Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers

Cross-phase modulation (XPM) of a frequency comb (finite-duration optical pulse sequence) by an intense, long Gaussian pump pulse is theoretically investigated, and new effects, namely, frequency-domain self-imaging phenomena (integer and fractional Talbot effects), are reported. The conditions favorable for observing spectral self-imaging phenomena by XPM are derived and numerically confirmed. The effects of nonidealities in a practical experiment (e.g., group-delay walk-off and dispersion) are also evaluated. One can use spectral self-imaging to tune the free spectral range of a frequency comb (without affecting the shape and bandwidth of the individual passbands) simply by adjusting the pump power in a fiber XPM scheme.     

空-频域中自成像分析和解释

基于魏格纳变换和魏格纳分布函数,分析讨论一维物体在空域-频域空间的自成像及其形成过程。从成像过程中各衍射频谱分量的光程差出发,给出Talbot效应和Montgomery效应的统一解释。对于周期物的Talbot效应,得到了用杨氏双缝干涉解释自成像现象的理论依据。周期物的自成像是物平面上间距为2倍周期,光程差为波长的整数平方倍的各衍射频谱分量同相相干迭加的结果。Montgomery效应是物平面上间距为抛物线关系,光程差为波长整数倍的各衍射频谱分量同相相干迭加的结果。     

Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings

We show that a temporal effect that is equivalent to the spatial self-imaging (Talbot) effect applies to the reflection of periodic signals from linearly chirped fiber gratings. The effect can be used for multiplying the repetition frequency of a given periodic pulse train without distorting the individual pulse characteristics. The practical limit on the frequency-multiplication factor depends only on the temporal width of the individual pulse. Thus we demonstrate that a suitable combination of well-known techniques for short-pulse generation, such as pulse mode locking, and the technique proposed here allows us to obtain short-pulse trains with ultrahigh repetition rates (in the terahertz regime). Results from simulations show good agreement with those predicted by theory.     

Nonlinear Talbot self-healing in periodically poled LiNbO_3 crystal [Invited]

The nonlinear Talbot effect is a near-field nonlinear diffraction phenomenon in which the self-imaging of periodic objects is formed by the second harmonics of the incident laser beam. We demonstrate the first, to the best of our knowledge, example of nonlinear Talbot self-healing, i.e., the capability of creating defect-free images from faulty nonlinear optical structures. In particular, we employ the tightly focused femtosecond infrared optical pulses to fabricate Li Nb O_3 nonlinear photonic crystals and show that the defects in the form of the missing points of two-dimensional square and hexagonal periodic structures are restored in the second harmonic images at the first nonlinear Talbot plane. The observed nonlinear Talbot self-healing opens up new possibilities for defect-tolerant optical lithography and printing.     

WDM compatible and electrically tunable SPE-OCDMA system based on the temporal self-imaging effect

A coding/decoding setup for a spectral phase encoding optical code-division multiple access (SPE-OCDMA) system has been developed. The proposal is based on the temporal self-imaging effect and the use of an easily tunable electro-optic phase modulator to achieve line-by-line coding of the transmitted signal, thus assuring compatibility with WDM techniques. Modulation of the code is performed at the same rate as the data, avoiding the use of high-bandwidth electro-optic modulators. As proof of concept of the technique, experimental results are presented for a back-to-back coder/decoder setup transmitting a 10 GHz unmodulated optical pulse train within an 80 GHz optical window and using 8-chip Hadamard codes.     

Widely tunable optical delay generator

We propose and demonstrate a method for quasi storage of light based on periodic spectral filtering realized in the time domain by amplitude modulation using frequency-to-time conversion. The delay can be tuned in a wide range by changing the frequency of an electrical modulation signal. In our experiments, the delay of single 2.5 ps pulses varied by 66 pulse widths. The technique works equally well for more complex optical data packets. Contrary to known approaches, the method has a very large spectral bandwidth and can be implemented by either fiber or integrated solutions using existing technologies. Because of the large bandwidth, fractional delays up to several tens of thousands of pulse widths can be achieved potentially for subpicosecond pulses, which is a tremendous value regarding the implementation simplicity.     

自成像魏格纳函数分析

在空域-频域空间,基于魏格纳变换和魏格纳分布函数,分析讨论了一维物体的自成像及其形成过程.从成像过程中各衍射频谱分量的光程差,给出了Talbot效应和Montgomery效应的统一解释.对于周期物的Talbot效应,得到了用杨氏双缝干涉解释自成像现象的理论依据.周期物的自成像是物平面上间距为两倍周期、光程差为波长的整数平方倍的各衍射频谱分量同相相干迭加的结果.Montgomery效应是物平面上间距为抛物线关系、光程差为波长整数倍的各衍射频谱分量同相相干迭加的结果.     

Aberrations of self-imaged patterns with two-dimensional periodicity

We develop a geometrical theory of aberration in the self-imaged patterns with two-dimensional periodicity. The patterns are considered to be a crossed-line grating or a periodic array of finite apertures. We first derive the raytracing equations for determining the optical path of a self-imaging ray. We then find the third- and fifth-order contributions to the wavefront aberration which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We also derive the expression of the ray aberration from the wavefront aberration. The ray aberration is classified into five distinct types by analogy with the cases in a refracting lens system. We show that the overall ray aberration is entirely undercorrected and the aberrated image patch is decentered from an ideal image point in the direction parallel to the direction tangent vector of a chief ray. The image evaluation technique discussed here will be useful in various applications related to self-image formation of two-dimensionally periodic patterns.     

Geometrical aberrations of self-imaged line gratings

We analyze the properties of a self-imaging system from the point of view of aberration theory. We examine analytically and numerically the geometrical aberrations that are observed in the self-image of a parallel-line grating. We first derive the raytracing equations for determining the optical path of a self-imaging ray with the order of diffraction l. We then obtain the third- and fifth-order contributions to ray aberrations which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We show that the overall ray aberrations are entirely undercorrected. The ray aberrations approach zero as the ratio of the grating constant to the wavelength of light becomes large enough. In a case of unit magnification, no curvatures are observed in the self-imaged lines. If the magnification is bigger than unity, the light rays passing through the point in a positive or negative domain of the aperture variable contribute to the formation of the curved images. The image evaluation technique discussed here can be useful in the various applications related to the self-image formation of a parallel-line grating and it can also provide insight into the self-imaging of other periodic objects.     

Frequency shifting of microwave signals by use of a general temporal self-imaging (Talbot) effect in optical fibers

A simple and practical microwave frequency-shifting technique based on a general temporal self-imaging (GTSI) effect in optical fiber is proposed, formulated, and experimentally demonstrated. The proposed technique can be applied to an arbitrary periodic microwave signal (e.g., a microwave tone) and provides unparalleled design flexibility to increase the frequency of the input microwave signal up to the desired value (limited only by the photodetector's bandwidth). For instance, we demonstrate frequency upshifting of microwave tones from approximately 10 to approximately 50 GHz and from approximately 40 to approximately 354 GHz. These results also represent what is to the authors' knowledge the first experimental observation of GTSI phenomena.     

Ray aberrations of two-dimensional oblique lattices on the self-image plane

We extend the geometrical theory of aberration for a self-imaging system to the case of two-dimensional oblique lattices. In our approach, the fundamental translation vectors of the lattice are not restricted in both length and orientation. Evaluating the disturbance of light through the oblique lattice under coherent illumination, we find the conditions of constraint which limit the self-imaging of the oblique lattice. Various types of oblique lattices are shown to obey the self-imaging conditions. We derive the equations to trace the optical paths of self-imaging rays and then analyze the ray aberrations which arise from the difference between the optical paths of a self-imaging ray and of the corresponding actual ray. The ray aberrations are shown to disappear when the periods of the lattice are large compared with the wavelength of light. We find that the ray aberrations carried by self-imaged oblique lattices are totally undercorrected and the aberrated image patches are displaced along the direction tangent vector of a chief ray.     

Unveiling Talbot Effect under Fresnel Diffraction at a Fork-Shaped Grating

The near-field effect of diffraction image self-reproduction or self-imaging of a periodic grating illuminated by quasi-monochromatic wave is well-known as the Talbot effect. Introducing a dislocation to a periodic structure provides a fork-shaped modulation of the phase/amplitude, which produces discrete diffraction pattern in a far-field consisting of optical vortices. In this paper, Fresnel diffraction at amplitude fork-shaped grating is theoretically and experimentally studied. The coexistence of spatial ordering and local violation of translational symmetry of the structure manifests itself in a strict diffraction pattern consisting of optical vortices in the far-field, which is shown to be accompanied by formation of a spatially ordered intensity distribution in the near-field, reminiscent the Talbot carpets for periodic structures. These results demonstrate the first evidence of Talbot effect occurred under light diffraction at fork-shaped gratings, being promising for deep understanding of near-field singular optics phenomena.     

Self-image (autoidolon) techniques for the realisation of optical computing type operations

Self-images of a number of spatially periodic (e.g., gratings) and quasiperiodic (e.g., halftone picture) objects have been systematically studied. These studies indicate that self-imaging techniques could be useful in optical computing type operations. It is also established that the Rayleigh relation in the context of self-imaging is quantitatively in error.     

Minimization of dispersion in an ultrafast chirped pulse amplifier using adaptive learning

Minimizing residual frequency dispersion that accompanies pulse stretching, amplification, and recompression is an important consideration in ultrashort chirped-pulse amplifiers. Here we show how an adaptive learning algorithm can be used in conjunction with a pulse shaper to compensate for higher-order and nonlinear dispersion in a chirped-pulse amplifier. Using spectral blueshifting as a sensitive diagnostic for pulse shape, we implement a 'learning loop' comprised of the pulse shaper, strong field laser ionization, and a genetic algorithm to minimize dispersion through the amplifier. We verify our optimization results using frequency-resolved optical gating (FROG) measurements and also show theoretically and experimentally that spectral blueshifting is indeed a sensitive diagnostic for pulse shape, and specifically, for higher-order dispersion.     

Phase interpretation for polarization-dependent near-field images of high-density gratings

It has been described that the near-field images of a high-density grating at the half self-imaging distance could be different for TE and TM polarization states. We propose that the phases of the diffraction orders play an important role in such polarization dependence. The view is verified through the coincidence of the numerical result of finite-difference time-domain method and the reconstructed results from the rigorous coupled-wave analysis. Field distributions of TE and TM polarizations are given numerically for a grating with period d = 2.3λ, which are verified through experiments with the scanning near-field optical microscopy technique. The concept of phase interpretation not only explains the polarization dependence at the half self-imaging distance of gratings with a physical view, but also, it could be widely used to describe the near-field diffraction of a variety of periodic diffractive optical elements whose feature size comparable to the wavelength.     

Multistage Fiber Amplifier with Spectral Compression Elements for High-Energy Laser Pulse Generation

We consider general principles of spectral compression of parabolic pulses in nonlinear optical fibers. It is known that the variance analysis provides a way to describe correctly the evolution of the pulse parameters under the spectral compression. We propose a computational model of multistage amplifier with spectral compression segments incorporated. The model provides the possibility to amplify middle-power input pulses to obtain pulse energies of hundreds of nanojoules with such output spectral characteristics that provide the possibility for further effective enhancement. We also discuss a modification of the model that provides the formation of a parabolic envelope and linear frequency modulation of the output pulse.     

The generation of terahertz radiation via optical rectification in the self-induced transparency regime

The possibility of efficient terahertz generation via optical rectification in electro-optic crystal doped with the two-level resonant impurities is demonstrated. Under conditions of the self-induced transparency regime the laser pulses slows down to achieve the phase matching condition. The terahertz generation is accompanied by periodic modulation and spectral deformation of the laser pulse. The most efficient generation occurs under condition of small negative detuning from the resonant frequency of the two-level system.     

Analysis of optical rib self-imaging multimode interference (MMI) waveguide devices using the discrete spectral index method

We have developed a semianalytical approach to study the propagation characteristics of multimode interference (MMI) waveguide structures. The effect of transverse crosssection geometry on self-imaging length and optical power throughput are investigated. We present results confirming that the nonquadratic modal dispersion encountered in stripe-loaded waveguides impairs self-imaging performance. Results indicate that the image degradation is more predominant for asymmetrically fed MMI waveguides.     

Self-imaging and modulational instability in an array of periodically curved waveguides

Linear and nonlinear light propagation in an array of waveguides with a periodically bent axis is theoretically investigated. In the linear propagation regime, it is shown that a self-imaging effect at periodic planes may occur, a phenomenon analogous to that of dynamic localization observed when an electron in a periodic potential is subjected to an ac field. In the nonlinear propagation regime, it is shown that periodic waveguide bending under the self-imaging condition inhibits the phenomenon of discrete modulational instability.     

High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb

We demonstrate high resolution coherent control of cold atomic rubidium utilizing spectral phase manipulation of a femtosecond optical frequency comb. Transient coherent accumulation is directly manifested by the enhancement of signal amplitude and spectral resolution via the pulse number. The combination of frequency comb technology and spectral phase manipulation enables coherent control techniques to enter a new regime with natural linewidth resolution.