Integer and fractional Talbot effect : theoretical study and practical considerations

Departamento De Física Aplicada Integer and Fractional Talbot Effect: Theoretical Study and Practical Considerations Integer and Fractional Talbot Effect: Theoretical Study and Practical Considerations Area de Óptica Laura Chantada Santodomingo Santiago de Compostela, Junio 2006 UNIVERSIDADE DE SANTIAGO DE COMPOSTELA UNIVERSIDADE DE SANTIAGO DE COMPOSTELA D. Carlos G´omez-Reino Carnota y D. Carlos Rodr´ıguez Fern´andez- Pousa, Catedr´atico del ´ Area de ´ Optica del Departamento de F´ısica Aplicada de la Universidad de Santiago de Compostela y Contratado Doctor del ´ Area de Teor´ıa de la Se˜nal y Comunicaciones del Departamento de F´ısica y Arquitectura de Computadores de la Universidad Miguel Hern´andez de Elche, CERTIFICAN que la presente memoria, titulada “Integer and Fractional Temporal Talbot Effect: Theoretical Study and Practical Considerations” ha sido realizada por D˜na. Laura Chantada Santodomingo bajo su direcci´on y constituye la Tesis que presenta para optar al Grado de Doctor en Ciencias F´ısicas. Santiago de Compostela, 8 de Junio de 2006. V◦. B◦. Director de la tesis V◦. B◦. Director de la tesis Carlos G´omez-Reino Carnota Carlos Rodr´ıguez Fern´andez-Pousa La doctoranda Laura Chantada Santodomingo Este trabajo ha sido financiado por los proyectos TIC99-0489 y TIC2003-03041 del ministerio de Educac´ıon y Ciencia y el proyecto PGIDT04PXIC20605PN de la Xunta de Galicia Gracias al ministerio de Educaci´on y Ciencia por la ayuda econ´omica recibida a trav´es de la beca FPU. Resumen Las t´ecnicas y dispositivos que permiten la manipulaci´on de las propiedades de las ondas guiadas, constituyen la base tecnol´ogica de la espectroscop´ıa de se˜nales moduladas, del procesado fot´onico, y de los m´etodos para el an´alisis temporal de fuentes pulsadas. Para implementar todas estas t´ecnicas se pueden utilizar tanto procesos lineales como no lineales. Dentro del contexto de la ´optica lineal, la llamada analog´ıa espacio-temporal se ha convertido en los ´ultimos a˜nos en la una gu´ıa muy fruct´ıfera para el dise˜no de nuevos dispositivos. La dualidad espacio-temporal permite trasladar conceptos e ideas bien conocidas en procesado de se˜nales espaciales al dominio temporal, proporcionando de este modo nuevas formas para el procesado, manipulaci´on y control de se˜nales temporales. Dicha analog´ıa est´a basada en la equivalencia matem´atica entre la ecuaci´on que describe la difracci´on paraxial de haces y la ecuaci´on de dispersi´on de primer orden que gobierna la evoluci´on de pulsos ´opticos cuando ´estos se propagan en un medio diel´ectrico o en una gu´ıa de onda. En ambos casos se trata de una ecuaci´on diferencial parab´olica con coeficientes imaginarios, y por tanto admite soluciones ondulatorias. Las ecuaciones que describen la evoluci´on de envolventes ´opticas lentamente variables, correspondientes a ondas planas o modos (en el caso de gu´ıas de onda) modulados temporalmente, son formalmente id´enticas a las ecuaciones paraxiales que describen la propagaci´on de ondas monocrom´aticas de extensi´on finita. Existe por tanto una correspondencia entre la variable temporal de los problemas dispersivos y la variable espacial transversal de los problemas difractivos. Asimismo, en el dominio de Fourier esta correspondencia se da entre los espectros de ambos problemas. Esta dualidad representa el marco conceptual de la compresi´on lineal de pulsos, as´ı como de muchos otros dispositivos que han sido propuestos en los ´ultimos a˜nos. Entre ellos podemos mencionar, sistemas de formaci´on de imagenes temporales, filtrado temporal por medio de lentes temporales, reduci´on de jitter, la transformada de Fourier temporal, la versi´on temporal del teorema de Van Cittert-Zernike, lentes iii iv Resumen temporales de gradiente de ´ındice, t´ecnicas de trasformaci´on tiempofrecuencia y frecuencia-tiempo y el efecto Talbot temporal entre otros. El efecto Talbot fue descubierto en 1836 por H.F. Talbot, cuando iluminaba una red de difracci´on rectangular de agujeros min´usculos con una fuente de luz blanca de peque˜no tama˜no. A ciertas distancias a lo largo de la direcci´on de propagaci´on, pod´ıan observarse r´eplicas de las estructuras peri´odicas que hab´ıa iluminado. Este efecto no fue satisfactoriamente explicado hasta que Lord Rayleigh, en 1881, lo atribuy´o a la interferencia entre haces difractados por las estructuras peri´odicas. Mostr´o que, en el caso de una red lineal iluminada con un frente de onda plano, las im´agenes de la red se repet´ıan a lo largo de la direcci´on de iluminaci´on, localizadas a distancias zγ=γd2/λ, donde γes un n´umero entero, del per´ıodo de la red yλla longitud de onda de la fuente. Sin embargo hubo que esperar largos a˜nos hasta que Cowley y Moodie (1957-1960), en sus estudios pioneros sobre las propiedades de la difracci´on de Fresnel de objetos peri´odicos, dieron una explicaci´on del efecto Talbot en el contexto de la teor´ıa difraccional de la formaci´on de im´agenes. A˜nos m´as tarde, Montgomery (1967) en un trabajo fundamental, desarroll´o las condiciones generales que debe satisfacer un objeto para obtener r´eplicas de su distribuci´on de amplitud compleja a lo largo de la direcci´on de iluminaci´on. Fue entonces cuando Montgomery introdujo el t´ermino autoimagen que a partir de ese momento se us´o conjuntamente con el t´ermino efecto Talbot en los trabajos que sobre este tema se han publicado hasta la actualidad. Adem´as de la reconstrucci´on de imagenes, se pueden observar patrones con estructuras mucho m´as ricas. Por ejemplo, en ciertos planos a lo largo de la direcc´ıon de propagaci´on, se poducen patrones cuya celda unidad est´a formada por la superposici´on coherente de las celdas originales desplazadas y con una diferencia de fase entre ellas. De este modo, si las celdas est´an suficientemente confinadas como para evitar la superposici´on, se obtienen r´eplicas del patr´on original pero con una periodicidad que es un cierto de n´umero de veces menor que la de la red ilumnada. Este efecto es conocido como efecto Talbot fraccionario. La contrapartida temporal del efecto de autoimagen o efecto Talbot tiene lugar cuando una se˜nal temporal peri´odica, como por ejemplo un tren de pulsos ´opticos, se propaga a trav´es de un medio dispersivo lineal bajo condiciones de primer orden de dispersi´on. Cuando la dispersi´on acumulada es un m´ultiplo de una cierta escala b´asica, el tren de pulsos se reproduce a s´ı mismo tras la propagaci´on, en lo que se conoce como efecto Talbot entero. Por otra parte, cuando la dispersi´on acumulada es una fracci´on de dicha Resumen v escala b´asica, a la salida del medio dispersivo se obtiene un tren de pulsos con una frecuencia de repetici´on que es un n´umero entero de veces la del tren original, siempre y cuando los pulsos sean suficientemente estrechos como para evitar la superposici´on. Este efecto es conocido como efecto Talbot fracionario. El efecto Talbot temporal entero fue descrito por primera vez por Jannson y Jannson quienes propusieron su uso para la transferencia de informaci´on contendia en una se˜nal peri´odica a lo largo de una fibra ´optica. Estos autores establecieron por vez primera las longitudes de la fibra ´optica en las que ten´ıa lugar la reconstrucci´on de la se˜nal peri´odica. Posteriormente, Andrekson aport´o la primera prueba experimental del efecto Talbot, al propagar un tren de pulsos generado por un l´aser de sincronizaci´on de modo (mode-locking) a trav´es de una fibra ´optica. A finales de los a˜nos noventa Shake y Arahira probaron experimentalmente el efecto Talbot fraccionario en fibras momomodo y posteriormente Aza˜na y Muriel hicieron lo propio en fibras de periodo progresivo (LCFG -linearly chirped fiber grating). El efecto Talbot temporal contin´ua atrayendo el inter´es de los investigadores, en especial el efecto Talbot fracionario, ya que puede utilizarse para el incremento de la frecuencia de repetici´on de una secuencia de pulsos peri´odica como la generada por l´aseres de sincronizaci´on de modo. La generaci´on de trenes de pulsos ´opticos con frecuencia de repetici´on ultrarr´apida es de vital importancia en campos como sistemas de computaci´on ´optica, porcesado de datos ultrarr´apido o sistemas de comunciaciones ´opticas entre otras ´areas cient´ıficas. A pesar de que se ha progresado mucho en la generaci´on directa de trenes de pulsos con altas frecuencias de repetici´on a partir de l´aseres de semiconductores o de fibra, resulta ventajoso tener t´ecnicas sencillas que permitan incrementar a´un m´as la frecuencia de repetici´on fuera de la cavidad del l´aser. Por otra parte, el efecto Talbot es un fen´omeno lineal y por tanto permite su integraci´on con otras t´ecnicas de procesado lineal tales como compresi´on o reconstrucci´on (reshaping) de pulsos. El efecto Talbot tambi´en ha sido usado junto con fen´omenos no lineales, as´ı como con sistemas de imagen temporal. Aunque el an´alisis experimental de la formaci´on de r´eplicas Talbot de trenes de pulses se centra t´ıpicamente en el dominio temporal, se puede extraer mucha m´as informaci´on mediante el an´alisis espectral de la intensidad del tren de pulsos. En los filtros convencionales dise˜nados para aumentar la frecuencia de repetici´on mediante filtrado ´optico, un filtro lineal suprime los arm´onicos ´opticos del tren de entrada que no pertenecen al tren de salida de alta frecuencia. Por lo tanto, el funcionamiento de estos sitemas se puede analizar por medio de la funci´on de transferencia ´optica vi Resumen del filtro lineal. Por el contrario, y debido al hecho de que la dispersi´on es un filtro de fase puro en el dominio ´optico, el an´alisis espectral de filtros Talbot ha de realizarse en t´erminos de intensidades, lo que da lugar a un an´alisis espectral no lineal. En este trabajo se presenta un estudio del efecto Talbot temporal entero y fracionario, teniendo en cuenta consideraciones pr´acticas tanto a nivel del tren de pulsos como del medio dispersivo utilizado. La memoria consta de tres cap´ıtulos. En el primero de ellos, introducimos de forma breve la dualiad espacio temporal que radica en la equivalencia matem´atica entre las ecuaciones que describen la difraci´on paraxial unidimensional y la propagaci´on de pulsos en un medio dispersivo de primer order con atenuaci´on despreciable. Dentro del contexto de esta analog´ıa presentamos la contrapartida temporal del efecto Talbot. La mayor parte del an´alisis se realiza en el dominio espectral y se basa en el c´alculo exacto de la densidad espectral de potencia de un tren de pulsos coherentes tras su propagaci´on en un medio dispersivo con coeficientes de dispersi´on de primer y segundo order arbitrarios. Hemos considerado que la se˜nal temporal est´a formada por un tren de pulsos gaussianos con modulaci´on lineal de fase (linear chirp). Estos resultados anal´ıticos permiten identificar las componenetes espectrales que representan la dispersi´on individual de los pulsos, as´ı como los t´erminos que corresponden la interferencia entre pulsos. Desde el punto de vista espectral las lineas Talbot se pueden interpretar como un filtro pasabanda m´ultiple, donde cada pasabanda corresponde a la interferencia entre pulsos separados un cierto n´umero de per´ıodos. El filtro elimina los arm´onicos de intensidad del tren de entrada que no pertenecen a la salida. La supresi´on de los arm´onicos s´olo es exacta para filtros Talbot semi-enteros, con coeficientes de dispersi´on de segundo orden y mayores despreciables, y pulsos en el l´ımite de transformada, ya que en este caso las distancias Talbot coinciden con las distancias de supresi´on de portadora. Los filtros Talbot no son est´aticos sino que dependen de la anchura espectral de la se˜nal de entrada y de las caracter´ısiticas dispersivas de la l´ınea. En particular, hemos deducido las condiciones que garantizan que el filtro Talbot proporciona un tren cuyas fluctuaciones en intensiad son despreciables incluso cuando el medio de propagaci´on presenta coeficiente de dispersi´on de segundo orden. Finalmente, hemos analizado la estabilidad de las l´ıneas Talbot frente a variaciones de frecuencia o de la longitud de la fibra. Cuando la anchura espectral del tren es grande, la inestabilidad de la dispersi´on tiende a ensanchar los pulsos empeorando el funcionamento del filtro debido a la interferencia entre pulsos. Por el contrario, cuando Resumen vii la anchura espectral es moderada, variaciones en la dispersi´on debilitan la supresi´on de arm´onicos. La estabilidad ´optima se alcanza cuando la anchura espectral de el tren, medida como FWHM (full width half maximum), es el doble de la frecuencia de repetici´on del tren de salida. El objetivo general del segundo cap´ıtulo es la descripci´on espectral de las propiedades de autoregeneraci´on del efecto Talbot temporal. Consideramos que los pulsos tiene forma Gaussiana y est´an linealmente modulados en fase. Adem´as suponemos que el dispositivo Talbot es ideal, es decir, es un medio dispersivo lineal de primer orden sin atenuaci´on, y con un ancho de banda superior al del tren de pulsos de entrada. En la primera secci´on analizamos la densidad de potencia espectral de un tren cuyos pulsos est´an aleatoriamente distribuidos antes y despu´es de un filtro Talbot. Hemos considerado que las variables que describen la presencia o ausencia de los pulsos en el tren no est´an correladas. Como consecuencia de la interferencia coherente entre los pulsos dispersados, la envolvente del ruido de banda ancha del espectro de radiofrecuencia despu´es de la detecci´on del tren se ensancha por causa de la modulaci´on de fase de los pulsos. La principal caracter´ıstica del espectro es la presencia de bandas pasantes que filtran el ruido de banda ancha, y cuyos centros aparecen desplazados respecto a los arm´onicos debido a la modulaci´on de fase. Por tanto, midiendo la desviaci´on relativa de estos centros con respecto a la anchura de las bandas pasantes es posible determinar el signo y la magnitud de la modulaci´on de fase de los pulsos del tren de entrada. Las bandas pasantes son tanto m´as estrechas cuanto mayor es la dispersi´on del dispositivo, menor es la anchura temporal de los pulsos y mayor es la modulaci´on de fase. El m´aximo valor de estas bandas pasantes depende de la cantidad relativa de pulsos presentes en el tren. Para dispositivos de baja dispersi´on y pulsos anchos las bandas pasantes aparecen moduladas por una funci´on coseno cuadrado. La anchura de esta modulaci´on es una fracci´on del valor del arm´onico fundamental de la entrada, y es independiente de la cantidad relativa de pulsos en el tren y de sus propiedades. En la segunda secci´on se analiza la influencia del timing jitter de un tren de pulsos gaussianos con modulaci´on de fase lineal en un dispositivo Talbot. A pesar de que en nuestro estudio no hemos tenido en cuenta posible fluctuaciones de amplitud (amplitude jitter) del tren de pulses, el analisis del timing jitter puede generalizarse a problemas con amplitude jitter o una combinaci´on de ambos. El an´alisis se lleva a cabo tanto en el domino temporal como en el espectral. En el an´alisis temporal consideramos que las variables que describen el timing jitter de los pulsos viii Resumen no est´an correladas. El estudio presentado muestra como los valores de la varianza del tren se homogeneizan tras su paso por el dispositivo Talbot, ya que tanto los pulsos individuales como su varianza se ensanchan debido a la dispersi´on, extendiendose por los intevalos temporales de los pulsos vecinos. Presentamos adem´as expresiones anal´ıticas aproximadas para la varianza de la entrada y de la salida, analizando la validez de dichas aproximaciones. Para dispositivos Talbot enteros, el valor de la varianza es constante e independiente del valor de la dispersi´on. Sin embargo para dispositivos Talbot fraccionarios, la varianza a la salida podr´ıa presentar una modulaci´on, dependiendo del cociente entre la anchura temporal del pulso y el per´ıodo del tren original. En cualquier caso, para los dispositivos Talbot dise˜nados para aumentar la frecuencia de repetici´on de un tren de pulsos, la aproximaci´on de la varianza a una constante contin´ua siendo v´alida. El problema del timing jitter se analiza tambi´en desde el punto de vista espectral. Ahora, y a diferencia del anterior an´alisis en t´erminos de varianza, consideramos que las variables aleatorias que describen el jitter de cada pulso est´an en general correladas. La principal cararter´ıstica del espectro resultante es, como ocurr´ıa en el caso de trenes de pulsos aleatoriamente distribuidos, la presencia de un mecanismo de filtrado del ruido de banda ancha, dando lugar a una reducci´on de la potencia de ruido integrado alrededor de los arm´onicos en el dominio de radiofrecuencia. Tambi´en se analiza la influencia de la modulaci´on de fase y de la anchura de l´ınea ´optica de los pulsos individuales. Estos resultados se utilizan para analizar la reducci´on de ruido de las r´eplicas Talbot bajo diferentes condiciones de propagaci´on. La reducci´on del ruido causado por el timing jitter depende principalmente de la modulaci´on de fase de los pulsos del tren, de la correlaci´on entre pulsos adyacentes y de la dispersi´on acumulada de la l´ınea, pero sin embargo no depende del valor de la desviaci´on est´andar del timing jitter. La modulaci´on de fase es la principal causa de deterioro de la reducci´on de ruido, ya que provoca el desplazamiento de las bandas pasantes con respecto a la posici´on de los arm´onicos. Sin embargo, un dise˜no adecuado de la l´ınea dispersiva permite de un modo sencillo compensar dicha modulaci´on de fase y al mismo tiempo reducir la anchura temporal de los pulsos. Debido a que la correlaci´on controla el ancho de banda del ruido antes de que el tren de pulsos entre en la l´ınea dispersiva y debido tambi´en a que el suavizado del timing jitter es m´as pronunciado a escalas del orden de la frecuencia de repetici´on, el ruido inicial ha de ser de banda ancha para Introduction 3 rate of 196 GHz (four times the original one) and 400 GHz (eight times the original one) respectively. Integer and fractional Talbot effect have been also demonstrated in linearly chirped fiber gratings (LCFG) [27, 28] and multimode fibers [29]. The temporal Talbot effect is still attracting the interest of researches [30–33], especially the fractional Talbot effect, due to the fact that it can be applied for multiplying the repetition rate of periodic pulse sequences. This property has also been demonstrated for multiwavelength optical pulse trains [34,35]. The generation of optical pulse trains with ultrahigh repetition rates is of vital importance in fields such as ultrafast data processing optical computing systems or ultra-high-bit-rate optical communications among others. Despite it has been much progress on direct generation of high-repetition rate pulse trains from semiconductor or fiber lasers, it is always advantageous to have simpler methods for further increasing the pulse rate [36–39]. To day the highest repetition rate reached by means of Talbot devices is 2.5 THz, which was achieved by multiplying 250×a 10 GHz train of pulses which had been previously compressed [40]. Talbot effect has also been applied to demostrate new types of modelocked lasers [41, 42]. Moreover and since Talbot effect is a linear phenomenon, it allows integration with other linear processing techniques, such as compression [43, 44] or reshaping [28, 45]. Its tunability has also been demonstrated by the use of tunable dispersive lines [46, 47], and by switching to soliton regimen [48]. Talbot dispersive lines have also been used together with nonlinear phenomena [42,47,49] and temporal imaging systems [32]. Although the experimental analysis of the Talbot-imagined trains usually focuses on the time domain, much information ca be inferred from a spectral analysis of the intensity of the train [24,25]. In conventional filters designed to increase the repetition rate based on optical spectral filtering, a linear filter suppresses the optical harmonics of the input train that do not belong to the high-frequency output train [47]. Therefore the performance of these systems can be analyzed from the optical transfer function of the linear filter. In contrast, and because dispersion is a phase-only filter in the optical domain, Talbot dispersion filtering is nonlinear and requires a spectral analysis in the intensity domain. Here we present a theoretical study of Talbot devices, taking into account practical considerations regarding both the incoming sequence of pulses and the dispersive medium. The study is mostly carried out in spectral domain by means of intensity spectrum, which leads to present Talbot lines as a nonlinear filter. The work is divided in tree chapters. In 4Introduction the first one the basis of space-time duality and the temporal Talbot effect are briefly reviewed. Likewise we analyze in terms of intensity spectrum how dispersive lines with arbitrary first and second order dispersion act on a sequence pulses. Subsequently, these results are particularized to a Talbot line, thus showing the mechanism of image formation depending on Talbot index. The deterioration due to second-order dispersion as well as the stability and tolerance of Talbot devices against timing and frequency variations are also tackled. In the second chapter we investigate the restoration capability of Talbot effect. This property is well-known in diffractive optics and has been exploited to smooth imperfections of a grating. Here we study non-ideal trains after its propagation in a Talbot line. In particular we analyze first trains of on-off-keyed pulses and later trains of pulses suffering from timing jitter. In both cases the imperfections of the pulse sequence are random and therefore the presented analysis of Talbot lines is not deterministic but stochastic. Chapter 3 deals with the influence of the coherence of the wave on Talbot devices. We first study the behavior of an arbitrary signal in a linear time-invariant system under partially coherence conditions. When the linear system is a lowest-order dispersive line the effect of the coherence on the temporal signal can be expressed as a low pass filter. These results are particularized for Talbot devices. A matrix analysis of Talbot lines is also reported, presenting the information entropy of the system as a measured of the overall degree of coherence of the system. Chapter 1 Temporal Talbot effect 1.1 Introduction The Talbot effect is widely known as a coherent phenomenon, which takes place in the paraxial diffraction region when a periodic object is illuminated by a coherent source [21,22]. It consists in the formation of self-images or replicas of the object at certain distances along the propagation direction, and has its origin in the interference among diffraction orders of the grating. At certain distances from the object exact replicas are found, which are called integer Talbot images. Between two integer Talbot planes, patterns whose unit cells are formed by the coherent superposition between shifted cells. These patterns are called fractional Talbot images. If cells are confined enough to avoid overlapping between neighboring cells, replicas of the grating but with a periodicity lower than that of the original pattern are achieved, which is known as fractional Talbot effect. Both integer and fractional Talbot effects have been proved and applied not only in free space but in inhomogeneous [50, 51] and waveguide media [52], where the propagating modes play the role of the diffracted orders in free space. The same phenomenon is found in the time domain when a periodic signal propagates in a dispersive medium [19, 27, 28, 53]. This fact, is a consequence of the well-known mathematical analogy between the equations governing one-dimensional paraxial diffraction and lowest-order dispersion for narrowband signals [1, 2]. In this context, the pulse envelope is equivalent to the complex amplitude distribution, dispersion is equivalent to diffraction and the periodicity of the grating corresponds to the periodicity of the temporal sequence. Thus, when the accumulated dispersion of the medium is a multiple of a basic scale replicas of the input 5 61.2. Space-Time Duality train of pulse are achieved [20]. Furthermore if accumulated dispersion is a fraction of the mentioned basic scale, a sequence of pulses with a repetition rate higher than that of the original train is obtained, as long as pulse are narrow enough to avoid overlapping [1]. This property has been widely exploited and nowadays Talbot devices constitute a well a well-established all-optical technique to increase the repetition rate of a train of pulses [20,25–28,54–56]. This chapter deals with the analysis of the temporal Talbot effect, and is organized as follows. In section 1.2, we present the basis of the space-time duality. The integer and fractional Talbot conditions in both space and time domain are established in section 1.3. The analytical study of the intensity of an arbitrary train of pulses after integer and fractional Talbot device is also reported. Section 1.4 is devoted to the analysis of a train of linearly chirped Gaussian pulses which propagates through an arbitrary second-order dispersion medium. This analysis is carried out in the Fourier domain by means of the intensity spectrum. These results are particularized for a dispersive line satisfying Talbot condition in section 1.5. Some examples are presented to show how the reconstruction of the train takes place, as well as the deleterious of the resulting train due to the presence of second order dispersion. Sections 1.6 and 1.7 are respectively devoted to the analysis of the rejection capability and stability of Talbot lines under fiber length and timing variations. Finally we end the chapter presenting our conclusions. 1.2 Space-Time Duality The so-called space-time duality involves the equivalent behavior of spacelimited beams propagating in free space and narrow band signals in a first-order dispersion medium. This analogy is based on the mathematical equivalence between the equations describing one-dimensional paraxial diffraction and narrow-band dispersion [1]. Both equations are derived from the wave equation by assuming two straightforward approximations. They are: monochromatic waves and paraxial rays for the spatial case; and narrowband fields for the temporal case [2]. In a linear non-magnetic medium with dielectric constant ǫthe spacetime evolution of the electric field, e, is governed by the wave equation, ∇2e=µ0ǫ∂2e ∂t2,(1.1) 1. Temporal Talbot effect 7 ∇2and µ0being the Laplacian operator and the magnetic permeability, respectively. Since the field components in Eq. (1.1) are uncoupled, the wave equation can be expressed without loss of generality in scalar form. From the wave equation we will derive the paraxial diffraction and first-order dispersion equations by setting the above mentioned approximations. 1.2.1 Paraxial diffraction equation In order to achieve the paraxial equation, the electric field is assumed to be a monochromatic wave with frequency ω0,e(x, y, z, t) = e0(x, y, z) exp(iω0t). The time derivations are now multiplicative factors (∂/∂t →iω0), converting the wave equation (1.1) into the Helmholtz equation, (∇2+k2)e0= 0,(1.2) where k=µ0ǫω02is the wave number. We are interested in the study of paraxial rays, i. e., rays confined mostly along the propagation direction z. Therefore, the most rapid spatial phase variations take place in zdirection, and the electric field can be rewritten as e0(x, y, z) = ψ(x, y, z) exp(−ikz).(1.3) ψ(x, y, z) is the complex amplitude distribution and varies slowly compared with the wavenumber k. The paraxial approximation implies that the curvature of the electric field in propagation direction is much less than the curvature of the transversal profile, ∂2/∂z2≪∂2/∂x2,∂2/∂y2and 2k|∂/∂z|.(1.4) Under this conditions the Helmholtz equation results in ∂ψ ∂z +i 2k∇2 tψ= 0,(1.5) ∇2 tbeing the transversal Laplacian operator. Eq. (1.5) is known as parabolic or paraxial equation and governs the evolution of the electric field of a monochromatic beam propagating down z-axis. 81.2. Space-Time Duality 1.2.2 Dispersion Equation The temporal analogy for the paraxial equation appears by considering narrowband waves propagating through a first-order dispersive medium. In order to derive the narrowband dispersion equation we suppose the electric field to be narrowband centered at ω0. Then, it can be conveniently rewritten as e(x, y, z;t) = e0(x, y, z;t) exp(iω0t) (1.6) By using expression (1.6), the wave equation, Eq. (1.1), in scalar way is expressed in the Fourier domain as ∇2E0(x, y, z;ω−ω0) + µ0ǫω2E0(x, y, z;ω−ω0) = 0,(1.7) where E0(x, y, z;ω−ω0) is the Fourier transform of e0(x, y, z;t) evaluated at ω−ω0, that is, the frequency measured with respect to the central frequency ω0. Thus, E0(x, y, z;ω−ω0) represents the electric field at baseband. Equation (1.7) can be solved by using the method of separation of variables [5]. If we assume a solution of the form E0(x, y, z;ω−ω0) = F(x, y)A(z, ω −ω0) exp(−iβ0z),(1.8) where A(z, ω −ω0) is a slowly varying function of zand β0is the wave number at the carrier frequency to be determined later. Equation (1.7) leads to the following two equations for 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 obtaining Eq. (1.10), the second derivative ∂2A(z, ω −ω0)/∂z2was neglected since A(z, ω −ω0) is assumed to be a slowly varying function of z. The wave number β(ω) as well as the transversal profile of the electric field, F(x, y), are determined by solving the eigenvalue equation (1.9). F(x, y) represents the the modal distribution, and for single-mode fibers, corresponds to the modal distribution of the fundamental fiber mode given by equations (1.8), and (1.9). We are interested in the second equation (1.10) which governs the time profile evolution of the electric field. It can be written as 1. Temporal Talbot effect 9 ∂A(z, ω −ω0) ∂z =−i[β(ω)−β0]A(z, ω −ω0) (1.11) where we have approximated β(ω)2−β2 0by 2β0(β(ω)−β0). This equation represents a linear time-invariant system with transfer function [3] H(z, ω) = exp[−i(β(ω)−β0)z].(1.12) The interpretation of equation (1.11) is clear. Each spectral component within the pulse envelope acquires, as it propagates, a phase shift whose magnitude is frequency dependent. Assuming a narrowband signal centered at ω0, the propagation constant, β(ω), can be expanded in Taylor series around the central frequency ω0, as follows β(ω)∼ =β0+β1(ω−ω0) + β2 2(ω−ω0)2+β3 6(ω−ω0)3,(1.13) where β0=β(ω0) is the propagation constant at the central frequency, β1=dβ(ω)/dω|ω=ω0is the inverse of the group velocity and β2= d2β(ω)/dω2|ω=ω0and β3=d3β(ω)/dω3|ω=ω0are the first and second-order dispersion coefficient respectively. By using Eq. (1.13), Eq.(1.11) is given, in the temporal domain, by ∂ ∂z +β1 ∂ ∂t −iβ2 2 ∂2 ∂t2−β3 6 ∂3 ∂t3a(z, t) = 0.(1.14) where a(z, t) is the optical slowly varying pulse envelope. It corresponds to the Fourier transform of A(z, ω −ω0), so that the optical envelope a(z, t) is expressed in baseband. By introducing a change of variables to travelingwave coordinate system, τ=t−β1z, Eq. (1.14) results in ∂ ∂z −iβ2 2 ∂2 ∂t2−β3 6 ∂3 ∂t3a(z, t) = 0.(1.15) This is the narrowband dispersion equation expressed in the proper reference frame. If the propagation medium is pure first-order dispersion (β3= 0), Eq. (1.15) is written: ∂a(z, τ) ∂z =iβ2 2 ∂a(z, τ) ∂τ2.(1.16) It is apparent now the mathematical equivalence between first-order dispersion equation (1.16) and the paraxial diffraction equation (1.5) when only one transversal dimension is relevant. They are parabolic equations 10 1.3. Temporal Self-Imaging Effect Space domain Time domain z z x τ k−1/β2 ψ(x, z)a(τ, z) Table 1.1: Conversion table between space and time domain which describe the evolution of the envelope field in space and time domain respectively. The comparison between dispersion and one dimension paraxial equation leads to the conversion table 1.1. Therefore, any medium enable to provide flat amplitude and linear group delay over the bandwidth of the signal, is adequate to reproduce the time-domain equivalent to paraxial diffraction effects. The most common mediums used are: single mode fiber (SMF) and linearly chirped fiber grating (LCFG) [57]. SMF is a good first-order dispersion medium as long as the the required fiber length is not too long and the bandwidth of the signal is narrow. Otherwise the second-order dispersion coefficient becomes relevant and must be taken into account [25]. In these cases, LCFG are more adequate since they can be specifically designed for the system requirements in terms of bandwidth and group delay [58, 59]. Nowadays the fabrication techniques of Bragg gratings are well established [57] and the resulting device is compact (a few centimeters) compared with a single mode fiber (several kilometers of fiber). However, the Bragg gratings could also present some disadvantages respect to SMF, such as deviations in the average dispersion as well as high frequency ripples in the group delay response [60]. 1.3 Temporal Self-Imaging Effect The spacial Talbot effect entails the formation of self-images of a periodic object along the propagation direction [21, 22]. The self-images or Talbot images are the result of the coherent interference between all diffraction orders. For uniform illumination the Talbot planes are those satisfying the condition, zγ/α =γ α d2 λ0 ,(1.17) 1. Temporal Talbot effect 11 where dis the period of the object, λ0= 2π/k0the wavelength of the illuminating source and γand αare coprime and integer numbers. For α= 1, exact images of the periodic object are found shifted by half a period when γis odd. These planes are called integer Talbot planes. For α6= 1 periodic patterns with a richer structure are found. In these cases, each cell in the pattern is composed of the coherent superposition of αreplicas of the original one mutually shifted by d/α, with a phase difference between them and with power reduced by 1/α. These are called fractional Talbot images. In case of the structure of cells is sufficiently confined to avoid superposition between neighboring cells, the pattern results in a replica of the original one but with a period αtimes smaller, shifted again by half a period if αγ is odd. According to the space-time duality the temporal analogy of the self-imaging effect appears when a coherent periodic temporal signal is propagated through a dispersive line, under first-order conditions. Here we will centered our attention on the temporal Talbot effect applied to the increasing of the repetition rate. We will describe it in terms of intensity since it is in this domain where the multiplication of the repetition rate of the train takes place. We consider an input signal with a repetition period t0. Notice that this periodic signal is the temporal equivalent of the periodic complex amplitude distribution, in such a way that the repetition time t0corresponds to the spatial period d. Mathematically, the optical envelope of the input periodic signal can be expressed as a(t, 0) = f(t)⊗ +∞ X m=−∞ δ(t−mt0) = +∞ X m=−∞ amexp(i2πmt/t0),(1.18) where ⊗denotes the convolution operation. f(t) represents an arbitrary pulse shape confined within |t|< t0/2, so that the incoming optical intensity is I(t, 0) = |f(t)|2⊗Pkδ(t−kt0). The set of variables amrepresents the harmonics of the signal, amt0=F(ω=m2π/t0), F(ω) being the Fourier transform of the pulse profile f(t). As has been shown in previous section a linear first-order dispersion medium is a linear time-invariant system, so that it is completely characterized by its transfer function, H(ω, z) [3] H(ω, L) = exp (−iβ2zω2/2),(1.19) or equivalently by its impulse response, h(t, z) which is the inverse Fourier transform of the transfer function 12 1.3. Temporal Self-Imaging Effect h(t, z) = FT−1[H(ω, z)] = (i2πβ2z)−1/2exp(it2/2β2z).(1.20) trepresents the proper time, that is, the time expressed in the reference frame moving with the pulse, t=tphy −β1z,tphy being the physic time. After traveling a distance zin a dispersive medium described by Eq. (1.19)-(1.20), the resulting optical envelope, a(t, z) can be expressed as the convolution of the complex envelope of the input pulse with the impulse response, a(t, z) = a(t, 0)⊗h(t, z). According with the convolution theorem, the optical signal can be equivalently computed in the Fourier domain by directly multiplication of the spectrum of the incoming signal, A(ω, 0) with the transfer function, A(ω, z) = H(ω, z)A(ω, 0). Thus, the dispersed optical envelope results in: a(t, z) = +∞ X m=−∞ amexp(i2πmt/t0) exp(−i2π2m2β2z/t2 0) (1.21) We now proceed to the analysis of the optical intensity for dispersive lines satisfying Talbot condition. We firstly derive the resulting optical intensity under integer Talbot conditions, and subsequently in subsection 1.3.2 fractional Talbot lines are analyzed. 1.3.1 Integer Talbot effect The integer Talbot condition, in time domain, is directly derived by setting α= 1 in Eq. (1.17) and making use of the conversion table 1.1: |β2|zγ=γt2 0 2π,(1.22) where γis an integer number. Under this condition the optical envelope, Eq. (1.21) is expressed as a(t, zγ) = +∞ X m=−∞ amei2πmt/t0e−iπm2γ(1.23) The second phase element in Eq. (1.23), e−iπm2γcan be reexpressed as a linear phase in the index m,e−iπmγ. It means that integer Talbot lines with odd values of γintroduce a phase difference of πbetween consecutive harmonics which is translated into a shifting of half a period in the time domain. The output optical intensity, I(t, zγ/α) = |a(t, zγ/α)|2, results in 1. Temporal Talbot effect 19 on the right, the corresponding intensity profiles. The broadband spectral envelope |J(ω)|2/|ζ(ω)|is depicted with a thin continuous curve. The thick continuous curve represents the product |J(ω)Z(ω)|2/|ζ(ω)|that controls the power carried by each harmonic of the output intensity train. The power of the harmonics is the value of this thick line at the multiples of the fundamental frequency Ω0= 2π×10 GHz, which are depicted with dots. On the right hand side, the thin continuous curve represents two periods of the intensity of the input train, and the thick curve two periods of the output intensity. The intensity PSD has been normalized to 0 dB at the carrier, whereas the temporal signals have been normalized to unity in the center of the period to help in visualizing the pulse interference patterns. −150 −120 −90 −60 −30 0 β2L=0 ps2/rad β3L=0ps3/rad2 0 20 40 60 80 100 120 140 160 −150 −120 −90 −60 −30 0 β2L=−90 ps2/rad β3L=0.557ps3/rad2 Frequency (GHz) Intensity 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) Normalized intensity Figure 1.1: Intensity PSD(left) and temporal intensity (right)of a 10-GHz train before(top) and after (bottom) its propagation in a standard SMF of length L= 4.29 km (compression length) In Fig. 1.1 the trains are propagated through a SMF with two lengths: L= 0 (initial train, top) and L= 4.29 km (bottom). β3is negligible in both examples (ǫ= 0 and ǫ= 0.0034 rad, respectively). The intensity PSD of the input train (top) has two lobes that group the harmonics of the intensity. They are centered at DC and 140 GHz, and are associated to the 20 1.4. Intensity Spectrum in Dispersive Lines first two contributions to the sum (1.41) in Eq. (1.43). Since their spectral overlapping is negligible, the sum can be approximated to |g0(ω)|2+|g1(ω)|2. The first lobe |g0(ω)|2, together with the overall envelope, represents the intensity spectrum of a single pulse, see Eq. (1.45). The second lobe |g1(ω)|2centered at 140 GHz is due to the slight interference between neighboring Gaussian pulses in the input train. In practice no pulse- to-pulse interference is present in the input train, so that the intensity harmonics should be grouped in a single lobe centered at DC. Additional interference lobes only appear as dispersion causes pulse broadening and subsequent interference [68]. The existence of interference lobes is a limitation of the representation of the input train as a coherent sum of Gaussian pulses: their infinite temporal extension causes pulse-to-pulse overlapping and interference even in the input signal. However, the value of the power of the harmonics describing the neighboring-pulse interference term, more than 90 dB below the carrier, is sufficiently small to rely on the validity of the model. −150 −125 −100 −75 −50 −25 0β2L=−210 ps2/rad β3L=1.3 ps3/rad2 0 20 40 60 80 100 120 140 160 −150 −125 −100 −75 −50 −25 0β2L=−273 ps2/rad β3L=1.69 ps3/rad2 Frequency (GHz) Intensity 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) Normalized intensity Figure 1.2: Intensity PSD(left) and temporal intensity (right)of a 10-GHz train after its propagation in a standard SMF of length L= 10 km (top) and L= 13 km (bottom) 1. Temporal Talbot effect 21 The second example in Fig. 1.1 (bottom) shows the signal after the compression length LC=−C/σ2 ωβ2= 4.29 km, where pulses reach the minimum temporal width, 1/σω= 6.7 ps. At this length the equivalent dispersion, Deq defined in Eq. (1.37), vanishes. This means that |Z(ω)|2∼ = |g0(ω)|2= 1, see (1.42) and the definition of ωn, so that the spectrum follows the envelope |J(ω)|2/|ζ(ω)|. In other words, this envelope is the single-pulse envelope of compressed pulses. This fact can also be derived by setting Deq = 0 in Eq. (1.46). Therefore, the factor g0(ω) in the singlepulse spectrum (1.46) accounts for the difference between compressed and non-compressed pulse spectra. In figure 1.2 the fiber length is increased beyond the compression condition up to L= 10 km (top) and L= 13 km (bottom). The second order dispersion is still negligible (ǫ= 0.0080 rad and 0.0104 rad, respectively). In first example (top figure) the interference between pulses yields an increase in power of the second lobe |g1(ω)|2. In temporal domain pulses are broaden and the superposition between neighboring pulse is now apparent. In second example (bottom figure) the power spectrum is composed of tree lobes: |g0(ω)|2,|g1(ω)|2and |g2(ω)|2which correspond to single pulse dispersion, interference between neighboring pulses, and interference between alternate pulses, respectively. The lobe |g1(ω)|2has raised in power and its peak is now centered at 80 GHz leading to an increase in power of the harmonics around that frequency. It can be observed in its temporal counterpart as an interference patter with eight maxima per period. 1.5 Intensity Spectrum in Talbot Lines In section 1.3 we have shown that self-imaging effect can be achieved when a periodic train of pulses, such as (1.34), is propagated in a first-order dispersion medium, whose lowest-order accumulated dispersion is a multiple of the basic scale t2 0/2π. This effect can be also achieved by matching the second-order accumulated dispersion or even higher order dispersion [19,69,70]. Here we are considering a dispersive line with first and second order dispersion. The Talbot condition will be defined by matching the accumulated first-order dispersion, whereas the second-order dispersion will be considered as a deviation from the ideal behavior. Then, according with Eq. (1.26), the self-images are achieved when the fiber length Lsatisfies the condition: 22 1.5. Intensity Spectrum in Talbot Lines |β2|LΩ2 0= 2πγ α.(1.47) where γand αare integer and coprime numbers. The quotient γ/α is called Talbot index. Due to the structure of the quadratic phases in (1.35) and (1.36), another value of total dispersion is of interest: DeqΩ2 0=|β2L+C/σ2 ω|Ω2 0= 2πγ α.(1.48) We will refer to (1.48) as the compressing Talbot condition. To interpret this condition, let us first consider that the length of the dispersive medium is adjusted to verify (1.48), the output signal is a Talbot image (integer or fractional, depending on α) of a train of compressed pulses. Let us further suppose that phase modulation and dispersion have opposite sign, C·β2<0, so that the length of the medium can be split in two positive contributions, L=LC+LT, the first being the compression length and the second the Talbot distance. This first part of the line is used to compress individually the pulses in the train [5], and the remaining length of the fiber, LT, produces the Talbot image of the train of compressed pulses. If both phase modulation and dispersion have the same sign, C·β2>0, the compression condition (1.48) produces the same overall result because the final structure of the spectral phases is equal to the previous case, C·β2<0. The only difference is that the distance of compressing Talbot condition when C·β2>0 is shorter than the distance when C·β2<0. Between Talbot and compressing Talbot conditions (1.47) and (1.48), a continuum of partial compression is obtained. Now we analyze the intensity spectrum of the temporal Talbot effect by means of some representative examples. Let us first consider in Fig. 1.3 an integer self-imaging dispersive line with index γ/α = 1. Pulses have a width of 15 ps and a chirp C= 2 rad, as in Figs. 1.1 and 1.2. The dispersive medium is a SMF at λ= 1550 nm, and compressing conditions (L= 80.1 km) are shown in the figures at the top, while non-compressing conditions (L= 75.8 km) are depicted at the bottom. In both examples second-order dispersion is negligible (ǫ= 0.064 rad and 0.061 rad, respectively). The thick continuous curve represents again the function |J(ω)·Z(ω)|2/|ζ(ω)| that accounts for the power carried by the output harmonics. At the output of the fiber we get a train of compressed pulses in the first example and a train of non-compressed pulses in the second one, both of them shifted by half a period because γis odd (γ= 1). The intensity PSD follows two 1. Temporal Talbot effect 23 −150 −120 −90 −60 −30 0β2L=−1.68x103 ps2/rad β3L=10.4ps3/rad2 0 20 40 60 80 100 120 140 160 −150 −120 −90 −60 −30 0β2L=−1.59x103 ps2/rad β3L=9.85ps3/rad2 Frequency (GHz) Intensity 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) Normalized intensity Figure 1.3: Intensity PSD(left) and temporal intensity (right)of a 10-GHz train after its propagation in a standard SMF of length L= 80.1km (top) and L= 75.8km (bottom), which correspond to the compressing Talbot and standard Talbot conditions with index γ/α = 1, respectively. different curves. The first one (thin continuous curve) corresponds to the intensity spectrum of a compressed pulse, while the second one (dashed curve) follows the spectrum of non-compressed pulses. From Fig. 1.3 we can interpret how Talbot effect creates the self-images of the train. Multiple pulse-to-pulse interference is reflected in the structure of dips and bumps of the function |J(ω)·Z(ω)|2/|ζ(ω)|. This structure is due to the sum over gn(ω) in Z(ω), see Eq. (1.41). In the absence of second-order dispersion the spectral separation between consecutive functions gn(ω) and gn+1(ω) is t0/Deq, while their spectral width goes as 1/σωDeq. Then, for fiber lengths of the order of magnitude of the Talbot distances, Deq ∼1/Ω2 0, the spectral overlapping between functions gn(ω) can be neglected if σω≫Ω0, i.e., if the pulse width is smaller than the period of the input train. Then, |Z(ω)|2can be approximated by Pn|gn(ω)|2and the intensity PSD can be rewritten as: 24 1.5. Intensity Spectrum in Talbot Lines S(ω, L)∼ =Ω0|J(ω)|2|ζ(ω)|−1X s Gs(ω)X k δ(ω−kΩ0) (1.49) where Gn(ω) = |gn(ω)|2. Dispersion thus acts as a multiple bandpass filter, the positions and width of the passbands being determined by Gn(ω). Their centers are located at: ωn=nt0/Deq,(1.50) and their 30-dB decay half width is, if second-order dispersion is negligible: ∆ω= 3.72/σωDeq.(1.51) Then, from Fig. 1.3 (top), it can be observed that the output harmonics of the compressing Talbot replica of the original train lie at the center of the n-th passband that describes the interference between pulses separated ntime slots. In other words, the n-th harmonic of the intensity train is created by interference of pulses separated ntime slots. Under noncompressing Talbot condition the behavior of the intensity PSD is similar, but the output harmonics do not lie on the passband’s centers, see Fig. 1.3 (bottom). However, the overall picture is similar. This point of view can be further explored by considering the examples presented in Fig. 1.4. The same train is propagated through a SMF to the compressing Talbot lengths with indices 1/2 and 3/2. The output train is similar, since in both examples the output is a 20-GHz train of transformlimited pulses. Again, second-order dispersion is negligible (ǫ= 0.034 rad and 0.094 rad, respectively). In both examples, the increase of the repetition rate is manifested in the suppression of the odd harmonics of 10 GHz. However, the interference structures are different. In the first example, above, the h-th output harmonic is created by the interference of pulses separated by htime slots, while in the second example, below, it is created by the interference of pulses originally separated by 3htime slots. In general, the passbands associated to compressing Talbot lines are centered at ωn=nαΩ0/γ,nbeing the index determining the interference between pulses separated by ntime slots. When nis a multiple of γ,n=hγ for certain integer h, the passband is centered at the h-th output harmonic hαΩ0. Then, if the compressing Talbot line is integer (α= 1) the h-th harmonic lies in the center of the passband with index n=h, so that all the harmonics survive and between two consecutive harmonics there are γ−1 passbands. Conversely, if the compressing dispersive line is fractional (α6= 1), only the frequencies multiples of the α-th input harmonic lie over 1. Temporal Talbot effect 25 −150 −120 −90 −60 −30 0β2L=−886 ps2/rad β3L=5.48ps3/rad2 0 20 40 60 80 100 120 140 160 −150 −120 −90 −60 −30 0β2L=−2.48x103 ps2/rad β3L=15.3ps3/rad2 Frequency (GHz) Intensity 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) Normalized intensity Figure 1.4: Intensity PSD(left) and temporal intensity (right)of a 10-GHz train after its propagation in a standard SMF of length L= 42.2km (top) and L= 118 km (bottom), which correspond to the compressing Talbot conditions with index γ/α = 1 and 3/2, respectively. the center of a passband. These surviving harmonics originate the spectral content of the output train, the remaining harmonics being filtered. The next example in Fig. 1.5 explores the suppression of harmonics in Talbot devices with index 1/3 and its dependence on the optical linewidth. A 10-GHz train with pulses of width 15 ps is launched into a SMF compressing dispersive line with index 1/3. Above, the chirp parameter of the pulses is C= 2 rad, so that the width of the compressed pulses is 6.7 ps. The 1/3 compressing Talbot line has a length L= 29.5 km. Below, the chirp parameter of the pulses is C= 1 rad, the compressed pulses are 10.6 ps wide, and the corresponding length is 30.6 km. Second-order dispersion is still negligible (ǫ= 0.0237 rad and 0.0062 rad, respectively). We observe that the suppression of harmonics that do not belong to the output 30-GHz train is not exact. These harmonics will be referred to as residual harmonics. The partial suppression of residual harmonics is due to the finite width of the bandpass Gn(ω), which are, according to (1.51), inversely proportional to the spectral width. 26 1.5. Intensity Spectrum in Talbot Lines −150 −120 −90 −60 −30 0β2L=−621 ps2/rad β3L=3.84ps3/rad2 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/rad β3L=3.84ps3/rad2 Frequency (GHz) Intensity PSD (dB/Hz) −100 0 100 0 0.2 0.4 0.6 0.8 1 Time (ps) Normalized intensity Figure 1.5: Intensity PSD(left) and temporal intensity (right)of a 10-GHz train after its propagation in a standard SMF of length L= 29.5km (top) and L= 30.6km (bottom), which correspond to the compressing Talbot conditions with index γ/α = 1/3. Pulses have a width of 15 ps and chirp C=2 rad (top) and C=1 rad (bottom) and 3/2. In Fig. 1.5, below, we observe that a reduction of the spectral width implies, from the temporal point of view, that pulses overlap in the output train causing pulse pedestal and intensity fluctuation. As a result, the temporal profile shows a periodic structure with frequency Ω0, reflecting the absence of exact repetition-rate multiplication. From the spectral point of view we notice, on one hand, the reduction of the spectral width, and, on the other, the increase of the power carried by residual harmonics, due to the broadening of the passbands Gn(ω). The inefficient filtering of residual harmonics is clearly shown at 10 and 20 GHz. It is important to underline that the factional Talbot lines with indices γ/α = (2k+ 1)/2 differ form any other fractional Talbot device. In Fig. 1.4 the odd residual harmonics are totally suppressed, independently of the value of the optical spectral width, but this fact is not general [25]. This exact cancelation occurs if the input pulses are transform-limited, i. e., if the input envelope is a real function so that its Fourier coefficients in (1.35) 1. Temporal Talbot effect 27 −150 −120 −90 −60 −30 0 β2L=−143 ps2/rad β3L=20 ps3/rad2 0 40 80 120 160 200 240 280 −150 −120 −90 −60 −30 0 β2L=−275 ps2/rad β3L=38.5 ps3/rad2 Frequency (GHz) Intensity 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) Normalized intensity Figure 1.6: Intensity PSD(left) and temporal intensity (right)of a 20-GHz train after its propagation in a standard DSF of length L= 285 km (top) and L= 551 km (bottom), which correspond to the compressing Talbot conditions with index γ/α = 1/3and 2/3, respectively. verify that ϕm=−ϕ−m. This, together with the spectral phases caused by dispersion, produces a relative phase shift of ±πbetween optical spectral lines with indices ±m, which implies the total suppression of the residual harmonics. This observation is similar to the well-known carrier suppression effect in analog communications [71]. In fact, compressing Talbot distances with indices γ/α = (2k+ 1)/2, and only these distances, coincide with the distances of carrier suppression. To conclude this section we consider the last example in Fig. 1.6 where second-order dispersion is not negligible. A 20-GHz train of pulses with width 5-ps and chirp C= 2 rad is propagated through a dispersion shifted fiber (DSF; β2= 0.5 ps2/km ·rad and β3= 0.07 ps3/km ·rad2 at λ= 1550 nm), whose length is adjusted to verify the compressing Talbot condition γ/α = 1/3 (top) and 2/3 (bottom). The thin continuous and dashed curves represent the envelope of the intensity PSD that would correspond to the compressing Talbot and non-compressing Talbot condition, respectively, if the second-order dispersion coefficient was zero. 28 1.6. Rejection Properties of Talbot Filters The thick continuous and the thin dotted curves are the envelope |f(ω)· J(ω)|2/|ζ(ω)|. Finally, the dots represent the harmonic content of the output train. The top figure 1.6 (ǫ= 3.32 rad) shows a slight decrease of overall bandwidth, as compared with the case β3= 0, and a progressive broadening of the passbands’ width along the RF spectrum. These widths can be approximated by ∆nω∼ =3.72|ζ(ωn)|1/2 σωDeq .(1.52) The width enlargement is due to the factor |ζ(ω)|1/2, and weakens the suppression or filtering of residual harmonics. Therefore, the output signal contains high frequencies that are manifested in the time domain as the oscillatory structure in the pulse’s trailing edge. For future use, we compute here the maximum value of the 30-dB passband half width in the significant part of the spectrum, ω∼3.72σω: ∆ωmax ∼ =3.72(1 + ǫ2)1/4/σωDeq.(1.53) where Eq. (1.44) has been used. Notice that, despite the presence of significant β3dispersion, the train of Fig. 1.6 (top) has no intensity fluctuations, broadening or pedestal, the only impairment being the distortion induced by second-order dispersion. If the influence of β3is sufficiently large the passbands overlap, causing the fading of the multiple passband structure. The approximation assumed in Eq. (1.49) is no longer valid and it is required to resort to (1.43) for the correct description of the intensity PSD. This is shown at the bottom Fig. 1.6 where β3influence is now larger (ǫ= 6.41 rad). Notice that in this regime the output signal shows periodic variations in pulse intensity in addition to pulse distortion. Pulse intensity fluctuations are periodic, and they are due to the underlying non-filtered input harmonics. 1.6 Rejection Properties of Talbot Filters As has been illustrated in the previous section, the dispersive mechanism of Talbot effect is described in the spectral domain by the passbands Gn(ω). This filtering is exact only when the dispersion line is adjusted to halfinteger compressing Talbot indices γ/α = (2k+ 1)/2 and higher-order dispersion is negligible. In any other situation, Talbot filtering is not ideal. 1. Temporal Talbot effect 35 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 10−2 10−1 Normalized spectral width W Relative length deviation δL/L γ/α=1 γ/α=1/4 γ/α=1/2 Eq. (1.64) Eq. (1.62) Figure 1.8: Relative length deviation allowed by the bounds (1.62) and (1.64) with respect to the normalized spectral width Wfor different Talbot filters with negligible second-order dispersion. dispersive filter. Using the definition of W, and assuming negligible secondorder dispersion, optimal tolerance condition (1.65) can be rephrased in terms of the spectral width of the input train. When it is measured as the FWHM, optimal conditions require that the FWHM is twice the repetitionrate frequency of the output train, 2αΩ0. From the temporal point of view, and assuming transform-limited pulses, this condition states that 99% of the energy of a pulse is contained in half a period of the multiplied train, t0/2α. The optimal value of tolerance is then: |δL| L<1 K(4αγ + 1) ∼ =0.10 αγ(1 + ǫ2)1/4.(1.66) Notice that it is inversely proportional to the product γα and, therefore, to the complexity of the filter. When we consider the usual fractional Talbot filters of the series γ/α = 1/α , Eq. (1.66) states that the expected optimal tolerance in length is about 10% divided by the repetition-rate factor α. For instance, in the examples in Fig. 1.8, W= 4.4, and the tolerance in length given by (1.66) is 5%. Notice finally from Fig. 1.8 that optimal tolerance 36 1.8. Conclusions (1.66) is located at the edge of a plateau. Nearly optimal tolerances can be obtained in its neighborhood. 1.8 Conclusions In this chapter we have briefly reviewed the well-known space-time duality which is based on the mathematical equivalence between equations describing one-dimensional paraxial diffraction and pulse propagation in a linear first-order dispersive medium with negligible attenuation. Under this analogy, the pulse envelope is equivalent to the complex amplitude distribution of light in diffractive optics. In this context, we have presented the temporal counterpart of the self-imaging or Talbot effect. We have analyzed, in terms of intensity, an arbitrary train of pulses after propagation through a dispersive medium under Talbot condition. For integer Talbot devices the output intensity is a replica of the input train, whereas for fractional Talbot lines an intensity train with higher repetition rate is obtained provided that input pulses are sufficiently narrow to avoid overlapping. In both cases the output intensity is shifted by half a period depending on the parity of Talbot index. Talbot devices are also analyzed from the spectral point of view. This study is based on the computation of the exact intensity PSD of a coherent train of pulses after its propagation in a dispersive medium with arbitrary first and second order dispersion and negligible attenuation. We have considered trains composed with linearly chirped Gaussian pulses. These analytical results enable the identification of the spectral components that represent the dispersion of individual pulses, as well as the terms where the interference between pulses is enclosed. In this way we have made connection with the existing results in the literature [63, 65–67]. From the spectral point of view, Talbot dispersive lines can be interpreted as a multiple bandpass filter where each passband corresponds to the interference between pulses originally separated a number of times slots. The filter essentially rejects the intensity harmonics of the input train that do not belong to the output. The rejection is exact only in semi-integer Talbot filters, with negligible higher-order dispersion, and transformlimited pulses, since in this case the filter is adjusted to the dispersion of carrier suppression [71]. The filter is not static, but depends on the spectral width of the input train and the dispersive characteristics of the line. In particular, conditions have been derived that guarantee that the Talbot filter provides output intensity trains with negligible intensity fluctuations, 1. Temporal Talbot effect 37 even in the presence of higher-order dispersion. In these cases, however, the pulses may present distortion. The derived condition, Eq. (1.59), essentially states that the pulses of the output train do not overlap. Finally, the stability under timing and length variations of the Talbot line has been analyzed. We have shown the properties of the two regimes of operation of Talbot filters, described by two bounds, Eqs. (1.62) and (1.64). When the spectral width of the train is large, dispersion instabilities tend to broaden the pulses and the filter’s performance worsens by pulse-to-pulse interference. In contrast, when the spectral width is moderate, dispersion variations decrease the rejection ability of the filter. Optimal stability is reached when the spectral width of the input train, measured as FWHM, equals twice the repetition-rate frequency of the output train. The expected length tolerances are about 10% divided by the repetition-rate factor . Chapter 2 Non-ideal coherent Temporal Talbot effect 2.1 Introduction As has been shown in the previous chapter, the Talbot effect is caused by the interference between all diffracted orders of a grating in space domain and by the multiple interference among all pulses of a train in its temporal counterpart. It means that all diffraction orders or all dispersed pulses get involved in the reconstruction of each unit cell or pulse of the resulting Talbot image. In this sense, it is a collective phenomenon, since the pattern is a consequence of the whole periodic structure of the grating or of the train of pulses. It is expected from a collective phenomenon like this some kind of stability under small variation of the parameters defining the cells of the periodic structure. In diffractive optics, several works have reported on Talbot effect as a method to smooth imperfections of a grating, such as the absence of a unit cell, or localized modifications of the basic period [21,73]. The degradation of the effect due to the finite orders from the optical axis is also known [22]. This walk-off effect progressively reduces the paraxial region where Talbot effect takes place. In the temporal domain the study of the degradation by the finite width of pulse trains has reported in ref [74] The aim of this chapter is to describe the tolerance of Talbot devices against train imperfections. Unlike the works previously cited, our approach to the problem is not deterministic but stochastic. Although the presented analysis is valid in both diffractive and dispersive domain, we will describe it from the temporal point of view. Among all possible deviation of the ideal periodic signal, we will treat first the randomly on- 39 40 2.1. Introduction off keyed pulses in the train and later the jittery pulses in the train. In both cases Talbot devices are shown to have filtering properties around frequencies of the order of the repetition rate of the train. The use of this property is then limited to broadband noise reduction. This chapter is split into two large sections. Section 2.2 is devoted to the analysis of a randomly distributed pulses after a Talbot device. This study is carried out in the Fourier domain by means of the power spectrum density, in the same way as the analysis of exact Talbot effect in the previous chapter. By assuming linearly chirped Gaussian pulses together with the independence of the variables which determine the presence or absence of a pulse in the train, an exact stochastic result of the RF-spectrum is achieved. The noise spectral after Talbot device consists of a set of passbands, each one being cosine-squared modulated. The centers of the passbands are shifted depending on the value of the chirp. This fact, allows an indirect characterization of the pulse in the train through the analysis of the spectrum. In particular, measurements of the positions and the width of the bandpass results in the determination of the chirp of the initial pulses. Moreover, the resulting noise spectral varies continuously with respect to the parameters that define the model. It means that, in order to observe the spectral features of noise, no fine tuning of the initial repetition rate of the train is necessary to match the Talbot conditions, in contrast with the temporal domain where the trace of the Talbot effect is the increase of the repetition rate of the train. However, for clarity, all results are derived under exact Talbot conditions. Section 2.3 deals with the behavior of Talbot devices when they are fed with a train suffering from timing jitter. Despite the fact we have centered our study in timing jitter, the formalism we present can be easily expanded to amplitude jitter of a combination of them [78]. Firstly, we approach the problem in time domain by means of the variance, in subsection 2.3.1. Analytical expressions for the input and output variance are reported, showing that the variance of the original train is flatted after passing a Talbot device. The presence of pedestal in the output train is also studied. In subsection 2.3.2 we analyzed the intensity spectrum of the jittery trains before and after the propagation of the train through the Talbot filter. This spectral analysis is more general than the previous one in terms of variance, since correlations between variables describing timing jitter are now allowed. The results presented here are not exact but rely on a small-signal approximation of jitter source. This technique has been widely applied in the analysis of the RF intensity spectrum of timing jitter noise 2. Non-ideal coherent Temporal Talbot effect 41 in mode-locked lasers [75–78]. The results are exemplified for two models of jitter pulse correlation, one describing the noise of fundamentally modelocked lasers and the other one describing harmonically mode-locked lasers. In subsection 2.3.3 we numerically investigate the conditions for a decrease in the noise power around the harmonics. Finally, we end the chapter presenting our conclusions. 2.2 Talbot imaging of On-Off-Keyed trains In this section we analyze the behavior of Talbot filters when they are fed with a coherent train composed by randomly distributed pulses. The optical envelope which will be launched in the Talbot line can be expressed as, a(t, 0) = X k bkf(t−kt0) (2.1) where f(t) denotes the pulse form, t0is the unit interval of the train, and the symbols bkare a set of random variables which will be assumed uncorrelated. These variables describe the presence (bk= 1) or absence (bk= 0) of pulses in the train with probability pand 1 −prespectively. Under ideal conditions all pulses are present (p= 1) so that the train is t0-periodic. The average over all realizations will be denoted by brackets h···i, so hbki=pand its standard deviation hb2 ki=σ2 b=p(1 −p). Unless otherwise stated all sums and integrals runs from −∞ to +∞. For simplicity the pulse form is assumed to be linearly chirped Gaussian: f(t) = exp[−t2(1 −iC)/2t2 p] (2.2) The parameter tpis the rms width of every pulse and Cis the linear chirp or phase modulation. In order to avoid overlapping between incoming pulses and also to allow repetition rate multiplication at the output of the Talbot device, we assume that the pulse width is smaller than the unit interval tp< t0 If the randomly train of pulses, Eq. (2.1), is propagated along a fistorder guided dispersive medium with negligible attenuation, Eq. (1.16), the optical envelope results in a coherent superposition of dispersed pulses. a(t, ξ) = "t2 p ρ(1 −iC)#1/2X k bkexp −(t−kt0)2(1 −iC) 2ρ.(2.3) 42 2.2. Talbot imaging of On-Off-Keyed trains where ρ= 1/σ2 ω+iξ +iC/σ2 ω,(2.4) σω= (1 + C2)1/2/tpis the linewidth of the incoming pulses and the parameter ξaccounts for the accumulated dispersion in the medium. This is the relevant parameter of the propagation medium. If the guided medium is chosen a fiber, ξ=β2L,Lbeing the fiber length; whereas for a linearly chirped fiber grating ξ=¨ Φ, ¨ Φ being the slope of the LCFG response in reflection mode around the carrier frequency ω0. This slope is designed to be constant within the bandwidth of the input signal. In this way, the study will be presented here, is manifestly independent on the high dispersive medium used as Talbot device. The dispersed train, Eq. (2.3), can equivalently be expressed in the Fourier domain, as follows A(ω, ξ) = tp2π 1−iC 1/2 exp(−ρω2/2) X k bkexp(−iωkt0),(2.5) −1000 −800 −600 −400 −200 0 200 400 600 800 1000 0 2 4 6 8 Amplitude (a.u.) −1000 −800 −600 −400 −200 0 200 400 600 800 1000 0 1 2 3 4 5 Amplitude (a.u.) Time (ps) Figure 2.1: Simulation of the form of the pulse envelope of 20-unit intervals of a 10-GHz, p=0.5 random train of unchirped, 12-ps rms Gaussian pulses before (above) and after (below) its pass through a Talbot device with γ/α = 1. 2. Non-ideal coherent Temporal Talbot effect 43 In Fig. 2.1, we present a simulation of the dispersion of a portion of a 10-GHz train (t0= 100 ps) randomly on-off keyed, after its pass through the first integer Talbot device, γ/α = 1. The total dispersion of the device is ξ= +1592 ps2/rad. The probability of appearance of pulses in the train is p= 0.5. Pulses are unchirped and have a rms width of 12 ps. The ideal Talbot would reproduce the incoming train shifted by half a unit interval. The temporal spreading of pulses due to dispersion creates Talbot images formed by interference of pulses. This phenomenon is local, depending on the number of neighboring pulses around a given time slot. This leads to an unsatisfactory temporal signal, since the temporal characteristics of the output train depend on the local behavior of the input. Nevertheless, wherever two pulses interfere in Fig. 2.1, they tend to form the desired ideal output train. Then, a spectral analysis of the signal can unveil the ability of the Talbot device to reconstruct output harmonics in a non-ideal situation. 2.2.1 Intensity Spectrum As has been shown in previous chapter, dispersive lines leave the power spectrum of the optical envelope, |A(ω, ξ)|2, invariant. However, since detection is a non linear transformation of the stochastic variables, a(t, ξ) or A(ω, ξ), the pulse spreading caused by the dispersion-induced phases, can be observed in the detected train. The power spectrum of the detected train i(t, ξ) = |a(t, ξ)|2, is computed by Fourier transforming the mean correlator, which is the temporal average of the correlator [79]: S(ω, ξ) = Z+∞ −∞ dτ exp(−iωτ)lim T→∞ 1 2TZT −T dt hi(t+τ, ξ)i(t, ξ)i,(2.6) where the photocurrent correlator is hi(t, ξ)i(t′, ξ)i=Z+∞ −∞ Z+∞ −∞ dω1 2π dω2 2πexp[i(ω1t−ω2t′)] ×hI(ω1, ξ)I(ω2, ξ)∗i. (2.7) 44 2.2. Talbot imaging of On-Off-Keyed trains The Fourier form of the photocurrent I(ω, ξ), is related to the Fourier form of the optical envelope A(ω, ξ) by an autocorrelation, I(ω, ξ) = Z+∞ −∞ dt exp(−iωt)i(t, ξ) = Z+∞ −∞ dω′ 2πA(ω+ω′, ξ)A(ω′, ξ)∗. (2.8) Then, using Eq. (2.5), the Fourier form of the photocurrent can be expressed as I(ω, ξ) = J(ω)X k Bk(ω) exp(−iωkt0),(2.9) where J(ω) = tp√πexp(−ω2/4σ2 ω),(2.10) Bk(ω) = bkX s bk+sgs(ω) exp(−iωst0/2) (2.11) and gs(ω) = exp −σ2 ω 4ξω +Cω/σ2 ω−st02.(2.12) To compute the power spectral density, Eq. (2.6), let us suppose that the expected value at different frequencies of the correlator of functions Bk(ω) is of the form hBk(ω1)Bm(ω2)∗i=hBk(ω1)ihBm(ω2)∗i +δkmΩ(ω1, ω2) + ∆k−m(ω1, ω2),(2.13) where δkm is the Kronecker delta, the asterisk denotes complex conjugation, and Ω(ω1, ω2) and ∆k−m(ω1, ω2) are adimensional functions to be evaluated a posteriori. The first term in Eq. (2.13), hBk(ω1)ihBm(ω2)∗igives rise to the intensity spectrum of the signal which is composed of a set of harmonics multiples of the fundamental frequency Ω0. The behavior of the signal power spectrum has already been analyzed in detail in the previous chapter, so here we will focus in the noise spectral density. The second and third terms in Eq. (2.13) originate the noise contribution to the microwave spectrum, resulting in 2. Non-ideal coherent Temporal Talbot effect 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 Power spectrum (dB/Hz) Frequency (GHz) Figure 2.3: The same as in Fig. 2.2, but with chirped pulses with C= 1 rad. ∆Wω=4.3 σωξ+C/σ2 ω−1∼ =4.3 Ω0 1 t0σω α γ,(2.24) where, in the last approximation, we assume again that |ξ| ≫ σ2 ω. Thus chirp reduces the width of the passbands irrespectively of its sign, through the factor σω. Finally, the peak noise level in the top of the passbands of a train of chirped pulses can be raised or lowered depending on the value of p, as in the unchirped case. But this peak noise level also suffers the bandwidth enlargement of the wideband output noise envelope, |J(ω)|2, which always raises the peak noise level over the initial broadband noise. In Fig. 2.3, we present a simulation of a train composed of chirped pulses. The chirp parameter is chosen as C= 1 rad. The remaining parameters are as in Fig. 2.2. The output noise envelope is broader than the input level owing to the existence of chirp. The passbands’ width is ∆Wω= 2π×1.32 GHz, and the relative redshift of the centers is 8.85×10−3. For instance, the fourth output harmonic at 80 GHz is redshifted 708 MHz. We see a modulation of high-order harmonics, as in Fig. 2.2, and the lowering of the noise peak level of passbands, corresponding to nonoutput 52 2.2. Talbot imaging of On-Off-Keyed trains harmonics in the left part of the plot, owing to nonmaximal modulation. From relations (2.23) and (2.24), it is straightforward to realize that the magnitude and chirp of the individual pulses can be extracted from the values of the passband width and the passband shift. To be precise, C=−2.9 sign(ξ)Ω0 s ∆(s) peakω (∆Wω)2.(2.25) This measurement can be performed for any passband, not necessarily one of those surrounding the harmonics (smultiple of γ). Notice also that it is independent of the concrete Talbot device used, since it does not depend on the value of the Talbot index. This expression is also valid when modulation of the passband is present, although in those cases a fitting of the whole passband can be necessary to obtain precise values of width and shift. 2.2.3 Analysis of the Modulation The passband functions Gi(ω) are modulated by a cosine-squared term both in the chirped and in the unchirped cases. Given a passband index s, the maximum value of the modulation factor Ms(ω) = 4p2cos2(ωst0/2) + σ2 b occurs for ω=mΩ0/s, (2.26) with many integer. It is easy to show that in trains composed of unchirped pulses, the modulation and passband maxima coincide for the output harmonics, and only in these cases. Therefore nonoutput harmonic passbands (s6=hγ) can show tips on top of the passbands, or lowering of the peak noise level if they are sufficiently narrow. When the pulses are chirped, the modulation maxima still remain over the harmonics, but the passbands are shifted. The spectral modulation full width in the passband of index sis ∆Mω= Ω0/s. (2.27) In particular, the output harmonics correspond to indices s=hγ so that ∆Mω= Ω0/hγ. This width becomes smaller as the harmonic or passband index becomes larger, but it does not depend on the train characteristics, such as the temporal width, tp, the value of the chirp, C, or the value of p. Then, modulation of the spassband appears when ∆Mω < ∆Wω, or equivalently when 2. Non-ideal coherent Temporal Talbot effect 53 4.3tp>γ α t0 s.(2.28) Then, modulation is present when the passband index sis high, the dispersion γ/α is small, or the relative width of the pulses tp/t0is high. The limit described in relation (2.28) is exemplified in Fig. 2.4, where we simulate a random 10-GHz train composed of 400 pulses with tp= 10 ps, γ/α = 1, ξ= 1592 ps2/rad, and p= 0.9, so that the output train also has 10 GHz. The 10-dB passband full width is 4.3 GHz, while the modulation width in output harmonics is 10/h GHz, so that modulation is expected to be observable after the third output harmonic. The conventions in this plot are as in the previous figures. The peak noise level is raised 5.2 dB owing to the simulation of a nearly full train with 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 Power spectrum (dB/Hz) Frequency (GHz) Figure 2.4: Power spectrum of a detected 10-GHz, p=0.9 random train of unchirped pulses with 10-ps rms width after its pass through a Talbot filter width γ/α = 1. The upper part of the plot shows (continuous curve) the power spectrum of the detected output train, (dashed curve) the power spectrum of the detected input train, and (dotted curve) the mean signal power spectrum of both input and output trains. The lower part shows analytical approximations to the noise power spectrum: (continuous curve) output noise; (dashed curve) input noise. 54 2.2. Talbot imaging of On-Off-Keyed trains 0 10 20 30 40 50 60 70 80 90 100 −240 −220 −200 −180 −160 −140 −120 −100 −80 −60 −40 −20 0 Power spectrum (dB/Hz) Frequency (GHz) Figure 2.5: Same as in Fig. 2.4, but with chirped pulses with C=−1 modulation of a passband appears, the decay around the peak value has to consider both ∆Wωand ∆Mω. However, since the modulation occurs for high harmonics, small dispersion, and broad pulses, the decay due to the structure of the passband is usually negligible, since ∆Wωis larger than ∆Mω. Using relation (2.22), it is straightforward to compute the decay Γ between noise level in the harmonic and the first modulation zero due only to modulation, Γ(dB) = 10 log(1 + 3p)/(1 −p). In Fig. 2.4, the decay Γ is 16 dB. In the worst case (p= 0.5), Γ attains a minimum value of 7 dB. This is also the typical value of decay between consecutive maxima and minima of modulation in the same passband. When the approximation fails because ∆Wωand ∆Mωare of the same order, or when the center of the passband is shifted owing to the chirp of the basic pulses, the value of the decay Γ has to be reevaluated from relation (2.21). To complete our examples, in Fig. 2.5 we plot our final simulation where chirp is taken into account. We choose the same parameters of simulation as in Fig. 2.4, except the value of the chirp, whose value is C=−1 rad. The chirped train blueshifts the center of the passbands, while the modulation does not change the position of its maxima. This results in asymmetric 2. Non-ideal coherent Temporal Talbot effect 55 patterns of modulation. The other features of the spectrum are as in Fig. 2.4. 2.3 Talbot imaging of mistimed pulses This section deals with the study of a Talbot device, when it is fed with a jittery train of pulses. The analysis will be carried out in the temporal domain by means of the variance of the resulting train as well as in the Fourier domain by means of the power spectrum of the detected train. Let us consider a train of pulses suffering from timing jitter, which can be represented by a(t, 0) = X k f0(t−kt0−ak).(2.29) The function f0(t) denotes the pulse form and t0is the unit time of the train. The center of pulses in the train occurs at tk=kt0+ak(k=−∞, . . . , +∞). The set of variables akis an infinite collection of random numbers with dimensions of time representing the timing jitter of each pulse. If the random numbers are set to zero, the train of pulses has a period of t0. Notice that we have considered that jitter shifts pulses from its ideal position but does not affect to their shape. The random variables are assumed to have a Gaussian probability density function with zero mean and standard deviation σj: p(ak) = 2πσ2 j−1/2exp −a2 k/2σ2 j.(2.30) The average over all the realizations of each one of jitter parameters will be denoted by brackets, h···i. Therefore, haki= 0 and ha2 ki=σ2 jfor any pulse index k. As in the previous section, we will consider linearly chirped Gaussian pulses with rms width equal to tp: f0(t) = exp −t2(1 −iC)/2t2 p,(2.31) Cbeing the phase-modulation parameter or linear chirp. In order to avoid overlapping between incoming pulses we will assume that the pulse widths are smaller than the exact period, tp< t0, and the standard deviation of pulse center is smaller than pulse width σj< tp. If this jittery train, Eq. (2.29) is propagated through a dispersive line, the resulting optical envelope can be expressed as a coherent interference 56 2.3. Talbot imaging of mistimed pulses of dispersed pulses: a(t, ξ) = X k fξ(t−kt0−ak) =X kZ+∞ −∞ h(t−t′, ξ)f0(t′−kt0−ak)dt′, (2.32) where ξis the accumulated dispersion of the system, and h(t, ξ) the impulse response or propagation kernel of a lowest-order dispersion medium with negligible attenuation. Using Eqs (1.20) and (2.31) the dispersed signal (2.32) results a(t, ξ) = "t2 p ρ(1 −iC)#1/2X k exp −(t−kt0−ak)2 2ρ,(2.33) with ρ=t2 p (1 −iC)+iξ. (2.34) We also compute for future use the dispersed signal in the Fourier domain, which can be written as A(ω, ξ) = tp2π 1−iC 1/2 exp(−ρω2/2) X k exp(iωkt0+iωak).(2.35) Averaged fields can be computed in the time domain as follows. The mean value of the optical envelope is a sum of averaged individual pulses, or the result of the propagation through the linear system of the initial averaged field: ha(t, ξ)i=X khfξ(t−kt0−ak)i=Z+∞ −∞ h(t−t′, ξ)ha(t′,0)idt′(2.36) A simple computation of the dispersed field yields to: ha(t, ξ)i=X kZ+∞ −∞ fξ(t−kt0−ak)p(ak)dak ="t2 p ˜ρ(1 −iC)#1/2X k exp −(t−kt0−ak)2 2˜ρ, (2.37) 2. Non-ideal coherent Temporal Talbot effect 57 where we have introduced the notation ˜ρ=ρ+σ2 j. On the other hand, the initial averaged field is recovered by setting ξ= 0 in the value of ˜ρin Eq. (2.37): ha(t, 0)i="t2 p t2 p+σ2 j(1 + iC)#1/2 ×X k exp "−(t−kt0)21 + iC 2(t2 p+σ2 j(1 + iC))#. (2.38) Then, the pulse width of the average input pulses can be expressed as: thf0i=t2 p1 + ǫ21 + ǫ2(1 + C2)−C2 1 + ǫ2(1 + C2),(2.39) where the parameter ǫis the ratio between the standard deviation of the timing jitter and the original pulse width, ǫ=σj/tp. Thus, the averaged input field is affected by random jitter only as change in the pulse width. For unchirped trains, the incoming pulses are broaden by the factor (t2 p+σ2 j)1/2. However, when input pulses are phase modulated, they may broaden or get narrow depending on chirp value. From Eq. (2.39) the transition occurs when C=Climit ≡1 + ǫ2 1−ǫ21/2 .(2.40) Then, if C < Climit pulses are broaden by timing jitter while C > Climit yields to narrower pulses than the original ones. The computation and interpretation of the output field is similar to a deterministic signal, and the details of the computation are widely known [20–22] and are equivalent to those shown in chapter 1. In this section we will simply quote the results. 2.3.1 Variance We are interested in the statistical variations of a concrete realization of a train of pulses affected by timing random jitter, rather than in mean values. To characterize them, it is necessary to calculate the corresponding variance. This quantity is defined as the averaged squared deviation of the field with respect to the mean for each temporal instant: 58 2.3. Talbot imaging of mistimed pulses V(t, ξ) =h |a(t, ξ)−ha(t, ξ)i|2i =ha(t, ξ)a∗(t, ξ)i−ha(t, ξ)iha∗(t, ξ)i,(2.41) and it has the usual interpretation as a measure of the variation of the output field at each sampling instant. Expressed as a coherent sum of dispersed pulses, Eq. (2.36), the variance is V(t, ξ) = X kX mhfξ(t−kt0−ak)fξ(t−mt0−am)∗i −X khfξ(t−kt0−ak)i 2 . (2.42) Here we will assume that random variables akare mutually independent, which implies that hg(ak)h(ap)i=hg(ak)ihh(ap)iwhen k6=pfor any functions gand h. Taking this into account, the variance can be easily cast as a sum over dispersed pulses: V(t, ξ) = X kh|fξ(t−kt0−ak)|2i−|hfξ(t−kt0−ak)i|2,(2.43) where the first part of the sum corresponds to the averaged square of each dispersed pulse and the second to the square of the average. It is a simple task to compute both contributions. The result is the difference of two trains of Gaussian functions: V(t, ξ) = tp √2η1(ξ)X k exp −(t−kt0)2 2η1(ξ)2 −tp √2η2(ξ) tp tX k exp −(t−kt0)2 2η2(ξ)2, (2.44) where we have introduced the following notations: η1(ξ)2=t2 pbt2+ξ2(1 + C2) 2t2 p ,(2.45) η2(ξ)2=t4+ξ(1 + C2) + Ct2 p2 2(1 + C2)t2,(2.46) 2. Non-ideal coherent Temporal Talbot effect 59 t2=t2 p+σ2 j(1+C2) and bt2=t2 p+2σ2 j. Then the width η1(ξ) represents the rms widths of the averaged squared dispersed pulses, and η2(ξ) denotes the rms width of the square of the average dispersed pulse. Therefore, variance is composed of two contributions coming from individual dispersed pulses, which will be called of h|f|2iand |hfi|2type, respectively. Since V(t, ξ) is a periodic series, it can also be expanded in a Fourier series of the form V(t, ξ) = +∞ X k=−∞ gkexp(2πikt/t0) = g0+ 2g±1cos(2πt/t0) + ... (2.47) with gk=g−kand gk=√πtp t0 exp(−2π2k2η1(ξ)2/t2 0) −√πt2 p t0texp(−2π2k2η2(ξ)2/t2 0). (2.48) Both forms of the variance, Eqs. (2.44) and (2.47), will be used in what follows. From Eq. (2.47) it is worth noting that 1 t0Zt0/2 −t0/2 V(t, ξ)dt =g0=√πtp t01−tp t,(2.49) that is, the mean value is constant and there is no dependence on the specific value of the dispersion parameter ξ. The variance of the input field V(t, 0) can be recovered by setting ξ= 0 in Eq. (2.44) through the corresponding values of the widths, Eqs. (2.45) and (2.46). Since the jitter parameter is small compared with the width of the original pulses, σj< tp, both widths become of the order of the original pulses, η1(0), η2(0) ≈tp. Moreover, since the pulses do not overlap, only two Gaussian functions contribute appreciably to V(t, 0) in the sum Eq. (2.44) for each unit interval. For instance, for the central period, −t0/2< t < t0/2, the following approximations applies: V(t, 0) ∼ =h|f(t−a0)|2i−|hf(t−a0)i|2(2.50) =tp √2η1(0) X k exp −t2 2η1(0)2−tp √2η2(0) tp tX k exp −t2 2η2(0)2 60 2.3. Talbot imaging of mistimed pulses On the other hand, the variance of the output envelope cannot be approximated in this way: when ξis different from zero, the dispersed widths [Eqs. (2.45) and (2.46)] are larger than t0, and then the Gaussian functions in Eq. (2.44) overlap between adjacent periods. Indeed, for a general Talbot condition, ξ≈t2 0, so that the order of the period of the pulses, η1(ξ), η2(ξ)≈t0. In order to obtain a simpler expression for the variance V(t, ξ) after the dispersive device, it is necessary to return to the Fourier expansion (2.47), whose coefficients are given in Eq. (2.48). Now, since η1(ξ), η2(ξ)≈t0, all harmonics in this expansion are damped. The leading-order approximation is just the DC term: V(t, ξ)∼ =g0.(2.51) This, together with Eq. (2.49), means that the output variance equals the mean value along the unit interval, and it is insensitive to 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 Normalized time Input contributions −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08 Normalized time Output contributions Figure 2.6: Contributions of type h|f|2i(solid curve) and |hfi|2(dashed curve) to the input (top graph) and output (bottom graph) variances due to a signle pulse In fig. 2.6 we present the central pulse contributions h|f|2iand |hfi|2 to input and output variances in an example. The period of the train has been chosen as t0= 1 in suitable units; then, the central unit interval is −0.5< t < 0.5. Moreover, the γ/α = 1/4, tp= 0.050, C= 0 and 2. Non-ideal coherent Temporal Talbot effect 67 With this convention we can compute the damping coefficients 2π2η2 1(ξ)/t2 0 and 2π2η2 2(ξ)/t2 0of the first harmonic in the expansion (2.47): 2g±1=2√πtp t0 exp(−2π2η1(ξ)2/t2 0) −2√πt2 p t0texp(−2π2η2(ξ)2/t2 0). (2.53) Here we will focus only on the second one, which in this example is smaller; see Fig. 2.6. Nevertheless, both have in general the same order of magnitude. This damping coefficient can be bounded for γ/α = 1/N as follows: 2π2η2(ξ)2 t2 0 =π2t2 (1 + C2)t2 0 +t2 0(1 + C2) 4Nt 2+t4 pC2π2 t2 0t2(1 + C2)+πC t2 p t2 >t2 0 t2 p 1 + C2 8N2+Cπ 2= (1 + C2) ln 10 + Cπ 2≥ln 10 (2.54) where we have used the Talbot condition, Eq. (1.26), and notations (2.45) and (2.46), and, in the last inequality, that σj< tp. Then, the value of 2π2η2(ξ)2/t2 0decreases as 1/N2, but since the ratio tp/t0is also reduced by a factor Nto avoid overlapping [see Eq. (2.52)], this damping coefficient is constant for any N. Moreover this coefficient is higher for phase modulated signals. In the most unfavorable case, C= 0, the value in Eq. (2.54) corresponds to a conservative damping factor of exp(−2π2η2(ξ)2/t2 0)<0.10. This results in a maximum modulation of the constant variance of 20%. The actual value of modulation is typically lower, especially if jitter is negligible. For the parameters presented here, with σjand tpof the same order, the modulation is of 6%. Numerical results in Fig. 2.12 illustrate these values. Thus, we conclude that when the sequence of pulses is sufficiently narrow to avoid overlapping in the fractional Talbot series, the contribution of higherorder harmonics to expression (2.51) is to a good approximation negligible. If overlapping is allowed such that the ratio tp/t0is kept constant, the constant approximation fails as γ/α decreases. A new feature appearing in this last simulation is the increase of pulse pedestal in fractional Talbot devices. This fact is obscured in Fig. 2.11 because of the overlapping between adjacent pulses. In Fig. 2.13 we present a new simulation for the fractional device corresponding to γ/α = 1/2, with tp= 0.040, C= 0 and σj= 0.007. The presence of pedestal here is clear. To 68 2.3. Talbot imaging of mistimed 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 Normalized time Amplitude Figure 2.13: Superposition of ten sequences of output pulses in a unit interval in the fractional Talbot device characterized by the index γ/α = 1/2, leading to a 2×repetition rate. The pulses are unchirped and their rms width is tp= 0.040. The standard deviation of random timing jitter is σj= 0.007. analyze this feature, let us consider a general Talbot device characterized by an index γ/α. The repetition-rate series correspond to the fractional Talbot series γ/α = 1/N. From Eq. (2.41) the power of the mean detected train ha(t, ξ)a(t, ξ)∗iis composed of the constant variance V(t, ξ) = g0plus the squared mean train ha(t, ξ)iha(t, ξ)∗i, which vanishes between pulses. In turn, the peak value of the output pulses in a general Talbot device is damped [2,15] in amplitude by a factor of 1/√αwith respect to the original train, Eq. (2.38). Then, for Gaussian pulses the ratio between the peak amplitude of the detected train (Apeak) to the value between pulses (σ2 ξ=g0) is Apeak σξ =1 σξ"t2 p αt√2η2(0) +σ2 ξ#1/2 ∼ =tp 21/4ptαη2(0) ∼ =√2 π1/4 1 √αtp t01/2t0 σj(1 + C2)1/2, (2.55) 2. Non-ideal coherent Temporal Talbot effect 69 where in the last approximation we have retained the leading order in the ratio σj/tpfrom Eq. (2.49). Therefore, for a given amount of jitter the pedestal increases as √α=√N. The chirp of the pulse increases the pedestal equivalently to an increase of the timing jitter by the factor (1 + C2)1/2. Moreover, the ratio decreases when the input pulses are wider. For the simulation in Fig. 2.13, this ratio is 21.2, with Apeak = 0.70 and σξ= 0.033. 2.3.2 Intensity spectrum In this section we approach the problem of a mistiming train from a spectral point of view, by analyzing the power spectral density of the detected jittery train before and after a Talbot device. Here, and in contrast to the previous time-domain study, in which independent pulse-to-pulse jitter variations were assumed, now arbitrary pulse-to-pulse timing jitter correlation is allowed. We will consider that correlation between pulses are stationary, and described by the function R, defined as R(k) = R(−k) = hamam±ki. As in the previous section the jitter variables are assumed to have zero mean and standard deviation σj. Furthermore we will also assume that σjσω≪1. The approximate formulas for the power spectrum that we derive are based on a second-order expansion of σjσωor equivalently, on a small-signal approximation in the variables ak. For deriving the expression of the power spectral density of the detected train, Eq. (2.6), we follow the same steps than in section 2.2.1, where the intensity spectrum of a random distributed pulse train was calculated. Let us assume that the correlator of the Fourier transformed detected train can be factorized as: hI(ω1, ξ)I(ω2, ξ)∗i=hI(ω1, ξ)ihI(ω2, ξ)∗i + 2πK(ω1, ω2)X s δ(ω1−ω2−2πs/t0).(2.56) where the function Kwill be derived below. From Eq.(2.56), the power spectral density can be split into two contributions: S(ω, ξ) = S(S)(ω, ξ) + S(N)(ω, ξ). S(S)is the signal contribution, previously analyzed in chapter 1 and S(N)is the noise one. Using Eqs. (2.6) and (2.7), we can straightforwardly show that the noise spectral density is the diagonal part of regular function K: S(N)(ω, ξ) = K(ω, ω). Thus the problem of computing S(N)(ω, ξ) reduces to computation of the Fourier-domain correlator and the mean Fourier intensity. Using 70 2.3. Talbot imaging of mistimed pulses Eqs. (2.35) and (2.8) yields the photocurrent by Gaussian integration. Introducing the notation µm(ω) = (ωρ −imt0)σ2 ω/2 results in I(ω, ξ) =tp√πexp(−ρω2/2) X pk exp[−(ap−ak)2σ2 ω/4] ×exp(−iωkt0+µ2 p−k/σ2 ω−iµ∗ p−kak−iµp−kap). (2.57) Now we substitute Eq. (2.57) into Eq. (2.56) and expand the exponentials in Eq. (2.57) to second order in jitter variables ak, performing the expectation values according to small-signal approximation. The result is proportional to the correlations of timing jitter variables: hI(ω1, ξ)I(ω2, ξ)∗i∼ =hI(ω1, ξ)ihI(ω2, ξ)∗i+πt2 pexp(−ρω2 1/2−ρ∗ω2 2/2) ×X k exp[−i(ω1−ω2)kt0]X rsq exp(−iω2qt0) exp[(µ2 r+µ∗2 s)/σ2 ω] ×[µ∗ sµrR(q−s+r) + µ∗ sµ∗ rR(q−s) + µsµ∗ rR(q) + µrµsR(q+r)] (2.58) where µris evaluated at ω1and µsat ω2. From Eq. (2.58) the diagonal part of function Kcan be directly read: t0S(N)(ω, ξ)∼ =πt2 pexp(−ω2/σ2 ω)X rsq exp(−iωqt0) exp[(µ2 r+µ∗2 s)/σ2 ω] ×[µ∗ sµrR(q−s+r) + µ∗ sµ∗ rR(q−s) + µsµ∗ rR(q) + µrµsR(q+r)] (2.59) where the functions µrand µsare now evaluated at ω. Notice that in Eq. (2.59), |exp(µ2 r/σ2 ω)|= exp(ω2/4σ2 ω)Gr(ω)1/2,Gnbeing the passbands shown in previous section: Gn(ω) = exp[−(ωξ +ωC/σ2 ω−nt0)2σ2 ω/2].(2.60) The passbands have Gaussian form with peaks at ω≃t0/|ξ|and rms width 1/σ2 ω|ξ|. For Talbot lines, |ξ| ≈ t2 0≫t2 p≈1/σ2 ω, so that overlapping between passbands can be neglected. Under this conditions, exp[µ2 r(ω)/σ2 ω]∼ =exp(ω2/2σ2 ω)Gr(ω)δrs and substitution into Eq. (2.59) yields 2. Non-ideal coherent Temporal Talbot effect 71 t0S(N)(ω, ξ)∼ =πt2 pexp(−ω2/2σω)X n Gn(ω) (2.61) ×X q exp(−iωqt0)[2|µn|2R(q) + µ∗2 nR(q−n) + µ2 nR(q+n)] By reordering the sums over q, using the definition of µnthe power spectral density can be expressed in the following more compact way: t0S(N)(ω)∼ =ω2J(ω)Φ(ω)X n Gn(ω)Mn(ω) (2.62) where ω2J(ω) = πt2 pω2exp(−ω2/2σ2 ω),(2.63) Φ(ω) = X k R(k) exp(−iωkt0) (2.64) and Mn(ω) = cos(ωnt0/2) −(ξ+C/σ2 ω−nt0/ω)σ2 ωsin(ωnt0/2)2.(2.65) The power spectrum is composed of a sum of four types of contribution. The first one is a smooth broadband function, ω2J(ω), which represents the envelope function of noise in the spectral domain that has bandwidth of the order of σω. The second function, Φ(ω), is the power spectrum of the timing jitter, which is the Fourier transform of the jitter correlation function. This function is manifestly Ω0periodic, and, in general, it is narrowband about the harmonics [75, 78]. However, in the noncorrelated timing noise limit, R(k) = σ2 jδk,0, the jitter noise becomes white, i.e., Φ(ω) = σ2 j. The terms in the sum in formula (2.62), however, are dispersion dependent and jitter independent. Each term in this sum is composed of a narrowband Gaussian function Gn(ω), to which we refer as the n passband, and an oscillatory function Mn(ω), to which we refer as the modulation function of the npassband. The functions Gn(ω) account for the portions of noise at which the previous noise contributions, ω2J(ω) and Φ(ω), are damped, thus providing the mechanism of noise filtering by multiple interference. Firstly we analyze the noise power spectrum of the incoming train by 72 2.3. Talbot imaging of mistimed pulses setting ξ= 0 in Eqs. (2.60) and (2.65). This is the RF noise of the intensity of the train before it enters the dispersive line, measured by direct detection of the train with a high bandwidth photodetector. From Eq. (2.60), the passbands become centred at ω=nt0σ2 ω/|C|. Because t0>1/σωand the bandwidth of the spectrum of the individual pulses is of the order of σω, the peak contribution of Gn(ω) when n6= 0 lies outside the pulse spectrum. Therefore we neglect the contributions of all the passband functions except that with index n= 0. In addition, the corresponding modulation function is unity, M0(ω) = 1, and thus the input noise spectrum reduces to t0S(N)(ω, 0) ∼ =ω2|F(ω)|2Φ(ω),(2.66) where |F(ω)|2=πt2 pexp(−ω2t2 p/2) is the spectrum of the intensity of a single pulse, |f0(t)|2. It is well known in the context of characterization of noise in fundamentally mode-locked lasers [75–78]. It corresponds to the noise spectrum of a detected jittery train Pk|f0(t−kt0−ak)|2, where we have neglected pulse overlap in the incoming train. It shows again the fact that to neglect the contributions of Gnfunctions is equivalent to neglect overlapping among pulses, since Gnrepresents the interference among n-th neighboring pulse. Notice that Eq. (2.66), is chirp-independent owing to the fact that direct detection of original train misses phase information. When a replica of the train is created after a Talbot dispersive line, the basic features of the noise spectrum are similar to those found in the analysis of noise induced by on-off keying of an exact train (see Eq. (2.21) in section 2.2). First, the broadband envelope of the spectrum, ω2J(ω), extends along the optical linewidth of a single pulse of the train such that, if the pulses are chirped, the noise bandwidth is broader than the bandwidth of the input noise envelope ω2|F(ω)|2. Second, noise is filtered around the maxima of the passbands Gn(ω). These maxima are located at ωn=nt0 |ξ+C/σ2 ω|∼ =αΩ0 n γ1−1 2πsign(ξ)Cα γ Ω2 0 σ2 ω.(2.67) According to formula (2.67), when the initial pulses are unchirped and passband function index nis a multiple of γ, these passbands are centered at the harmonics of the output train. When Talbot index γis not unity, between two consecutive output harmonics there are γ−1 passbands uniformly separated. When the initial pulses are chirped there exists a deviation of the maxima of the passband functions Gn(ω) that depends on the relative sign between dispersion and chirp. When dispersion is positive, ξ > 0, the maxima [formula (2.67)] are redshifted (blueshifted) if the chirp 2. Non-ideal coherent Temporal Talbot effect 73 is positive (negative). The passband with index nwill therefore exhibit a maximum in the neighborhood of a harmonic of the output train when n=hγ, where his the harmonic index. The noise maximum can be raised with respect to the noise value before dispersion. Nevertheless, the precise value of noise at the harmonic frequencies hαΩ0does not change. The full width of the 10-dB decay of these passbands is computed from Eq. (2.60), providing a simple estimate of the bandwidth of the resultant noise spectrum about the harmonics: ∆Wω=2√2 √log e 1 |ξ+C/σ2 ω|∼ =4.3 σω|ξ|=4.3 t0Q.(2.68) This width is common to all passbands. In formula (2.68) we have defined the parameter Q: Q=γ α σω Ω0 .(2.69) Because the temporal width of a pulse after a dispersive line of high dispersion ξgoes as t0Q[5], Qis the temporal dispersion-induced spread of an individual pulse with respect to the pulse separation and represents the number of basic time periods of the original train that occupies a pulse after dispersion. Finally, each of these passband functions, Gn(ω) is modulated by the functions Mn(ω). In this case the structure of the modulation is richer than that found in the analysis of on-off-keyed trains after Talbot devices. First, notice that on the top of the passbands that correspond to output harmonics (n=hγ) the value of Mn(ω) is unity, showing ripples as the offset from the top increases. When the passband index does not correspond to an output harmonic, however, the modulation function does not reach a local maximum on top of the passband. Then these passbands show tips on top that are due to this nonmaximal modulation. Moreover, modulation maxima are in general bigger than unity, so noise can rise above the broadband envelope [Eq. (2.63)]. We present an example of this property in next subsection below. Noise vanishes when modulation reaches zero, Mn(ω) = 0; near the npassband that corresponds to the hharmonic, the equation Mn(ω) = 0 can be represented by means of a variable xthat accounts for the offset from the value of the harmonic, ω(x) = Ω0(hα +x). Then the zeros of the modulation are the solutions of cot(πhγx) = sign(ξ)C+ 2πQ2α γ x x+hα.(2.70) 74 2.3. Talbot imaging of mistimed pulses A graphic analysis of this function shows that there exist two different sets of solutions, which correspond to equal numbers of positive and negative values of x. There exists a zero in each incremental offset of width Ω0/n on the left and the right sides of the harmonics that correspond to the branches of the cotangent function. The location of the zeros is not exactly symmetric with respect to the harmonic, even in the absence of chirp. Therefore, noise skirts around harmonics will always be asymmetric. Because of the decreasing character of the branches of the cotangent function, when both chirp and dispersion have different sign the zeros of modulation are located at lower values of the offset than those with unchirped pulses. If chirp and dispersion have the same signs, the offset of the zeros increases. For large values of Qthe two sets of left and right solutions approximate the solutions of tan(πhγx) = 0. These properties are also exemplified and analyzed in the next subsection. To present examples of the power spectrum it is necessary to introduce a concrete model for the correlation between pulses. It is not our objective to review the possible characteristics of noise and its origin. They ultimately depend on the concrete systems that generate the trains. Here we show two simple models derived from the noise theory of mode-locked lasers [78], which permits a general illustration of the traces in the intensity spectrum by consideration of two different types of pulse-to-pulse timing jitter correlation. 2.3.2.1 Pulse-To-Pulse Partially Correlated Noise The noise spectral density functions of fundamentally mode-locked lasers, before dispersive propagation, present identical peaks at multiples of the pulse repetition frequency [78]. This partially correlated noise can be portrayed by a recursive relation between adjacent pulses. The deviation of the center of the pulses is a fraction η > 0 of the deviation of the previous pulse, augmented by an additional term ǫm: am=ηam−1+ǫm.(2.71) It is assumed that both quantities have zero mean, hami=hǫmi= 0, and that the additional terms are mutually independent and thus uncorrelated, hǫmǫni=T2δm,n. Therefore ηcontrols the correlation between pulses and depends on the underlying mechanism of pulse generation. The jitter variations described by parameter ǫm, however, represent noise induced by spontaneous emission or vacuum fluctuations. 2. Non-ideal coherent Temporal Talbot effect 75 The relation of the two parameters, ηand T, to the rms value of jitter is T2=σ2 j(1 −η2). The correlation function is R(k) = σ2 jη|k|. Thus jitter noise is uncorrelated and therefore white when η= 0, whereas totally correlated noise corresponds to η= 1. The timing jitters spectral density is straightforwardly computed from the correlation, resulting in a Fabry- Perot-type function: φ(ω) = σ2 j 1−η2 1−2ηcos(ωt0) + η2.(2.72) Notice that this function is manifestly periodic, with a period equal to the fundamental harmonic of the train, and is peaked in the harmonics. Thus the noise spectrum’s density before it enters the Talbot dispersive line has symmetric noise RF sidebands or skirts around each harmonic. Expansion of Eq. (2.72) around a harmonic hΩ0reduces the form of the skirts to a Lorentzian shape [78]. In all the examples presented the pulse-to-pulse-temporal separation is t0= 100 ps, corresponding to trains with repetition rate Ω0= 2π×10 GHz. The temporal width of the pulses in the train is tp= 10 ps, and the jitter’s standard deviation is σj= 100 fs. The simulations presented here were generated from a string of 1024 basic periods, each with 64 sample points. The power spectrum is estimated numerically by use of the Bartlett algorithm [80], averaging 8 spectra, each obtained from a sequence of 128 time periods. The frequency separation of the points in the spectrum is therefore 78 MHz (= 1/128 ×100 ps). Jitter is created numerically by use of standard pseudorandom number generators and recursive relation (2.71). In Fig. 2.14 we represent the spectrum of a train of unchirped pulses (C= 0) that corresponds to the intensity of the jittery train before and after the Talbot dispersive line. Dispersion is adjusted to create the first integer Talbot image of the original train (γ/α = 1), so Q= 1.59. The parameter that describes correlation is η= 0.7. In the lower part of Fig. 2.14 we represent by a dashed curve our analytical approximation of the noise spectral power of the detected train before it enters this dispersive device, as given by formula (2.66). By a continuous curve we plot analytical formula (2.62) of the noise spectral power of same train detected after the Talbot device. In the upper part of the figure we depict the simulation of the spectra before and after the Talbot dispersive line, normalizing the DC component to 0 dB. These spectra show peaked contributions that correspond to the harmonics of a perfect train. To help 76 2.3. Talbot imaging of mistimed 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 Power spectrum (dB/Hz) Frequency (GHz) Figure 2.14: Intensity power spectral density (PSD) of a jittery train of pulses (σj= 100 fs, η= 0.7) before and after a temporal Talbot device with index γ/α = 1. Pulse width, tp= 10ps; chirp, C= 0; repetition rate, 10 GHz; Q=1.59. Above, numerical power spectral densities for the input and output trains (continuous curves) and numerical power spectral density of the mean signal (dotted curve). Below, analytical noise power spectral densities for the input (dashed curve) and output (continuous curve) trains. in visualizing the contributions of noise we represent the spectrum of the mean signal by using a dotted curve. The agreement between simulations and analytical formulas is excellent. To appreciate the different contributions to the RF noise skirts around the harmonics, we plot in Fig. 2.15 the contributions to noise in the right-hand offset near 50 GHz. The offset runs up to 5 GHz, the midpoint between harmonics. At the left in Fig. 2.15, and normalizing the power spectrum to 0 dB at zero offset, we represent jitter power spectrum Φ(ω) by using a dashed curve, modulation function Mn=5(ω) by using a continuous curve, and passband function Gn=5(ω) with a dotted curve. The different contributions to the total power spectrum are clearly observed. The jitter power spectrum has a 10-dB half-width of 1.71 GHz; the 10-dB decay half-width of passband function Gn=5(ω) is 2.15 GHz. 2. Non-ideal coherent Temporal Talbot effect 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 Power spectrum (dB/Hz) Frequency (GHz) Figure 2.21: The same as in Figure 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 Power spectrum (dB/Hz) Frequency (GHz) Figure 2.22: The same as in Figure 2.14 but γ/α = 1/2which leads Q= 0.80. 84 2.3. Talbot imaging of mistimed pulses consecutive harmonics. In Figure 2.21 the Talbot index is chosen to be γ/α = 1. The passbands functions are centered around the harmonics of the signal. However the supermode noise peaks are located at midpoint between two passbands, so that they are completely filtered by the Talbot device. It is worth to compare this figure with figure 2.14. The parameters of both figures are the same, except the jitter spectral noise of the incoming train. We can see the appearance of noise supernoise peaks due to lack of correlation between the timing jitter of consecutive pulses. Figure 2.22 shows the results of the noise spectrum after a fractional Talbot device with index γ/α = 1/2. The passband functions describing dispersion are now centered at the even harmonics, so that the odd harmonics are filtered and the multiplication rate of the signal is doubled. In this case the supermode noise peaks located between harmonics, are damped but not completely filtered by Talbot devices. This fact is due to the broadening of the passbands with respect to the previous case (γ/α = 1) weakening the supermode noise filtering. Notice that the damping of these peaks is more efficient for high frequencies, since the modulation helps in narrowing the passbands. 2.3.3 Jitter Smoothing The integration of the RF noise skirts in a symmetric domain of bandwidth Ω0about the hth harmonic is one of the standard operational tools used to obtain experimentally the rms value of the timing jitter of a train from noise analysis of the intensity before it passes through a dispersive line [75]. Since after a fractional Talbot device the spectrum is composed of harmonics multiples of α, we will compute the timing jitter around the hα-harmonic of the original train, which enables to compare the jitter before and after the dispersion line. The procedure that permits a simple determination of jitter is to approximate the broadband noise envelope in formula (2.66) by its value in the harmonic and to integrate along a passband of width Ω0 about the harmonic. This renders the total integrated noise proportional both to squared harmonic index α2h2and to standard deviation σ2 jof the 2. Non-ideal coherent Temporal Talbot effect 85 rms jitter: Z(αh+1/2)Ω0 (αh−1/2)Ω0 dω S(N)(ω, 0) ∼ =(2.75) ∼ =t−1 0Ω2 0h2α2|F(hαΩ0)|2Z(αh+1/2)Ω0 (αh−1/2)Ω0 dω Φ(ω) =t−1 0Ω3 0α2h2|F(αhΩ0)|2σ2 j. Then, from a comparative measurement of different integrated RF noise sidebands, the value of rms jitter is experimentally determined. As we illustrated in previous sections, the RF noise skirts after a Talbot dispersive line change as a result of to the presence of the passband and the modulation functions. Thus total noise, measured as the result of integration of the noise skirt spectrum along a passband around the harmonics, can get smoothed. Notice that, because of the structure of the noise skirts after the Talbot dispersive line, the output noise cannot be ascribed purely to the timing jitter but a combination of both amplitude and timing jitters in the pulses of the output train. The interpretation of output noise as a specific combination of both amplitude and timing jitters in the pulses of the output train will not be addressed here. The integrated noise along bandwidth Ω0about the hth harmonic of the output signal (αhΩ0) after the Talbot dispersive line is expressed as Z(αh+1/2)Ω0 (αh−1/2)Ω0 dω S(N)(ω, ξ) (2.76) ∼ =t−1 0Ω2 0α2h2J(αhΩ0)Z(αh+1/2)Ω0 (αh−1/2)Ω0 dωΦ(ω)Gn=hγ(ω)Mn=hγ(ω). In formula (2.76) we have also approximated the broadband envelope by its value in the harmonic and have retained in the sum only the passband function that corresponds to the harmonic αhΩ0, that is, n=hγ. According to the examples presented in previous sections, this is a good approximation when the dispersive line in adjusted to the first Talbot image, γ= 1, because no new passbands between harmonics appear in the spectrum. The extension of the integration to a bigger number of passband functions is nevertheless straightforward. Then the ratio ρ(αh)of the integrated RF noise power after dispersion [formula (2.76)] to that before dispersion [formula (2.75)] is expressed, from formulas (2.62)-(2.66) and (2.60), as 86 2.3. Talbot imaging of mistimed pulses ρ(αh)∼ =exp n2C2/2Q2Ln=hγ(η, Q, C),(2.77) where the first term accounts for the different noise envelopes 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 perform the integration by again using the variable ω(x) = Ω0(αh +x) and with the notation Φ(x)≡Φ[ω(x)]/σ2 j, Gn=hγ(x)≡Gn=hγ[ω(x)], and Mn=hγ(x)≡Mn=hγ[ω(x)]. From the structure of formula (2.77) the following consequences are derived: First, the ratio in formula (2.77) depends on correlation, not on the precise value of σ2 j. Second, if pulses are unchirped, noise is essentially symmetric about the harmonic, and the resultant ratio is always less than unity because of the presence of the passband functions. The reduction increases as the width of the passband narrows, so smoothing increases with Qand also as η→0, because the input noise is white in a bigger band. Finally, when the pulses are chirped the value of the ratio is raised through the exponential factor in formula (2.77) that accounts for the different noise envelopes. This increase becomes bigger at high harmonics. In Fig. 2.23 we plot the ratio ρ(αh)of the fifth harmonic (h= 5) of the first integer Talbot dispersive device (γ/α = 1) to correlation parameter ηand for three values of Q. The values of the ratio [formula (2.77)] were obtained by numerical integration of Eq. (2.78). First, notice that, owing to the quadratic dependence of the integrated noise power formula (2.75)] on the jitters standard deviation σj, a reduction of 10 dB in the ratio ρ(αh) amounts to a reduction of σjby a factor of 3, whereas a 6-dB decay is associated with a reduction of σjby a factor of 2. In Fig. 2.23(a) the ratio is computed for an initial train composed of unchirped pulses, whereas in Fig. 2.23(b) the pulses are slighted chirped (C= 0.5 rad). As the correlation parameter controls the white-noise bandwidth of the RF noise skirts of the harmonic, this is one of the most determinant parameters of the smoothing; see Fig. 2.23(a). For high values of η, noise is correlated, the noise skirts are narrow, and consequently the noise reduction induced by the presence of passbands is small. However, when the correlation is low, the presence of the passbands induces a significant decrease in the RF noise skirts, which increases with parameter Q. Chirp is the most deleterious factor for this smoothing. From Figs. 2.17 and 2.18, when pulses are chirped the noise skirts are asymmetric and the peak values are raised above the initial noise level, leading to the increase in the ratio in 2. Non-ideal coherent Temporal Talbot effect 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 Correlation parameter η Ratio ρ(h=5) (dB) Q=2 Q=5 Q=7 Q=2 Q=5 Q=7 Figure 2.23: Ratio ρ(h)for the fifth harmonic h= 5 (fifth passband, n= 5) and for the first integer Talbot replica γ/α = 1 as a function of correlation parameter ηfor three values of dispersion parameter Q: continuous curves, Q= 2; dashed curves, Q= 5; dotted curves 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 Dispersion parameter Q Ratio ρ(h) (dB) h=1 h=5 h=7 h=1 h=5 h=7 Figure 2.24: Ratio ρ(h)for the first integer Talbot replica γ/α = 1 and for jittery trains with correlation parameter η= 0.7as a function of dispersion parameter Qfor three harmonics h: continuous curves, h= 1; dashed curves, h= 5; dotted curves h= 7. 88 2.3. Talbot imaging of mistimed pulses formula (2.77) illustrated in Fig. 2.23(b). However, even in the presence of chirp, the combined effect of high Qand low correlation can lead to a significant decrease in the ratio ρ(αh). To visualize its dependence on the Qfactor, and also its small variations within harmonic index h, in Fig. 2.24 we represent again the ratio for the first integer Talbot replica, γ/α = 1, for jittery trains with correlation parameter η= 0.7 as a function of Qfor different harmonics hand for the same two values of chirp, C= 0 and C= 0.5 rad. Both the decrease in the ratio ρ(αh)with Qand its small dependence on the harmonic are observed. Again, small chirps in Fig. 2.24(b) spoil the noise reduction. This effect is more pronounced for high harmonics, because the passband shift and the subsequent increase in the noise peak value are proportional to the harmonic frequency. But notice that this increase in ratio ρ(αh)for chirped pulses can be compensated for by means of an increase in Q. In Figs. 2.25 and 2.26 we represent the ratio ρ(αh)in the second Talbot device, γ/α = 2. The presence of new passbands between harmonics has been taken into account in the numeral evaluation of integral equation (2.78). In Fig. 2.25 we plot ratio ρ(αh)for the fifth output harmonic, h= 5, with respect to the correlation parameter for different values of C and Q. When C= 0 rad, the curves in Fig. 2.25(a) are similar to those in Fig. 2.23(a), proving that the ratio depends basically on the value of Q through the width of the passbands located in the harmonic. The presence of new passbands adds a subsidiary contribution to ratio ρ(αh). Therefore the increase in Talbot order produces an increase in the reduction of smoothing. However, when the chirp is nonzero, the subsequent change in the passband position always increases the value of the ratio. In Fig. 2.25(b) the ratio becomes bigger than unity for low values of Q, but, again, increasing the value of Qcompensates for the increase in the ratio. Finally, in Fig. 2.26 we depict the evolution of ratio ρ(αh)for the second integer Talbot replica, γ/α = 2, for jittery trains with correlation parameter η= 0.7, as a function of Qfor three harmonics h. In Fig. 2.26(a), unchirped pulses are considered. These curves are similar to those of Fig. 2.24(a) except for the first harmonic, h= 1, whose values of the reduction are worse because of the presence of the new passband at 5 GHz; Fig. 2.19. In Fig. 2.26(b) we consider pulses with chirp, C= 0.5 rad. Again, ratio ρ(αh)increases owing to a shift in the passband maximum and can be compensated for by use of high values of Q. 2. Non-ideal coherent Temporal Talbot effect 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 Correlation parameter η Ratio ρ(h=5) (dB) (b) C=0.5 Q=2 Q=5 Q=7 Q=2 Q=5 Q=7 Figure 2.25: Ratio ρ(h)for the fifth harmonic h= 5 (tenth passband, n= 5) and for the first integer Talbot replica γ/α = 2 as a function of correlation parameter ηfor three values of dispersion parameter Q: continuous curves, Q= 2; dashed curves, Q= 5; dotted curves 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 Dispersion parameter Q Ratio ρ(h) (dB) h=1 h=5 h=7 h=1 h=5 h=7 Figure 2.26: Ratio ρ(h)for the first integer Talbot replica γ/α = 2 and for jittery trains with correlation parameter η= 0.7as a function of dispersion parameter Qfor three harmonics h: continuous curves, h= 1; dashed curves, h= 5; dotted curves h= 7. 90 2.4. Conclusions 2.4 Conclusions In this chapter we have analyzed the behavior of Talbot lines when they are fed with non-ideal coherent train of linearly chirped Gaussian pulses. We have assumed an ideal Talbot filter, i. e., a pure first-order dispersion medium without attenuation and non linear effects. Furthermore, we considered the bandwidth of Talbot line is large enough to cope with the optical bandwidth of the incoming train. In section 2.2 we have analyzed the power spectral density of a on-off-keyed train after passing a Talbot filter. We have assumed that the variables describing the presence or absence of pulses in the train are uncorrelated. As a consequence of the coherent interference of dispersed pulses, the microwave spectrum after the detection of the train has a broadband noise envelope that follows the linewidth spectrum of an individual pulse, i.e., the microwave envelope bandwidth is enhanced by the value of the chirp. The basic feature of the spectrum is the presence of passbands filtering this broadband noise, whose centers are shifted with respect to the harmonics due to the chirp of the incident pulses. Measurement of the relative deviation of these centers with respect to passband width allows a determination of the chirps sign and magnitude. The width of the passbands is narrower as the dispersion of the device decreases, the pulse width is shorter, and the chirp is larger. The peak value of these passbands depends on the relative amount of pulses in the train. For small dispersion devices, high passbands, and wide pulses, the passbands show a cosine-squared modulation. The width of this modulation is a fraction of the input harmonic value, being independent of the relative amount of pulses in the train or of the properties of the pulses in the train. In short, our analysis shows that the parameters describing the pulse train have a neat influence in the resulting microwave spectrum, allowing the indirect characterization of pulse trains through the random Talbot effect. In experimental situations, however, the random Talbot effect has to coexist with other types of broadband and narrowband noise due to the sources or modulators used, such as amplitude noise or timing jitter. For instance, as is well known, the spectrum of detected pulse trains from mode-locked lasers show typical noise skirts around output harmonics, whose width depends on the correlation properties of noise [76–78]. Moreover, deviations of the ideal behavior of the Talbot filter will also affect the characteristics of the spectrum. Although it seems unlikely that the random Talbot effect of pulse trains shows all the fine spectral 2. Non-ideal coherent Temporal Talbot effect 91 details contained in Eq. (2.18) or (2.21), the overall features should be still present. Anyway the noise features of Talbot devices can be enlarged when required by concrete experimental circumstances. In the section 2.3, we have studied the influence of random jitter of a linearly chirped Gaussian pulses in Talbot devices. The analysis has been performed in the both temporal and spectral domain. In the temporal analysis we have assumed the jitter to be uncorrelated. We have shown that the variance of the train is flattened by use of dispersion after passing a Talbot device. This is due to the dispersion of the individual pulses that form the train, as their variance is spread on neighboring intervals. We have also given approximate formulas for the input and output variances and discussed the validity of these approximations. For integer Talbot devices, the value of the variance is constant irrespective of the value of the dispersion. For fractional Talbot devices with γ/α < 1, a modulation of the constant value of the variance may be present, depending on the ratio between the pulse width and the period of the original train. Nevertheless, for Talbot devices leading to N×repetition rates, the variance still is, to a good approximation, constant. The presence of pulse pedestal in these devices has also been analyzed. From the spectral point of view we report on a wider analysis. Now an arbitrary correlation among jitter variables is allowed. The basic feature of the resultant spectrum is, like in the case of on-off-keyed trains, the presence of a broadband noise-filtering mechanism by multiple interference at the scales of a fraction of the trains repetition rate, which provides a reduction of the integrated noise power around the harmonics as measured in the RF regime. The influence of chirp and optical linewidth of the individual pulses in the train was also analyzed. We have used the results derived to explore the noise reduction of different coherent Talbot-imagined trains under different propagation conditions. The smoothing of jitter noise by Talbot interference depends basically on the chirp of the pulses of the train, on the correlation between adjacent pulses in the train, and on the accumulated dispersion of the line but not on the absolute value of the timing jitters standard deviation. Chirp has been shown to be the most deleterious factor because it induces a shift of the peak noise level with respect to the position of the harmonic that causes an increase in the total noise power. However, careful design of the dispersive line can easily compensate for it [43, 82] and compress the pulses simultaneously, as has been shown in first chapter. As correlation controls the jitter noise bandwidth before the jitter noise 92 2.4. Conclusions enters the dispersive line and because jitter smoothing is more pronounced at scales of the order of the repetition rate (typically less than 1 GHz for the 10 GHz trains), the initial noise must be broadband to produce significant jitter reduction. This fact limits the possible application of this procedure to trains composed of slightly correlated pulses. Ultimately, this application depends on the mechanisms that produce the trains. For instance, the use of a Talbot dispersive line cannot be successfully applied in fundamentally mode-locked lasers that have noise offsets up to 100 MHz; see, e.g., Ref. [77]. In this case the active locking in the cavity of the laser produces highly correlated timing jitter. However, even in these cases the Talbot replica of a mode-locked laser output can in principle help to reduce other types of non-narrowband noise, such as the supermode noise [81] in harmonically mode locked lasers, for which both phase noise and timing jitter are uncorrelated between adjacent pulses. [77,78]. The noise reduction depends on the accumulated dispersion of the line through the order of the Talbot replica. Basically, high Talbot orders imply the appearance of new noise windows between harmonics and the narrowing of the passband widths, which provides the bandwidth of the smoothed spectrum about the harmonics. Except for the first harmonic, the presence of new windows is a subsidiary effect when one is computing the decrease in noise ratio ρ(h). The narrowing of the passbands is controlled by dispersion parameter Q, which accounts for the width of the dispersed pulses relative to the repetition rate of the train. High Qvalues imply that the resultant pulses in a Talbot replica are formed from contributions of a high number of dispersed pulses, thus providing a coherently averaged output. The same smoothing mechanism has been found in the time domain. 3. Partially coherent Temporal Talbot effect 99 3.3 Partially Coherent Light in Talbot Lines In this section we specialize the previous results for a Talbot line. We now consider the partial light source is modulated by a periodic function, so that the optical envelope at the input of dispersive medium, a(t, 0) is given by a(t, 0) = ǫ(t)m(t) = ǫ(t)"f(t)⊗X k δ(t−kt0)#.(3.21) Then, using (3.12) and (1.32), we obtain the following expression for the average intensity at the Talbot distances zγ/α, Eq. (1.32): hI(t, zγ/α)i=α γ 1 t2 0 Sα γ t t2 0⊗|f(t)|2⊗1 αX k δ(t−τ−kt0/α).(3.22) This formula can be reduced to the results of [84] when dispersion is specialized to integer Talbot distances (α= 1) and f(t) is real. Notice that, according to (3.12), partial coherence results in a smoothing of the average intensity, irrespectively of the exact or approximated character of the (monochromatic) temporal Talbot effect due to pulse overlapping. If, however, the initial pulses are sufficiently narrow to avoid overlapping at the Talbot distance under consideration, the coherent intensity train is exactly periodic with period t0/α see Eq. (1.32). Then, formula (3.22) describes the smoothing of the average intensity profile in each exact period t0/α due to partial coherence. Formula (3.22) also describes an example of the temporal Collett-Wolf theorem [13,14]. This result states that two optical modulated light fields with different coherence properties may propagate in dispersive media with identical average intensity distribution. In our case, it is apparent from (3.22) that, given a Talbot distance zγ/α, a pulse train carried by a monochromatic wave of intensity I0, and with single pulse profile q(t) would provide the same average intensity of a train of pulses f(t) carried by a partially coherent wave if: α γ 1 t2 0 Sα γ t t2 0⊗|f(t)|2=I0|q(t)|2.(3.23) The dual behavior of single-pulse intensity and spectral content becomes clear through this convolution. Equivalently, partial coherence can be interpreted as defining the low pass filter (3.13). The equivalent bandwidth (3.15) of the filter at Talbot distances zγ/α is: 100 3.3. Partially Coherent Light in Talbot Lines Beq =1 2 τc t2 0 α γ,(3.24) Notice that the bandwidth increases with the inverse of the Talbot index γ/α. This means that, for the same type of pulses, coherence requirements are more restrictive for high values of the Talbot index γ/α and that, for a given value of the repetition rate α, the increase of γsmoothes the average intensity of the Talbot-imagined train. In the time domain, this observation can be interpreted as follows. The process of reconstruction of pulses at Talbot distances is multiple pulse-to-pulse interference due to dispersion. Partial coherence damps the interference pattern between neighboring pulses, since the underlying carrier misses progressively its own phase. This damping is more pronounced when the interfering pulses are initially more distant. Therefore, high values of the Talbot index γ/α, or, alternatively, highly dispersive devices, which require constructive interference between pulses that were initially more distant, are more sensitive to the coherence of the carrier. Formula (3.24) can also be used to determine coherence requirements for the recovery of the Talbot train of intensities. Since this train is composed of intensity harmonics of the fundamental frequency 1/t0, the quantity Beqt0represents the number of harmonics that are not filtered by partial coherence. The requirement of faithful reconstruction of the pulses is therefore Beqt0≫1. For instance, a typical DFB laser has a spectral line with FWHM of 50 MHz. Its coherence time is therefore τc∼20 ns. If, moreover, the repetition rate is 2.5 GHz (t0= 400 ps), and γ/α = 1/4, the resulting relative bandwidth is Beqt0= 32. On the other hand, the quantity Beqt0/α, which represents the number of intensity harmonics that constitute the output train, does not depend on the repetition-rate index α:Beqt0/α =τc/2t0. This is a consequence of the inverse relationship between equivalent bandwidth and dispersion: lower dispersion -or, equivalently, higher repetition-rate index implies higher bandwidth or a less severe smoothing of the single-pulse intensity, see (3.24). But the increase in index αalso implies the increase in the separation α/t0 of the intensity harmonics that conform the output train, so that the total number of the output intensity harmonics is independent of α. In fact, Beqt0/α is a measure of the coherence time relative to the period of the train. This conclusion can also be obtained by direct computation. Let us consider the harmonic expansion of the initial intensity train: 3. Partially coherent Temporal Talbot effect 101 |f(t)|2⊗X k δ(t−kt0) = X k Ckexp(i2πkt/t0),(3.25) where Ckare the harmonics of the periodic modulation function. Then, the corresponding expansion of the coherent output train with increased repetition rate only involves the harmonics multiples of α: |f(t)|2⊗1 αX k δ(t−τ−kt0/α) = X k Ckα exp(i2πk(t−τ)α/t0).(3.26) If we insert this expansion in (3.22) to take into account the partial coherence of the carrier wave, we obtain: hI(t, zγ/α)i=X k CkG(−kt0γ) exp(i2πk(t−τ)α/t0),(3.27) which shows that coherence limits the number of output intensity harmonics independently of the repetition-rate index α. Finally, we conclude the analysis of formula (3.22) with the description of its generalization to multiple carrier wavelengths [99, 101]. Let us assume that the carrier is composed of Nindependent partially-coherent wavelengths. Let us also assume that the detection conditions assure that the corresponding modulated waves are mutually incoherent, so that the final average intensity is a sum of the intensities associated to different wavelengths but with the same modulation. Then, the filter in (3.22) generalizes to a sum over the different spectral lines: hI(t, zγ/α)i=α γ 1 t2 0 N X k=1 Skα γ t t2 0⊗|f(t)|2⊗1 αX m δ(t−τ−mt0/α). (3.28) and thus the spectral content of the carrier waves determines the filter’s temporal response. In this sum, frequency shifts between different carriers translate into time delays. For instance, let us assume that the modulation is carried by two monochromatic waves with different intensities and frequencies ν0and ν0+ ∆ν0, respectively. Then, the sum in Eq. (3.28) contains two terms, associated to the power spectral densities S1(f) = I0δ(f) and S2=I′ 0δ(f−∆ν0). The resulting filter is: α γ 1 t2 0 2 X k=1 Skα γ t t2 0=I0δ(t) + I′ 0δ(t−γ αt2 0∆ν0),(3.29) 102 3.4. Matrix Theory and Entropy thus describing a weighted superposition of two replicas of the signal, with a mutual delay that depends on the frequency difference ∆ν0. A mapping of this type -from the spectral content of the carrier to a delay in the modulated waveformsfinds application in the construction of microwave transversal filters [102]. In formula (3.28), however, the filter acts over pulses created as Talbot replicas of the original train. Therefore, the wavelength-dependent delay introduces a relative delay between Talbotimagined trains, not between replicas of a dispersed train, as in (3.12). This observation represents the basis of the temporal Lau effect. 3.4 Matrix Theory and Entropy The first objective of this section is to express the output intensity of a Talbot device in a matrix form, separating the influence of partial coherence and the spectral content of the incoming train of pulses. To this end, we observe that the harmonics Cαk of the coherent output intensity in eq. (3.26) can be expressed in terms of the amplitude α-harmonics aαk : Cαk =αX r arαa∗ (r−k)α.(3.30) This can be derived from (3.26), since the absence of pulse overlapping leads to the following equality: Iu(t, zγ/α) =  f(t) √α⊗X k δ(t−τ−kt0/α) 2 =√αX s aαs exp[i2παs(t−τ)/t0] 2 . (3.31) On the other hand, in Eq. (3.27) we observe that the Talbot effect involves the sampling of the complex coherence function with sampling period ts= γt0. This permits the description of the influence of partial coherence in a matrix way. For this purpose, we define the coherence matrix Mwith elements: Mrs =M∗ sr =g(γ(r−s)t0).(3.32) The properties of the complex degree of coherence imply that the coherence matrix is a Hermitian Toelplitz matrix [103,104]. We also define a time- 3. Partially coherent Temporal Talbot effect 103 dependent column vector Awhose entries are given by: Ar(t) = arα exp(i2παrt/t0),(3.33) which corresponds to the α-harmonics of the incoming optical envelope. From now on, we will assume that N= 2n+1 is the number of α-harmonics describing the input pulse train, so that the indices in (3.32) and (3.33) run from −nto n. If the spectral content of the input pulses, f(t), is contained in a band [−Wp, Wp] around the optical carrier, then: n=⌊WpT/α⌋,(3.34) and ⌊x⌋is the largest integer smaller or equal to x. Using these definitions, the average intensity (3.27) can be rewritten as: hI(t, zγ/α)i=I0αA+(t−τ)·M·A(t−τ),(3.35) where A+is the Hermitian conjugate of vector A. Specializing this expression for t=τit follows that Mis a positive-definite matrix, since in this case Ais an arbitrary time-independent vector and the intensity is clearly a non-negative quantity. The coherence matrix is a Hermitian operator and therefore diagonalisable by means of a transformation matrix B: diag(λ1, λ2,···, λN) = 1 NB·M·B+,(3.36) where λrrepresent its eigenvalues and the factor 1/N has been introduced in order to normalize the trace of the diagonal matrix to unity. Since the coherence matrix is positive semi-definite, its eigenvalues are nonnegative [103]. With this decomposition the average intensity is: hI(t, zγ/α)i=I0Nα n X r=−n λr|br(t−τ)|2,(3.37) where: br(t) = n X q=−n Brqaαq exp(i2παqt/t0).(3.38) Eq. (3.37) represents the average intensity at Talbot distances as a weighted sum of intensity patterns, |br(t−τ)|2, the eigenvalues λrbeing the weights of the sum. Thus, the average intensity can be reinterpreted as an incoherent sum of amplitude patterns br(t−τ) whose number and relative intensity is 104 3.4. Matrix Theory and Entropy determined by the eigenvalues λr[85,86]. In the coherent limit the coherence matrix is unity Mrs = 1, ∀r, s, and there is a single eigenvalue different form zero, λK= 1 for certain index K. The corresponding eigenvector is BKq = 1/√N∀q, so that the only significant amplitude bK(t) represents the amplitude of the train of pulses with increased repetition rate under total coherence. On the other hand, the incoherent limit is represented by a coherence matrix Mrs =δrs. In this case all eigenvalues are equal to 1/N, and therefore all patterns in the sum (3.37) are equally weighted. The matrix of eigenvectors is Brq =δrq, and the intensity is constant in the incoherent limit. Therefore, the form of the amplitude patterns br(t) is a linear combination of the α-harmonics of the train that interpolates between total coherence and incoherence conditions. On the other hand, the number and relative magnitude of the eigenvalues enclose the influence of partial coherence in the performance of the temporal Talbot effect. In Appendix B we review several parameters that have been introduced as a measure of the partial coherence in similar matrix decompositions of partially-coherent optical effects from its eigenvalue decomposition [87, 88]. These can also be used to measure the overall impact of partial coherence in the intensity patterns arising at Talbot distances. Among them, we will focus on the informational entropy [85]: H=− n X k=−n λkln λk.(3.39) The entropy ranges from its minimum value Hmin =Hcoh = 0 that represents total coherence; and its maximum, Hmax =Hincoh = ln N, which corresponds to the incoherent limit. A somewhat similar measure of the coherence is the effective number Neff of uncorrelated random variables that represent a signal [89]. In our context this number is an equivalent measure of the number of significant non-zero eigenvalues that contribute to the expansion of the intensity (3.37). Since this expansion can be interpreted as the incoherent sum of the intensities associated to Ndifferent sources [86], we refer to Neff as an effective number of incoherent sources. Let us identify this expansion with an equivalent expansion associated to the intensity recorded at a certain Talbot distance with exactly Neff ≤ Nuncorrelated, equally-weighted sources, so that the entropy of this equivalent system would be H= ln Neff . Then, given a Talbot device operating under partial coherence with entropy H, we define the number 3. Partially coherent Temporal Talbot effect 105 of effective sources as: Neff =⌈exp(H)⌉,(3.40) where ⌈x⌉represents the minimum integer larger than x. The relation of this quantity with those introduced in [87–89] is also briefly reviewed in Appendix B. In order to present examples of the use of the expansion in eigenvalues (3.37) we consider Talbot devices fed by partially-coherent trains with a Lorentzian carrier linewidth, whose complex degree of coherence is an exponential function [3]: g(τ) = exp(−|τ|/τC),(3.41) where τCis the coherence time. Let us consider a modulation signal with period t0= 1 ns, described by 201 harmonics, which feeds a Talbot device with index γ/α = 1/2. From Eqs. (3.27) and (3.30), only those harmonics multiples of α= 2 have influence over the value of the average output intensity. Therefore, the output intensity is described by a coherence matrix with dimension N×N= 101 ×101. The ordered eigenvalues of this system (λ1≥λ2≥...) are presented in figure 3.1. On its left hand side, the largest ten eigenvalues of the coherence matrix are depicted for different values of the coherence time. The eigenvalues of a partially coherent source with τC= 100 ns are depicted by squares. Each pulse in the original train is able to interfere with the nearest 2τC/t0= 200 initial pulses, so that we expect that the output train is composed of faithful replicas of the initial pulses. The value of the entropy has increased from its value in the coherent limit but it is still far from its maximum value. If the temporal coherence is reduced up to 10 ns (circles) or 3 ns (triangles-down), pulses in the initial train can interfere only with the nearest 20 or 6 pulses, respectively. This leads to a progressive equalization of the magnitude of the eigenvalues. Correspondingly, the entropy is increased. Finally, for τC= 1 ns (trianglesup) we almost reach the incoherent limit, since each pulse can interfere with only their two neighbours. The entropy is close to its maximum value and the eigenvalues almost have the same magnitude. On the right hand side of figure 3.1, we depict the evolution of the two largest eigenvalues (λ1, λ2), together with the tenth (λ10) and lowest (λ101) eigenvalue, with respect to the coherence time. Their behavior is equivalent to the previous plot: the relative importance of the main eigenvalue decreases with the coherence time. When all eigenvalues are equally weighted with λr∼1/N = 1/101 ∼0.01 we reach the incoherent limit. In this case, this happens for coherence times of about τC∼100 ps. 106 3.4. Matrix Theory and Entropy 12345678910 10−3 10−2 10−1 100 Eigenvalues’ index Eigenvalues 10−1 100101102 10−3 10−2 10−1 100 Coherence time (ns) Eigenvalues H=1.06 H=2.94 H=3.90 H=4.47 λ1 λ2 λ10 λ101 Figure 3.1: Left hand side: the ten higher eigenvalues of coherence matrix associated to different coherence times: τc= 100 ns (squares), τc= 10 ns (circles), τc= 3 ns (triangles-down) and τc= 1 ns (triangles-up). Right hand side: evolution of the eigenvalues λ1,λ2,λ10 and λ101 as a function of the coherence time. The parameters of the Talbot device are t0= 1 ns, N=101, and γ/α = 1/2. In figure 3.2 we exemplify the influence of partial coherence in the output average intensity. We also explore the effective number of sources and their ability to reconstruct the output by truncation of expansion (3.37). In this plot we represent the output average intensity that results after taking into account all eigenvalues in (3.37) (continuous line) and the truncation to Neff eigenvalues (dashed line), for different values of the coherence time. The initial train is composed of Gaussian pulses of τP= 50 ps width (measured as the half-width at 1/e decay in power) and period t0= 1 ns. It is propagated through a Talbot device with index γ/α = 1/2. In spite of the fact that a Gaussian pulse train is composed of an infinite collection of harmonics, we will assume that the set of harmonics with 3. Partially coherent Temporal Talbot effect 107 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Intensity (a.u.) 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Intensity (a.u.) 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Intensity (a.u.) 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) Intensity (a.u.) H=0.17 → Neff=2 H=0.89 → Neff=3 H=2.07→ Neff=8 H=1.58 → Neff=5 Figure 3.2: Average intensity of a 1GHz train of 50 ps Gaussian pulses after propagation along a γ/α = 1/2-Talbot device. Continuous curve: exact average intensity. Dashed curve: approximate average intensity after truncation to Neff sources. power above −30 dB the maximum describes acceptably the train. Under this criterion and taking into account (3.34) the dimension of the matrix problem N= 2n+ 1 is determined by n=⌊0.419 τP t0 α⌋,(3.42) which in this example leads to N= 9. Note that this truncation of the number of harmonics does not assure that the initial pulses are timelimited to the interval [0, t0/2], as required for exact increase of the train’s repetition rate, but will be sufficient to describe the effect of partial coherence. In figure 3.2(a) we present an example of a highly coherent carrier with τC= 100 ns. The effective number of sources is two. The curve depicted by 108 3.5. Entropy in the Limit of Large N considering two sources is indistinguishable form the exact intensity output. We have also checked that the intensity associated to a coherent carrier is indistinguishable from those depicted here. In figure 3.2(b) we consider a carrier with τC= 10 ns, which is the second example in our previous figure 3.1. In the exact partially-coherent intensity pattern we observe the progressive smoothing of the intensity pattern, as expected. On the other hand, the effective number of sources is now 3, and the resulting average intensity after truncating the expansion (3.37) is still in good agreement with its numerical value. In figures 3.2(c) and (d) the coherence time of the carrier are 3 ns and 1 ns, so that the effective number of sources increases to 5 and 8, respectively. We observe the progressive fading of the output pulse train, which tends to a constant value in the incoherent limit. We also notice that the accuracy of the truncation by Neff effective incoherent sources decreases when sources become less coherent. This is due to the fact that the relative importance of the first neglected eigenvalue in the expansion, with index k=Neff + 1, with respect to the previous one, with k=Neff , tends to unity in the incoherent limit. Therefore, the truncation of the intensity expansion by the effective number of sources provides a good approximation of the average intensity as long as the coherence of the source is not extremely low. In general, it is necessary to use sources with high phase and amplitude stability to achieve a coherent phenomenon such as the Talbot effect, and therefore the approximation of expansion (3.37) by truncation to Neff terms is valid in practice. 3.5 Entropy in the Limit of Large N The application of the temporal Talbot effect to multiplying the repetition rate of pulse trains requires the absence of pulse overlapping in the output, or equivalently, narrow initial pulses. The description of narrow pulses requires a large number N, which is the number of α-harmonics in the train and also the dimension of the coherence matrix. Fortunately, the properties of Toeplitz matrices in the limit of large Nare well understood. They are based on Szeg¨o theorem on the distribution of eigenvalues [104], which states that: lim N→∞ N X k=1 Fτ(N) k=Z+π −π dΩ 2πF[P(Ω)].(3.43) Conclusions From the spectral point of view, a Talbot line is a multiple bandpass filter, where each passband represents the interference between dispersed pulses separated a certain number of time slots. Thus, fractional Talbot lines, act filtering the intensity harmonics of the input train which do not belong to the output, leading to an increase of the repetition rate of the train. The rejection is exact only for transform-limited pulses and semi-integer Talbot devices with negligible second-order dispersion. Talbot filters are not static but depend on the dispersive characteristics of the line as well as the bandwidth of the input train. In the particular case of linearly chirped Gaussian pulse train, we have derived the conditions which guarantee that the Talbot device provides an output train with negligible intensity fluctuations, even in presence of second order dispersion. However, in this case, the output train may suffer from single pulse distortion. The stability and the tolerance of Talbot filters under timing and length variations have been studied. If input trains have large bandwidth, small dispersion variations lead to notable broadening of pulses, worsening the performance of Talbot line by pulse-to-pulse interference. Nevertheless, when spectral width is moderate, dispersion instabilities deteriorate the rejection capability of the filter. Input Trains with a bandwidth measured as FWHM equals to twice the the repetition-rate of the output train yields to the optimal stability. The tolerance length are expected to be about 10% divided by the repetition-rate factor. We also have explored Talbot devices as smoothers of imperfections of the input signal. In particular, we have first studied randomly on-off keying of the pulses in the train, and then timing jittery trains before and after passing through a Talbot line. We have assumed pulses to be linearly chirped Gaussian. In both cases the basic feature of the spectrum is the presence of passbands which filter the broadband noise. If pulse are unchirped the passbands are centered in the harmonics of the perfect train, otherwise their centers are slightly shifted. Measurement of the 115 116 Conclusions relative deviation of this centers with respect to the passband width allows a determination of the chirp’s sign and magnitude. Passbands present a modulation, which increases in frequency along the RF spectrum. The timing jittery noise reduction has been analyzed from the variance and by the integration of the RF noise skirt about the harmonics. When jitter variables are uncorrelated, the variance of the train after passing Talbot device is flattened, becoming time-independent. It leads to the smoothing of the timing jitter but also to the appearance of pulse pedestal. The same result is achieved by means of RF noise integration. However when pulse jitter is correlated, the noise reduction is less significative. It is easy understood from the RF spectrum. In this case, jitter noise is narrowband around the harmonics and consequently the noise reduction induced by the presence of passbands is smaller. Chirp has been shown to be the most deleterious factor because it induces a shift of the peak noise level with respect to the position of the harmonic that causes an increase in the total noise power. The influence of the source coherence in Talbot devices has been analyzed. The partial coherent carrier of the modulated wave after propagation along a lowest-order dispersion medium is as a low pass filter acting over the average intensity associated to a monochromatic carrier. Under Talbot conditions, the coherence limitations are more restrictive for high dispersive lines, i.e, high Talbot index, since longer coherence time is need to produce faithfull pulse-to-pulse interference. Increasing dispersion or decreasing the coherence time results in a progressive decrease of the filter’s bandwidth and therefore, in loss of the low-time scale details when compared with the intensity obtained with a monochromatic carrier. We also approach the Talbot effect under partial coherence from a matrix point of view. The diagonalization of the matrix problem representing the average intensity after Talbot device yields to the interpretation of the average intensity as an incoherent sum of intensity patterns. In this context, we have computed the informational entropy and the number of effective sources as measures of the overall performance of the Talbot device. Appendix A Let be define the functions f1(t) and f2(t), from the arbitrary function f(t) as: f1(t) = f(t) exp(2πist/t0)⊗X m δ(t−mt0/α) (A.1) f2(t) = f(t) exp(2πi(s−q)t/t0)⊗X m δ(t−mt0/α) (A.2) If f(t) is confined into [−t0/2α, t0/2α], the product f1(t)f2(t)∗can be expressed as follows f1(t)f2(t)∗=|f(t)|2exp(2πiqt/t0)⊗X m δ(t−mt0/α) (A.3) which does not depend on the variable s. By using the Fourier expansion of f1(t)f2(t)∗and carrying out the multiplication among harmonics we obtain FT [f1(t)f2(t)∗](ω) = 2πα2X p δ(ω−2πpα/t0)X r arα+sa∗ rα+s−pα−q(A.4) From (A.3) and (A.4), it follows that the sum X r arα+sa∗ rα+s−pα−q(A.5) does not depend on s, so the proof is complete. 117 Appendix B Several parameters have been introduced for measuring the overall degree of coherence of the matrix representation of a partially-coherent spatial system. In the context of a study of the effect of partial coherence in the Wigner representation of a paraxial optical system [87,88], Bastiaans introduced a set of quantities µm(m > 1) as a measure of partial coherence. The parameters µmare based on the Neeigenvalues λkof the matrix representing the impact of partial coherence in the system’s performance: µm= Ne X k=1 λm k!1 m−1 .(B.1) Parameter µ1is defined as the limit µ1= limm→∞ µm. It can be shown that parameters µmare bounded by 0 and 1:µm= 0 corresponds to the totally incoherent case whereas µm= 1 represents the coherent limit. The value of parameters µmis a nondecreasing function of the index m, and therefore µ1is the smallest of the set µmand thus the most conservative measure of the degree of coherence. For m→ ∞,µ∞represents the maximum eigenvalue. The original definition of the effective number of uncorrelated random variables representing the optical signal is precisely its inverse, 1/µ∞[89]. The special case µ1can be expressed as a function of the entropy: µ1= exp Ne X k=1 λkln λk!= exp(−H).(B.2) Then, parameter µ1is the inverse of the effective number of sources as introduced in the text, Neff = 1/µ1, and represents the largest measure of the number of effective sources when considering the complete series µm. 119 Acknowledgments Esta tesis representa la culminaci´on del trabajo realizado en los ´ultimos cuatro a˜nos. Durante estos a˜nos han sido muchas las personas con las que me he cruzado y quienes, bien a nivel acad´emico, bien a nivel personal han contribuido a que este trabajo llegase a buen puerto. A todos ellos quisiera mostrar aqu´ı mi m´as sincero agradecimiento. Espero no dejarme a nadie en el tintero, pero por si eso ocurriera pido disculpas de antemano. En primer lugar, y como no podr´ıa ser de otra manera, quisiera agradecier a mis directores de tesis Carlos G´omez Reino y Carlos R. Fern´andez-Pousa, el haberme dado la oportunidad de incorporame a su grupo de investigaci´on, su continua atenci´on y seguimiento, as´ı como las inumerables palabras de apoyo, afecto y animo, que me han brindado a lo largo de estos a˜nos. Sin su ayuda la presentaci´on hoy de esta tesis no hubiera sido posible. Quisiera adem´as hacer extensivo mi agradecimiento a todos los miembros del grupo de ´ Optica GRIN, Mariv´ı, Carmen, Maite, Antonio, Dani y Miguel, por la ayuda prestada y por el distendido y agradable ambiente de trabajo que entre todos generan. Por supuesto no me olvido de mis compa˜neros de despacho, tanto de los que ya no est´an, en especial Jorge y Justo (¡os echamos de menos!) como de los que se han ido incorporando, Paula y la ”reci´en” llegada Bea. Con ellos he compartido mucho m´as que horas de trabajo entre las mismas cuatro paredes, y si bien, en los ´ultimos meses de redacci´on de la tesis hubiese pagado por un tranquilo y silencioso despacho para mi solita, nada hubiese sido lo mismo sin ellos. I also would like to thank to Hans Peter Herzig for giving me the opportunity to join the Applied Optics group of the IMT in Neuchatel. Thanks to Omar Manzardo for supervising my work and for all beers shared, and in general all the members of the Applied Optics group. Voglio ringraziare Piero, per tutto l’aiuto che mi ha dato nel lavoro svolto in laboratorio, ma soprattutto perch`e grazie a lui conservo uno splendido ricordo dei mesi trascorsi a Neuchˆatel. Un sentito ringraziamento anche 121 122 Acknowledgments a tutti i suoi amici che fin dal primo giorno mi hanno accolto benissimo. Tambi´en quisera recordar aqu´ı a toda la gente estupenda que he tenido la suerte de conocer durante estos a˜nos. En especial quisiera nombrar a Luc´ıa, Javi, Xoel, Sara e Iria quienes, a nivel personal, han sido para mi un gran apoyo, y con quienes he compartido tan buenos momentos. Por ´ultimo y no por ello menos importante tengo que agradecer a mis padres todo el cari˜no, la confianza, el apoyo me han mostrado simpre, especialmente en mis momentos m´as bajos. !Gracias! Publications Papers •L. Chantada, C.R. Fern´andez-Pousa, M.T. Flores-Arias, C. Bao, M.V. P´erez and C. G´omez-Reino, ”Stochastic description of timing jitter in temporal Talbot effect: variance and spectral power”, ´ Optica pura y aplicada,37, 11-16 (2004). •C.R. Fern´andez-Pousa, F. Mateos, L. Chantada, M.T. Flores-Arias, C. Bao, M. V. P´erez and C. G´omez-Reino, ”Broadband noise filtering in random sequences of coherent pulses using Temporal Talbot effect”, J. Opt. Soc. Am. B 21, 914-922 (2004). This article has been selected for the May 2004 issue of Virtual Journal of Ultrafast Science.(http://www.vjultrafast.org.) •C.R. Fern´andez-Pousa, F. Mateos, L. Chantada, M.T. Flores- Arias, C. Bao, M.V. P´erez and Carlos G´omez-Reino, ”Timing jitter smoothing by Talbot effect. I Variance”, J. Opt. Soc. Am. B 21, 1170-1177 (2004). This article has been selected for the June 2004 issue of Virtual Journal of Ultrafast Science (http://www.vjultrafast.org.) •C.R. Fern´andez-Pousa, F. Mateos, L. Chantada, M.T. Flores-Arias, C. Bao, M.V. P´erez and C. G´omez-Reino, ”Timing jitter smoothing by Talbot effect. II Intensity Spectrum”, J. Opt. Soc. Am. B 22, 753-763 (2005). This article has been selected for the May 2005 issue of Virtual Journal of Ultrafast Science (http://www.vjultrafast.org.). •L. Chantada, C.R. Fern´andez-Pousa and C. G´omez-Reino, ”Spectral analysis of temporal self-imaging in fiber dispersive lines”, J. 123 124 Publications Lightwave Technol. 24, 2015-2025 (2006). •L. Chantada, C.R. Fern´andez-Pousa and C. G´omez-Reino, ”Theory of the partially coherent temporal Talbot effect”, Opt. Commun. in press (2006). •L. Chantada, C.R. Fern´andez-Pousa and C. G´omez-Reino, ”Matrix theory and entropy of the partially coherent temporal Talbot effect”, submitted to J. Mod. Opt. in press (2006) Congress Contributions •L. Chantada, C.R. Fern´andez-Pousa, M.T. Flores-Arias, C. Bao, M.V. P´erez , ”Descripci´on estoc´astica del efecto Talbot temporal: Varianza y Potencia espectral”, 7th National meeting on Optics, Santander, Spain, September 8-11 (2003), page 64. •L. Chantada, C.R. Fern´andez-Pousa, M.T. Flores-Arias, C. Bao and C. G´omez-Reino, ”Broadband noise filtering in random sequences of coherent pulses using the temporal Talbot effect”, MOC’04, Jena, Germany, September 1-3, (2004) page 54. •L. Chantada, C.R. Fern´andez-Pousa and C. G´omez-Reino, ”Exact RF intensity Spectrum of OOK trains of coherent pulses after linear propagation under first and second GVD” Optical architectures for RF signal processing and signal mixing, Panticosa, Spain, September 27-29, (2004). •L. Chantada, C.R. Fern´andez-Pousa and C. G´omez-Reino, ” Intensity spectral analysis of the temporal Talbot effect in a second-order dispersive line”, 4th Spanish meeting on optoelectronics 13-15 july (2005), Elche (Spain), TFO4-195-200. •L. Chantada, C.R. Fern´andez-Pousa and C. 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