisie a adem´as hace ex ensi o mi ag adecimien o a odos
los miemb os del g upo de ´
Op ica GRIN, Ma i ´ı, Ca men, Mai e, An onio,
Dani y Miguel, po la ayuda p es ada y po el dis endido y ag adable
ambien e de abajo que en e odos gene an.
Po supues o no me ol ido de mis compa˜ne os de despacho, an o de los
que ya no es ´an, en especial Jo ge y Jus o (¡os echamos de menos!) como de
los que se han ido inco po ando, Paula y la ” eci´en” llegada Bea. Con ellos
he compa ido mucho m´as que ho as de abajo en e las mismas cua o
pa edes, y si bien, en los ´ul imos meses de edacci´on de la esis hubiese
pagado po un anquilo y silencioso despacho pa a mi soli a, nada hubiese
sido lo mismo sin ellos.
I also would like o hank o Hans Pe e He zig o gi ing me he
oppo uni y o join he Applied Op ics g oup o he IMT in Neucha el.
Thanks o Oma Manza do o supe ising my wo k and o all bee s
sha ed, and in gene al all he membe s o he Applied Op ics g oup. Voglio
ing azia e Pie o, pe u o l’aiu o che mi ha da o nel la o o s ol o in
labo a o io, ma sop a u o pe ch`e g azie a lui conse o uno splendido
ico do dei mesi asco si a Neuchˆa el. Un sen i o ing aziamen o anche
121
122 Acknowledgmen s
a u i i suoi amici che in dal p imo gio no mi hanno accol o benissimo.
Tambi´en quise a eco da aqu´ı a oda la gen e es upenda que he enido
la sue e de conoce du an e es os a˜nos. En especial quisie a nomb a a
Luc´ıa, Ja i, Xoel, Sa a e I ia quienes, a ni el pe sonal, han sido pa a mi
un g an apoyo, y con quienes he compa ido an buenos momen os.
Po ´ul imo y no po ello menos impo an e engo que ag adece a mis
pad es odo el ca i˜no, la con ianza, el apoyo me han mos ado simp e,
especialmen e en mis momen os m´as bajos. !G acias!
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