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Abstract
We introduce and experimentally demonstrate a new design for passive Talbot amplification of repetitive optical waveforms, in which the gain factor can be electrically reconfigurable. The amplifier setup is composed of an electro-optic phase modulator followed by an optical dispersive medium. In contrast to conventional Talbot amplification, here we achieve different amplification factors by using combinations of fixed dispersion and programmable temporal phase modulation. To validate the new design, we experimentally show tunable, passive amplification of picosecond optical pulses with gain factors from m = 2 to 30 using a fixed dispersive line (a linearly chirped fiber Bragg grating).
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Amplification of signal intensity, including short optical pulses, is crucial for initiating physical processes, sensing, communications, measurement, and information processing [1, 2]. In traditional amplification methods, external power sources are conventionally used to multiply incoming signal carriers in order to deliver a higher intensity signal at the output through an active gain process [3, 4]. In contrast, passive amplifiers have been shown to amplify repetitive input waveforms (e.g. optical pulses) without using active gain through coherent waveform addition [5]. Here, a repetitive input waveform train is stored and coherently added to itself in a high-finesse cavity to build up fewer, replica, amplified waveforms at the output. The new waveforms are strengthened by the amount of repetition rate reduction. Nevertheless, this method requires ultra–precise active phase control of the input signal envelope and carrier, which is often impractical outside of a laboratory environment. Recently, we reported on a simple, all-fiber passive amplification technique for repetitive waveforms without using active gain [6], and demonstrated input-to-output gain and temporal noise mitigation of picosecond optical pulses in a Talbot amplifier [7]. Our technique is based on lossless repetition-rate division of the input periodic waveform (pulse) train through a dispersion-induced Talbot effect, inherently leading to intensity amplification of the input individual waveforms. Similar to cavity-based methods, the output waveform is amplified by the rate-division factor (without considering practical insertion losses). Nevertheless, our previous design used a fixed length of dispersive medium for a given gain factor, requiring a change of dispersive device, and therefore a significant modification in the experimental setup, to tune the amplifier gain. As for any amplification approach, the capability of tuning the amount of gain provided by the amplifier is crucial. In amplified systems, unexpected changes in input signal power are often corrected simply by tuning amplifier gain, eliminating the need to physically change, tune, or adapt upstream components. Additionally, adaptive gain and gain-modulation are highly desirable for signal control and processing [8, 9].
Recently, we have introduced a new design for a passive Talbot amplifier, in which the gain factor can be electrically reconfigurable while using a fixed amount of dispersion [10]. In particular, similarly to our previous Talbot amplification design in [6], the amplifier setup is composed by an electro-optic phase modulator followed by an optical dispersive medium. In contrast to this previous work, by generalizing the equations for the required phase modulation and dispersion, we are able to obtain a range of various gain factors using a fixed dispersive element, i.e., a linearly chirped fiber-Bragg grating (LC-FBG) in our experiments. Programming the gain factor is then entirely achieved by the temporal phase modulation step. In this letter, we provide a comprehensive analysis of the theoretical background and design equations for programmable Talbot amplification. We further verify our model by amplifying repetitive, picosecond optical pulses generated from a pulsed fiber laser with tunable gain factors ranging from m = 2 to 30 using a fixed dispersive element. Specifically, the new design of the Talbot amplifier is demonstrated through successful proof-of-concept experiments, amplifying repetitive, picosecond optical pulses generated from a pulsed fiber laser with tunable gain factors ranging from m = 2 to 30.
2. Principle of operation
Figure 1 shows the operation principle of our amplification technique. In the standard temporal Talbot effect, a flat-phase repetitive input waveform with repetition period T0 (input signal at z = 0) is self-imaged after dispersive propagation through every integer Talbot distances (q + 1)zT, where q = 0, 1, 2, … is a free design parameter. This can be seen on both the Talbot carpets 1 (a) and 1 (b) where inputs at repetition periods T´ and T´´ show self-images every Talbot distance zT´ and zT´´, respectively. Recall that the fundamental integer Talbot distance depends on the input rate period and dispersion parameteras follows (e.g., for the top carpet in Fig. 1(a)): zT´ = T´2/(2πβ2), where β2 is the amount of second-order dispersion. There also exists an infinite amount of fractional distances, defined by the “Talbot Carpet” [11–15], where one can observe rate-multiplied self-images of the input waveform train (fractional Talbot effect), with correspondingly reduced individual waveform intensity. The examples in Fig. 1(a) at Talbot distances zT´/3 and zT´/2, have intensities reduced by the factors by which the repetition rate is multiplied, m = 3 and m = 2, respectively. Notice that in general, an input repetition rate 1/T0 is multiplied by a factor m ( = 2, 3, 4, …) at any fractional Talbot distance (s/m)zT, where s is any integer coprime with m. In an integer Talbot self-image, the uniform phase profile of the input is restored, as can be seen e.g., in the self-imaged pulses at a distance zT´ in the top carpet (phase profiles represented by the dashed blue lines). However, the rate-multiplied self-images are affected by deterministic pulse-to-pulse residual phase, such as those observed at zT´/3 and zT´/2.
Fig. 1 Concept for programmable Talbot amplification: (a) Talbot carpet when the input repetition period T´ = 3T (b) Talbot carpet when the input repetition period T´´=2T = 2T. Pulse trains with repetition period T, at z zT´/3 and z = zT´´/2, result in different amplification factors using the same amount of dispersive delay (purple shading). The dashed blue lines represent the phase profiles of each of the temporal pulse trains in the carpet.
In previous demonstrations of Talbot amplification [6], we precondition an input pulse train to mimic the repetition rate and temporal phase of rate-multiplied self-images, such as those at zT´/3 and zT´/2. By propagating these pulse trains to the nearest self-image distance zT´, we amplify the pulse train intensity by the amount of repetition rate reduction. In the cases where preconditioned inputs are made to match the pulse train at zT´/3 and zT´/2, amplification factors of the output train are m = 3 and m = 2, respectively. Figure 2 shows how a fiber-integrated electro-optic phase modulator can be used to electronically program the desired phase profile for intensity amplification. However, in our previous design [6], each amplification factor requires a new dispersion value, typically implemented by physically changing the dispersive element (spool of fiber, fiber Bragg grating, etc.). As illustrated in Fig. 2, our new design enables tuning the amplification factor by simply changing the temporal phase modulation profile, without modifying the dispersion amount. Figure 1 shows the concept of achieving programmable amplification factors using the same dispersive delay. The idea can be understood through a comparison of the two different Talbot carpets shown in Fig. 1(a) and Fig. 1(b), each corresponding to a different input repetition period, T´=3T and T´´=2T. Both carpets show the same repetition rate, T, at fractional Talbot distance zT´/3, and zT´´/2, respectively. If we precondition an input pulse train to match the waveforms at these fractional distances (same period, T, but different residual temporal phase), propagation through the same amount of dispersive delay (purple shading) will result in a flat-phase output pulse train with amplification factors m = 3 and m = 2, respectively. In this example, the same dispersive propagation distance required for both amplification factors is 2zT´/3=3zT´´/2=T´2/(3πβ2).. Notice that to ensure the same propagation length in the two cases, we select the first integer Talbot plane (q = 0) in the top carpet but the second integer Talbot plane (q = 1) in the bottom carpet.
Fig. 2 Method for phase preconditioning pulse trains for dispersion-induced temporal Talbot amplification. PM: phase modulator. DM: dispersive medium.
This example provides the needed guidelines to obtain the generalized equations for programmable Talbot amplification of an input waveform train, with a repetition period equal to T, by a gain factor of m ( = 2, 3, 4, …). For our derivations, we assume the general input plane location in the Talbot carpet to be at (s/m)zT, where zT is the fundamental integer Talbot distance corresponding to an input period equal to m times the period T, (T0 = m × T), and we recall that s is an integer and co-prime with m. Similarly, the general output plane location is at (q + 1)zT, with q = 0, 1, 2, … In general, in order to passively amplify a waveform train under the stated conditions, the required amount of dispersion can be calculated as that corresponding to propagation from the above-defined input location to the output in the Talbot carpet, namely
The input waveform train should be phase modulated according to the Talbot phase modulation function corresponding to the input location (s/m)zT in the carpet, namely [16]when (s∙m) is even, andwhen (s·m) is odd, and where [1/a]b is the inverse of a modulo b. In these equations, m = 2, 3, 4, … is the gain factor and n = 0, 1, 2, ... is the pulse number. The phase function in Eq. (2) is known to be periodic with fundamental period equal to m, namely φn = φn + m. Such a temporal phase modulation creates m times as many frequency tones in the spectrum of the input signal, leading to a comb frequency spacing that is reduced by m times with respect to the input, a process that is known as the spectral self-imaging effect [17]. If theses phase shifts are reduced to a 2π range, a periodic sequence of discrete phase steps in the range [0, 2π] is obtained. It is worth mentioning that setting s = m−1 and q = 0 in the generalized equations in Eq. (1) and Eq. (2) gives rise to the conventional dispersion equation (ϕ2 = mT2/2π) and application of the temporal phase-modulation profile (φn = (m−1/m)πn2) previously defined for conventional Talbot amplification, as reported in [6].Let us now consider the situation where we choose a maximum gain factor m = M. The corresponding required dispersion value by setting q = 0 and s = M−1 is ϕ2 = MT2/2π. Indeed, for a prescribed input temporal period T, we can now use the same dispersion value for a wider range of gain factors, satisfying ϕ2 = (qm + m−s)mT2/2π = MT2/2π, according to Eq. (1) above. In this equation, s and q are free design parameters, and as such, we can obtain a number of different gain factors m, being smaller or equal to M (m≤M), that satisfy this condition for a given s and q. The required phase modulation function can be then obtained using Eq. (2) with the corresponding m and s factors.
3. Experimental results
In what follows, we will provide experimental results for demonstration of a passive Talbot amplifier of picosecond optical pulses, providing a programmable set of gain factors. In particular, we demonstrate gain factors of m = 2, 3, 5, 6, 10, 15, and 30 by implementing a design with M = 30, with corresponding s = 1, 2, 4, 1, 7, 13, and 29, and q = 7, 3, 1, 0, 0, 0, and 0, respectively. As illustrated in Fig. 3, we use an actively MLL to generate uniform-phase, ~5-ps (intensity full-width at half maximum, FWHM), Gaussian-like pulses at a wavelength of 1550 nm.
Fig. 3 Experimental setup. MLL: mode-locked laser, BPF: band-pass filter, OA: optical amplifier, OD: optical delay line, PC: polarization controller, PM: phase modulator, LC-FBG: linearly chirped fiber Bragg grating, CLK: radio frequency clock source, AWG: arbitrary waveform generator, RF Amp: radio frequency (RF) amplifier.
In order to minimize the effect of group delay ripples in the LC-FBG used as a dispersive stage, the input pulse bandwidth is reduced to ~0.200 nm (FWHM) with a Gaussian-like band-pass filter, thereby increasing the input pulse width into the Talbot amplifier to ~17 ps (FWHM). The input optical pulse is phase-modulated by a 40-GHz bandwidth electro-optic phase modulator, which is driven by a 7.5-GHz bandwidth electronic AWG. After temporal modulation, pulses are input into the LC-FBG, providing a fixed second-order dispersion with coefficient ϕ2 = 12,926 ps2/rad (~10,134 ps/nm) over the entire input pulse bandwidth. All temporal intensity waveforms are measured by a 500-GHz optical sampling oscilloscope (OSO) and all spectra are measured by an optical spectrum analyzer (OSA) with a resolution of 0.4 pm.
Figure 4 shows experimental results for our system. Because our central aim is to leave the dispersion fixed, we choose the repetition rate of the input pulse train, using Eq. (1), to allow the best possible range of tunable amplification factors. In particular, we select the repetition rate to be 19.2 GHz ( = 1/T = (M/2πϕ2)1/2), for a target maximum amplification factor of M = 30 (enabling the programmable gain factors defined above). Figure 4(a) shows the prescribed electro-optic phase modulation profiles applied to the input optical pulses for the different gain factors demonstrated in the experiment (m = 2, 3, 5, 6, 10, 15, and 30, from top to bottom). The dashed black lines show the ideal prescribed phase profiles, calculated from Eq. (2), and the solid green lines show the actual phase drives delivered by the AWG, captured with a 40-GHz electronic sampling scope. Figure 4(b) shows the measured optical spectra of the input data signal before and after temporal phase modulation, showing the predicted spectral self-imaging effect, leading to the anticipated decrease in the comb frequency spacing of the input signal by the amplification factor of m. The dashed horizontal arrow shows the same spectral span of 0.154 nm, corresponding to the input repetition rate of 19.2 GHz. Figure 4(c) shows the temporal intensity waveforms of the input pulse trains (dashed grey) and the resultant amplified pulse trains (solid red) at the output of the dispersive medium (LC-FBG). Note that in each case, the repetition rate of the input is divided by the amplification gain factor m.
Fig. 4 Experimental results of the demonstrated programmable passive amplifier. (a) Prescribed temporal phase profile. The dashed black lines: Ideal temporal phase profile. The solid green lines: Actual phase profile drive generated from the AWG. (b) Measured optical spectra of the optical pulse trains after temporal phase modulation. Dashed, blue, double-arrow shows a 0.154-nm (19.2-GHz) span. (c) Temporal traces of the input and amplified output pulses (Input: dashed gray, Output: solid red), measured using a 500-GHz sampling oscilloscope.
As one can see in Fig. 5, correspondingly, the output pulses are amplified in intensity according to the predicted gain factor over an output power value that is lower with respect to the input by the insertion loss of the amplifier (phase modulator and LC-FBG), i.e., by ~6 dB. Input-to-output gain is achieved when the passive Talbot gain (m) exceeds the system insertion loss, namely for experimental Talbot gain values of m = 6, 10, 15, and 30. As expected, the output temporal pulses are nearly undistorted copies of the input pulses, with a measured FWHM time-width around 17 ps in all cases. Because our programmable amplifier is based on noiseless repetition-rate division of the input periodic waveform, the signal-to-noise ratio remains unchanged regardless of changing the amplification factor m [6]. In fact, any Talbot-based amplification technique will also show noise reduction in pulse-to-pulse intensity fluctuations and timing-jitter [7].
Fig. 5 Superimposed single temporal pulse waveforms measured at the system output for the different amplification factors achieved in the experiment.
The slight mismatch between input and output pulse shapes can be mainly attributed to the deviation of the actual dispersion value used in experiments from the ideal Talbot condition [18], as well as to the mismatch from the ideal temporal phase modulation drives, see Fig. 4(a). The gain factors that are achieved in the experiments for all cases up to m = 15 are very close to those theoretically expected with a slight deterioration for higher gain values. For instance, the measured gain factor (output to input pulse peak power ratio, after extracting insertion losses) for the case m = 15 is 12.54, just a 17% below the ideal gain value. The gain reduction is more significant in the case m = 30, for which the measured amplification factor is ~18.37. This deterioration in the actual gain with respect to the ideal one is associated with the pedestal that is observed in the output pulses, and it is mainly attributed to the intrinsic time-resolution limitation of the AWG, which fails to reproduce the ideal temporal phase drive for more complicated phase patterns (i.e., for higher values of m). Though the prescribed phase drive shows the same phase values over entire pulse bit-slots, we note that this technique only requires the correct constant phase over the pulse width, providing a high-tolerance to errors in the phase drive. This allows us to use a 7.5-GHz AWG to effectively manipulate a 19.2-GHz signal, pulse-by-pulse. Pulse pedestal can be reduced, thereby increasing energy in the main pulse, by using a higher-bandwidth AWG to better emulate the theoretical phase drive, as well as more carefully matching the temporal Talbot dispersion conditions.
4. Conclusion
In conclusion, we have demonstrated a new simple, flexible design for passive amplification of repetitive pulse waveforms. The key feature of this design is that it allows for tunable gain without physically changing the experimental setup. Unlike previous approaches, here the dispersive medium is fixed and the amplification factor can be programmed electronically by modifying the temporal phase modulation profile according to the analytic equation in Eq. (2). Our proof-of-concept experiments have successfully shown in-fiber amplification of picosecond optical pulses with tunable gain factors in situ from 2 to 30, indicating a robust technique with many potential advantages.
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