1. Introduction

With the advent of ultrashort optical sources, there is an increasing demand to monitor the various impairments that potentially degrade the quality of optical signals. Therefore, the field of optical performance monitoring has experienced those latest years a growing development with, for example, a wide range of techniques now available to quantify the deleterious consequences of pulse transmission in telecommunication links [1]. But it can be also of a great interest to develop reliable and accurate tools enabling the characterization of highly stable sources. Indeed, the practical measurement of low jitters still remains very challenging: the main issue is to design a device able to characterize relative fluctuations as low as a few percents. For such a reduced level of fluctuations, the noise contribution of the electronic detection chain should not be neglected as it may translate into serious artifacts. In this context, several dedicated methods have been proposed to evaluate the root mean square (rms) values of the timing and amplitude fluctuations: assuming an adequate mathematical treatment, those data can be extracted from the optical second harmonic autocorrelation function [2,3] or from the analysis of the harmonics constituting the electrical radio frequency (RF) spectrum of the periodic signal [4,5]. However, neither method can be used for any arbitrary repetition rate. Indeed, due to the optical delay that has to be induced, autocorrelation measurements are in practice restricted to repetition rates above ten gigahertz. On the contrary, for RF measurements, due to the high number of harmonics to be analyzed, repetition rate is strongly limited by the bandwidth of the detector and the associated electronics.

We report here an easy-to-implement technique that is suitable to any repetition rate and that is fully compatible with widely spread photodiodes, oscilloscopes and communication signal analyzer softwares. Our solution relies on a nonlinear fiber-based device that enables the optical magnification of the fluctuations in such a way that they become easily and reliably detectable by usual electronics. At first, we shall describe the experimental set-up and the principle of the optical amplitude jitter magnifier. We then present a set of experimental results assessing the performance of the proposed device at telecommunication wavelengths. A magnification by a factor ten is achieved and more remarkably, the probability distribution of the fluctuations is conserved and monitored.

2. Experimental set-up

The configuration we investigate is sketched in Fig. 1 . The experimental set-up relies exclusively on commercially available components commonly used for optical telecommunications. A low jitter fiber laser source is actively mode-locked by a 10-GHz RF signal and delivers Fourier-transform-limited picosecond pulses (2.4 ps). An intensity optical modulator driven by an arbitrary waveform generator artificially degrades the high quality pulse train. The resulting signal under test (SUT) is thus affected by a controlled level of amplitude jitter and is sent into an erbium-doped fiber amplifier (EDFA) with a moderate output average power of 14mW.

 

Fig. 1 Experimental set-up

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The amplified pulse train then propagates into a highly nonlinear fiber (HNLF). This fiber is 920 meter long and presents a low anomalous dispersion coefficient (D = 0.7 ps/km/nm) combined with a high value of nonlinearity (10 /W/km). The linear losses and dispersion slope are both reduced (0.5 dB/km and 0.01 ps².km/nm, respectively). The nonlinearity of the fiber leads to a spectral expansion of the pulses due to self-phase modulation. At the output of the fiber, an optical bandpass filter (OBPF) made of a fiber Bragg grating associated with an optical circulator (OC) is used to carve into the expanded spectrum. The OBPF is fixed at the same central wavelength as the incoming signal, has a Gaussian spectral intensity profile and a full spectral width at half maximum of 200 GHz The resulting signal is detected by means of a fast photodetector combined with a high speed sampling oscilloscope (50 GHz bandwidth).

3. Principle of operation and design guidelines

The key element in this set-up is the optical transfer function (TF) that links the input and output peak-powers. According to the choice of parameters of the fiber, the TF shape may significantly vary and three regimes have been previously isolated [6]: a regime where the TF is monotonously increasing (regime C), a regime where the TF is characterized by a rise and fall evolution (regime A) and a regime in-between where the TF exhibits a large plateau with an inflexion point (regime B). In most of the previously studied configurations, the goal of similar architectures was not to increase optical fluctuations but, to the opposite, to limit output pulse train jitter in a regeneration context so that the latest regime was favored [6–9].

On the contrary, the application we target here dictates the use of a TF presenting a very pronounced slope as illustrated in Fig. 2(a) , typical of very pronounced A-regime. Whereas the achievement of B-regime requires a very precise balance between the linear and non-linear properties of the fiber under use [6], we have found from systematic numerical simulations based on the standard nonlinear Schrödinger equation that it was rather easy to work in the expected A-regime. The main condition to be fulfilled deals with the length of the fiber, which has to be much shorter than the typical dispersive length [6,10]. This leads to a large tolerance on the experimental choice of the fiber parameters. Let us note however, the following general trend, shorter the fiber is, higher the working power of the device should be, which qualitatively maintains a constant nonlinear phase shift. Concerning the OBPF properties, we have checked that it was not a crucial parameter and that similar features can be observed as long as the spectral width of the OBPF is equal to or shorter than the initial pulse spectrum.

 

Fig. 2 (a) TF of the device and associated parameters. Inset: optical spectrum at the local minimum of the TF before the OBPF (black line) compared with the OBPF (green line). (b) Pulse train with 1% initial relative amplitude jitter: SUT (b1) is compared to the pulse train obtained after optical magnification (b2). (c) Comparison of the TF based on the peak-powers of the incoming and outcoming pulses (red curve) with the TF relying on pulse energy (blue curve) or average pulse energy when the train exhibits 5% relative fluctuations (dotted black line). (d) Probability distribution of the output parameter (peak-power: red ; pulse energy: blue) compared to the distribution of the initial peak-powers magnified by a factor 10 (dotted black line). The initial distribution of the peak powers is plotted in solid black-line.

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Regarding the shape of the TF, for the range of powers under consideration, the TF should be ideally monotonous and should be approximated over a wide part by its tangent. More precisely, if P_C is the abscissa at the origin of the tangent to the curve for a working power P_E0, it can be easily shown that the fluctuations of the pulse train are amplified if the simple condition 0 < P_C < 2 P_E0 is fulfilled. A magnification factor of the device can be defined as the ratio of the relative fluctuations of the outcoming signal over the relative fluctuations of the SUT, i.e. (σ_S/P_S0) / (σ_E/P_E0) with σ_E, σ_S being the standard deviations of the input and output pulse streams respectively.

In order to more concretely illustrate the operating principle, let us consider a working power located in the middle of second rising segment of the TF (Fig. 2(a)). The tangent to the TF accurately describes the TF over a wide range. It is interesting to notice that for such a configuration the power P_C physically corresponds to the power where the output optical spectrum splits into two lobes, leading to little energy remaining in the central filtered part (inset Fig. 2(a)). Remark that such a property has been beneficial in the past in the framework of optical analog to digital conversion [11]. Numerical simulations reported in Fig. 2(b) clearly illustrate how the initial pulse train with very low power fluctuations (Fig. 2(b1)) is converted into a train with much more noticeable fluctuations (Fig. 2(b2)).

Ideally, the proper analysis of the peak-power of the pulses requires to resolve the ultrashort pulses, which needs onerous devices. But from numerical simulations presented in Fig. 2(c), we can make out that this difficulty can be overcome by considering the energy of the output pulses instead of their peak power: as the nonlinear scheme does not induce significant fluctuations of the output temporal duration, nearly identical transfer functions will be recorded. We have also checked that considering the average energy of a pulse train with a reduced level of fluctuations does not induce any artificial distortions of the TF.

Finally, we compare in Fig. 2(d) the statistical distributions of the initial pulse train to the outcoming optical magnified one: superposition of the probability distributions based on output peak-power (red curve), on output energy (blue curve) and on the initial peak-power distribution magnified by a factor 10 (black dotted curve) demonstrates the ability of the device to stretch the fluctuations and to maintain the general shape.

4. Experimental results

Transfer function and magnification factor

Figure 3(a) reports the experimental TF (solid black circles). Quantitative agreement with the numerical simulations is remarkably achieved. Systematic measurements of the output amplitude jitter according to the input pulse energy (Fig. 3(b)) are also presented. In more details, between 0 and 0.5 pJ, due to the proportional nature of the section of the TF (in other words, P_C = 0 so that the required condition of jitter amplification is not fulfilled), the measurement gives a value close to one. Between 0.5 and 0.8 pJ, a small decrease of the fluctuations can be noticed, which can be explained by the previously mentioned optical limiting properties of the device [6]. By increasing initial average power (0.8-1.2 pJ), a significant increase of the jitter is detected. However, given the negative slope of the curve (corresponding to P_E0 < P_C < 2 P_E0), the resulting histogram will suffer from an inversion so that this range of powers is not the best suited for an accurate monitoring of the statistical properties. It is indeed more convenient to work with higher powers, typically between 1.3 and 1.5 pJ where the TF is monotonously increasing. Therefore, by positioning adequately the working power P_E0 compared to the corresponding threshold P_C, the condition 0 < P_C < P_E0 ensures both magnification of the amplitude jitter and the conservation of the distribution. Subplots 3(a1) and 3(a2) illustrate the benefits of the device: whereas the analysis of the SUT is severely compromised by electronic noise, after optical magnification by a factor slightly above 10, the fluctuations become clearly distinct and can be straightforwardly recorded.

 

Fig. 3 (a) Experimental transfer function of the device (black circles) compared to the numerical simulations (blue line). Subplots (a1) and (a2) illustrate the statistical distribution of the peak powers of the input and output pulse trains respectively (subplot (a1) is partly impaired by the electronic noise contribution). (b) Evolution of the magnification factor according to the working average power (an initial jitter of 4 percent is superimposed on the modelocked train). Experimental results (black circles) are compared with the results of numerical simulations (blue line) (c) Relative output jitter versus input jitter (rms values).

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Experimental magnification factors have also been compared to results based on the numerical integration of the standard nonlinear Schrödinger equation: from the simulated TF, the resulting jitter enhancement can be predicted using the simple following formula: 1/(1 - P_C /P_E0) (blue solid line). The excellent agreement that is once again observed confirms that dispersion, Kerr nonlinearity and spectral filtering fully explain the recorded trends.

Complementary experimental results

Complementary measurements are reported in Fig. 3(c). By varying the calibrated level of initial fluctuation that is superimposed over the initial signal, we have checked that the jitter magnification does not depend of the initial level of jitter as long as it remains below 5 percent, which is fully consistent with the purpose of the device. Another very important feature that we have experimentally checked is the ability of the set-up not to distort the statistical distribution of the fluctuations. We have indeed tested various kinds of statistical fluctuations that we have accurately controlled thanks to the electrical arbitrary waveform generator. Preliminary results based on a white Gaussian noise (Figs. 3(a1) and 3 (a2)) have been confirmed with more complex fluctuations such as modulation of the pulse train by low-frequency low-amplitude sinusoidal (Fig. 4(a) ) or triangular (Fig. 4(b)) waves. The resulting histograms exhibit the expected two peak structure or constant distribution respectively. In other words, after calibration, the linear nature of the TF enables not only a convenient measurement of the initial rms amplitude jitter but also preserves the shape of the statistical distribution of the fluctuations. As a consequence, and contrary to techniques detailed in refs [2,4,5], no assumption such as a Gaussian distribution is required.

 

Fig. 4 Signal recorded before (subplots 1) and after (subplots 2) the optical jitter magnifier. SUT may be impaired by the electrical noise contribution of the oscilloscope (leading to a relative amplitude jitter estimated to 1%). Eye-diagrams and associated histograms of a pulse train modulated by a low frequency sinusoidal wave or by a low frequency periodic triangular wave are plotted on subplots a and subplots b, respectively. Results dealing with a 40-GHz pulse train made of four time-interleaved 10 GHz pulse trains are given on subplots (c).

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As a final proof of the very high sensitivity of the device, we present an experiment where four trains at 10 GHz are time-interleaved in order to obtain a 40 GHz pulse train. Time interleaving is achieved through a widely spread optical bit rate multiplier made of optical delay lines and couplers and where a careful balance of the powers of the four channels has to be found. Visual level equalization made with the direct detection on the oscilloscope leads to the record presented on Fig. 4(c1) where no difference in the level of the interleaved channels can be distinguished. After evolution in the optical magnifier (Fig. 4(c2)), differences in the level of the 10 GHz trains is readily apparent, which may facilitate further fine level adjustment.

5. Conclusion

To conclude, we have demonstrated a practical and flexible fiber-based device that provides an all-optical magnification of the amplitude jitter by one order of magnitude. The quasi-instantaneous nonlinear Kerr response of silica facilitates the measurement of the jitter level of stable ultrafast pulse trains and also provides an easy way to get access to the associated statistical distribution of pulse trains exhibiting low fluctuations. Potential applications are not restricted to optical telecommunications: the proposed optical amplitude jitter magnifier may become essential for the design and performance analysis of all the applications based on a highly stable pulse sources.

Our set-up relies on an anomalous highly nonlinear fiber combined with central bandpass spectral filtering. However, other fiber based schemes could also provide alternative such as normally dispersive fibers with spectral offset filtering [10,12]. Nevertheless let us point out that the involved physical process should be dependant on the peak-power of the pulse. Consequently, schemes such as nonlinear optical loop mirrors [13] where the instantaneous power is taken into account are not suitable [14]. We may also anticipate that the proposed technique will soon benefit from the recent progresses in highly nonlinear waveguides [3,15] and photonic integration that will ultimately offer compact and low cost approaches.