Concept and operation principle

Figure 1 illustrates the concept of our passive amplification technique. We effectively exploit an ‘inverse temporal Talbot effect’ in which the final amplified temporal image is recovered from a previous multiplied self-image. As shown in Fig. 1, top, in the standard temporal Talbot effect, a flat-phase repetitive input waveform (signal at z=0) is self-imaged, after dispersive propagation through a distance z T (integer Talbot distances). There also exists an infinite amount of fractional distances, given by the ‘Talbot Carpet’ (a map of all possible coherence revivals resembling a Persian rug in intricate but repeating patterns)20 that give multiplied self-images; see examples at the fractional Talbot distances z T /2 and 2z T /3, Fig. 1, top. Dispersive propagation speeds up and slows down the different frequency components ‘colours’, originally in-phase that make up the waveform train, redistributing the original signal energy into the mentioned different temporal intensity patterns. An integer self-image exhibits the same repetition rate and individual waveform intensity as the input, whereas in the multiplied self-images, the repetition rate is increased, and the individual waveform intensity is correspondingly decreased by an integer factor. The repetition rate-multiplication (intensity division) factors for the multiplied self-images shown in Fig. 1 at z T /2 and 2z T /3 are 2 and 3, respectively. In an integer self-image, the uniform temporal phase profile of the input is restored. However, in the multiplied self-images, such as those observed at distances z T /2 and 2z T /3, there exists a waveform-to-waveform residual temporal phase structure (dashed black). This residual temporal phase represents instances where the waveform field-amplitude has been advanced or delayed in relation to the envelope centre.

Using a multiplied image at a fractional distance as the input instead of the conventional phase-free input at z=0, further dispersive propagation to the distance z T produces an output with an amplified intensity. As shown in Fig. 1 top, a waveform starting at the fractional distance z T /2 will be amplified in intensity by a factor of m=2 at the output z T . Likewise, a waveform starting at the fractional distance 2z T /3 will be amplified by a factor of m=3. This requires the application of a prescribed temporal phase-modulation profile to an input signal to make it appear as though it has already propagated from z=0 through an amount of dispersive delay equivalent to the target multiple image. For example, Fig. 1, bottom, shows how if we condition a typical flat-phase input to look like the waveform train at 2z T /3 by proper temporal phase modulation (dashed black line), subsequent propagation through z T /3 more of dispersive delay will give the output shown at z T , one-third the repetition rate and three times the intensity. Notice that the described processes involve only a suitable manipulation of the input signal temporal and spectral phase profiles, not magnitude, ensuring that the signal energy is ideally preserved. The full Talbot carpet provides an infinite amount of fractional self-image locations and corresponding phase profiles, so that any desired repetition rate division and corresponding amplification factor can be obtained, limited only practically by the control of temporal phase modulation and spectral phase filtering from dispersive propagation.

To passively amplify any arbitrary repetitive input signal by m-times (m=2,3,4,…), a temporal phase of

(where s=m−1) is applied on the n-th temporal pulse (n=0,1,2,…) of the input periodic signal. This is followed by propagation through a first-order dispersive medium, ideally providing a linear group-delay variation as a function of frequency over the signal spectral bandwidth. The dispersive medium should introduce a total dispersion value:

where T is the repetition period of the input pulse train, z A is the length of the dispersive medium and β 2 is the dispersion coefficient, defined as the slope of the group delay as a function of the radial frequency ω, per unit length. Mathematically, β 2 =[∂2β(ω)/∂ω2] ω= ω 0 , where β(ω) is the propagation constant through the medium, that is, τ g (ω)=z A [∂β(ω)/∂ω] ω= ω 0 is the medium’s group delay and ω 0 is the central (carrier) frequency of the considered signal (periodic waveform train). The temporal phase φ n can be assumed to be applied on time slots equal to the pulse repetition rate. In practice, however, it is sufficient to apply the same phase over the pulse duration. Notice that the phase profile φ n is periodic with a fundamental period equal to the gain factor m, namely φ n =φ n+m . If these phase shifts are reduced to a 2π range, a periodic sequence of discrete phase steps in the range [0,2π] is obtained. The example illustrated in Fig. 1 of the main text for m=3 gives a repeating temporal phase profile of {0, 2π/3, 2π/3, 0, 2π/3, 2π/3,…}, where the third phase level corresponding to n=2 has been obtained by taking modulo 2π.

The temporal phase shifts φ n are defined from the Talbot carpet20. These phase shifts induce a spectral self-imaging (Talbot) effect on the modulated pulse train24. In particular, the temporal phase-modulation process produces new frequency components, reducing the frequency spacing of the input signal discrete comb-like spectrum by an integer factor of m. This is consistent with the repetition-rate division (temporal period increase) by a factor m that is subsequently achieved on the temporal pulse train following dispersive propagation.

Experimental demonstration

We experimentally demonstrate passive amplification using optical pulses generated from a standard commercial pulsed fibre laser, which does not incorporate any carrier-envelope phase stabilization mechanism. The pulses from the laser are input directly into a dispersive optical fibre and delivered at the fibre output through conventional integer temporal self-imaging20. Next, by adding a suitable temporal phase through electro-optic modulation to waveforms prior to dispersion, the waveform intensity will be locally amplified according to the amount of repetition rate reduction. Because the absence of temporal phase-modulation produces an integer Talbot self-image of the original pulse train at the fibre output, we are able to compare the cases of passively amplified and unamplified waveforms that have propagated through the same optical system. Any loss from the system will show up in both the passively amplified and unamplified data, allowing us to isolate the effectiveness of passive amplification alone.

Figure 2 shows the prescribed electro-optic phase-modulation profiles to the ~7-ps Gaussian input optical pulses generated by the fibre laser for the cases when we target gain factors of m=2, 5, 15 and 27, respectively. The temporal phase functions are generated from an electronic arbitrary waveform generator (AWG). The solid red lines show the ideal temporal phase profiles, and the dashed blue lines show the actual phase drive delivered by the AWG. Figure 3 presents the optical spectra recorded with a high-resolution (20 MHz) optical spectrum analyzer after temporal phase modulation showing the predicted spectral Talbot effect, leading to the anticipated decrease in the frequency comb spacing by factors of 2, 5, 15 and 27, respectively. Figure 4a shows the experimental results of the demonstrated passive amplification, with gain factors of m=2, 5, 15 and 27, at the output of the dispersive fibre link (total dispersion ~2,650 ps nm−1 in the case of m=2 and m=5, and ~8,000 ps nm−1 for m=15 and m=27). Supplementary Movie 1 shows the case of m=15. In the case of m=15 and m=27, fibre losses from dispersion are significant, ~42 dB from a total of six dispersion-compensating fibre modules, so that we include active amplifiers, Erbium-doped fibre amplifiers (EDFAs), in our system to detect the signal at the end of the span to show proof-of-concept for these higher-amplification factors. This practical limitation can be overcome by using lower-loss dispersive devices, such as chirped fibre gratings for losses as low as 3 dB for the entire dispersive spectral phase employed25. Another low-loss alternative method is spectral line-by-line shaping26, for which the desired spectral phase shifts can be restricted to the range [0,2π].

Figure 2: Experimental-prescribed phase-modulation profiles. Temporal phase-modulation patterns required for amplification factors m=2, 5, 15 and 27, as determined by the Talbot carpet. The dashed blue curves show the experimental traces generated from an electronic arbitrary waveform generator (7.5-GHz analogue bandwidth), as measured by an electrical sampling oscilloscope (40-GHz bandwidth), whereas the solid red curves show the ideal phase traces. For comparison, the experimentally measured electrical traces have been normalized to match the ideal phase profiles. Full size image

Figure 3: Measured optical spectra of the optical pulse trains. Optical spectra of the optical pulse trains recorded with a high-resolution (20-MHz) optical spectrum analyzer after phase modulation with the phase modulator turned off (blue, top plot) and with the phase modulator turned on (red, bottom plot), demonstrating the expected decrease in the frequency comb spacing by factors of (a) m=2, (b) m=5, (c) m=15 and (d) m=27, respectively. Full size image

Figure 4: Experimental demonstration of passive waveform amplification. (a) Optical sampling oscilloscope (500-GHz measurement bandwidth) time trace of pulse trains at the dispersive fibre output before passive amplification (dashed blue, with the phase modulator turned off) and after passive amplification (solid red, with the phase modulator turned on) for the desired amplification factors of m=2, 5, 15 and 27. Experimental passive gain is measured as 2, 4.92, 14.4 and 20.2, respectively, on the optical sampling oscilloscope. (b) Passive amplification of an arbitrary waveform demonstrating the insensitivity of the Talbot method to temporal signal shape. Full size image

The original repetition rates of the input pulse trains in the four reported experiments are 9.7, 15.43, 15.43 and 19.75 GHz, respectively. The reduced rates after passive amplification are 4.85, 3.08, 1.03 and 0.73 GHz, as measured by the radio frequency (RF) spectra of the detected output optical pulse trains, Fig. 5, in excellent agreement with the desired amplification factors. Figure 4a shows the optical sampling oscilloscope (OSO) trace (500-GHz measurement bandwidth) of the pulse train after dispersion in the case with a phase-conditioned input (solid red) and without (dashed blue). For m=2, the pulse train intensity, as measured by the OSO, doubles as predicted. For m =5, 15 and 27 the ideal amplification factors are nearly obtained, 98% of the desired m=5, 96% of the desired m=15 and 75% of the desired m=27. In all cases, the amplified temporal waveforms are nearly undistorted replicas of the unamplified Gaussian pulses. The reduction in the fidelity of the Gaussian spectral envelope (Fig. 3) for amplification factors m=15 and m=27, and the associated slight decrease of the expected gain, is mainly due to the time-resolution limitations of the AWG, which fails to reproduce the ideal temporal phase drive for more complicated phase patterns, Fig. 2. Figure 4b further proves how Talbot passive amplification is achieved without affecting the temporal shape of the input waveform. Here, the laser pulses are reshaped before amplification to exhibit a sinc-like pulse waveform in the time domain. Using the same scheme that was applied to Gaussian pulses for amplification by 15 times, we similarly demonstrate passive amplification of the sinc-like pulses by a factor of ~12.6. The Talbot passive amplification approach can be applied on any arbitrary waveform with no fundamental limitation on the signal frequency bandwidth. It is only practically limited by the spectral bandwidth of the linear region of the particular dispersive medium. In the case of dispersive delay provided by optical fibre as employed in this work, linear dispersion around 1,550 nm greatly exceeds the pulse bandwidth. For other wavelength regions or wave systems, care may need to be taken to ensure a linear chirp provided by dispersive delay.

Figure 5: Experimental verification of repetition rate division for passive amplification. Traces show the RF spectra of the optical pulse trains after photodetection at the dispersive fibre output without passive amplification (dashed blue, with the phase modulator turned off) and with passive amplification (solid red, with the phase modulator turned on) for the desired amplification factors of 2, 5, 15 and 27. Full size image

Analysis of noise performance

The plots in Fig. 6 demonstrate the phenomenon of passive amplification without the injection of intensity noise. In these experiments, no active amplification was used in the dispersive span. The identical match of the spectra, in the presence and absence of passive amplification, particularly at the noise floor, indicates that our technique does not contribute any measurable intensity noise, consistent with what one would expect from a passive system. Using the spectral linear interpolation method, the data show that the optical SNR (OSNR) remains the same in the presence of amplification. To be more concrete, Fig. 6a shows the OSNR, in a 1-nm resolution bandwidth of the optical spectrum analyzer, to be 50 dB without passive amplification, phase modulator (PM)-OFF (dashed blue), and 50 dB with passive amplification, PM-ON (red), indicating a noise figure of 0 dB. Within our ability to measure OSNR, we show there is no injected intensity noise from passive amplification. On the other hand, when an active amplifier (EDFA) is used at the end of the same network instead of passive amplification, the OSNR degrades from 50 to 36 dB, PM-OFF+EDFA (dot-dashed green), due to the injected ASE noise from the active amplifier. Notice that all the spectra traces are normalized to their respective amplitude peaks. The injected noise from active amplification can also be inferred from the time traces shown in Fig. 6b. Although both active and passive amplification amplify the peak of the signal by ~7 dB (corresponding to passive gain of m=5 and the equivalent active gain of 5), the active amplification also raises the average noise floor by 7 dB (dot-dashed green), whereas passive amplification leaves the floor at its original level (red). Figure 6c shows optical oscilloscope traces with zero averaging, clearly confirming an increased fluctuation when active amplification is employed (top trace) as compared with passive amplification (bottom trace).

Figure 6: Experimental verification of noiseless amplification. (a) Optical spectra of the pulse trains measured at the dispersive fibre output before passive amplification (phase modulator turned off, PM-OFF), (dashed blue), after passive amplification (phase modulator turned on, PM-ON) with m=5 (red), and with active amplification (EDFA) with gain of 5 (dot-dashed green). Notice that all the spectra are normalized to their respective amplitude peaks. Measurement settings were fixed to a 1-nm resolution bandwidth for the optical spectrum analyzer. (b,c) Corresponding optical sampling scope time traces in averaging mode and sampling mode (no averaging), respectively. Full size image

Figure 7 shows enhancement of ER as the passive amplification factor m increases for a noisy input with OSNR=10. In this experiment, ASE noise was injected onto the pulse train in a controlled fashion using an EDFA, paired with a variable attenuator, at the input of the phase modulator and placing two more EDFAs in the dispersive span. We define ER enhancement as the ratio of the ER of the Talbot-amplified signal with respect to the ER of the input train, where ER is given by the ratio of the average peak intensity of the waveform to the average intensity of the noise floor. Figure 7a shows simulated data (blue squares) of how the ER scales linearly with m, and experimental data points for m=2, 5 and 15 overlaid (red triangles). The dashed blue line shows the expected linear trend for ER enhancement. Passive amplification increases the waveform peak intensity by m-times, while leaving the noise floor at its average value. After phase modulation, energy is distributed into m-times more frequency tones. After temporal redistribution by dispersive delay, the individual frequency components of the waveform will add coherently to give an average peak power that is m-times the original signal. The noise floor, however, will remain the same. This is shown clearly in the optical sampling scope trace in Fig. 6b, as well as Fig. 7b,c, which show the OSO time traces with passive amplification (red) and without (dashed blue) for m=5 and m=15, respectively. Note that traces in Fig. 7b,c have been normalized to the same peak power rather than the noise floor to better show ER enhancement. So long as the new frequency components generated during phase modulation are the correct ones (dictated by a correct phase drive), the noise floor will have the same average level from the destructive interference as before. Intensity noise present at the top of the waveform signal will also keep the same average value through the passive amplification process, in such a way that the ER enhancement actually increases linearly by a factor lower than m (approaching m for a higher input OSNR).

Figure 7: Extinction ratio enhancement. (a) Enhancement of extinction ratio (ERE) as the passive amplification factor m increases for a noisy input with OSNR=10. The experimental data points (red triangles) for m=2, 5 and 15 are overlaid with the simulation trend (blue squares). The solid violet line shows the experimentally measured ERE when a band-pass filter is employed without Talbot amplification. Solid green circles show experimental data points of ERE when a band-pass filter is used in conjunction with Talbot amplification. (b) Optical sampling oscilloscope time traces with passive amplification (red, PM-ON), without passive amplification (dashed blue, PM-OFF), with a BPF alone (dot-dashed green) and with both the use of a BPF and passive amplification (double-dot-dashed brown), for m=5. (c) Similar optical sampling oscilloscope traces as in b but with m=15. Full size image

Talbot amplification is not simply a time-domain equivalent of band-pass filtering (BPF) but does actually redistribute signal energy from in-band noise. To show this important feature of our technique, we additionally show data of Talbot amplification used in conjunction with BPF and compare it with Talbot amplification without BPF, and to BPF alone. Figure 7a shows the experimentally expected enhancement to ER when BPF is employed without Talbot amplification (solid violet). Here the filter used had a nearly flat-top spectral response with a 40-dB bandwidth of 2.8 nm. For this case of OSNR=10, Fig. 7a shows that BPF is better at enhancing the ER than passive amplification alone for m=5, and that it is equivalent for m=15. This can be also seen in the difference between the solid red and dot-dashed green curves in Fig. 7b,c. However, when a BPF is used in conjunction with Talbot amplification, the ER improves drastically since the BPF gets rid of the out-of-band noise, and the passive amplification additionally enhances the resulting ER as discussed above. In the case of m=15, the ER enhancement increases to 20.9 dB rather than ~11 dB from passive amplification alone. Likewise the ER enhancement for m=5 shifts to 17.7 dB rather than ~6.5 dB from passive amplification alone. We note that the ER enhancement from the BPF depends on the OSNR while the ER enhancement for Talbot amplification depends both on level of noise and amplification factor m. Talbot amplification becomes particularly effective for enhancing ER at low OSNRs, whereas BPFs are less effective.

Figure 8 shows how Talbot amplification behaves as a conventional averaging process, for example, scope averaging, on ASE-like intensity noise fluctuations. Figure 8a shows experimental data for the coefficient of variance (CV), the ratio of the s.d. to the mean for the top level, of a noisy pulse (OSNR=10) versus the inverse of the square root of amplification factor m=N (red squares). Also shown is the CV versus the inverse of the square root of number of scope averages N (blue circles), demonstrating the equivalence of Talbot amplification to averaging. The theoretical trendline for scope averaging, which scales as (N)1/2, is overlaid (dashed green). Experimental sampling oscilloscope traces in Fig. 8b,c show how the point-to-point fluctuation is nearly the same for scope averaging and Talbot amplification. Figure 8b (blue) shows results for a pulse without passive amplification and a regular scope average of N=15, and Fig. 8c shows results for a Talbot-amplified pulse by m=15 with no scope averaging. OSNR=10 for the experimental traces. Here the realignment of newly created frequencies with old ones, with just as much negative and positive fluctuation, creates an averaging effect. Because there are m-times as many frequency tones adding together, the reduction in fluctuation (CV of intensity noise) at the waveform top nearly follows the conventional counting rule and goes like the square root of m. Passive amplification is therefore equivalent to a real-time optical average. Said another way, Fig. 8c is the equivalent of Fig. 8b without the need for detection and post-processing. Such a real-time average could be particularly important where a clean pulse is needed directly in the optical domain. Notice also that whereas Talbot amplification is equivalent to scope averaging concerning its effect on white noise fluctuations, Talbot amplification additionally enhances the waveform train ER as the gain factor is increased. This is in sharp contrast to conventional scope averaging, where the noise floor and peak always average to their same respective levels, keeping ER constant.

Figure 8: Averaging effect of Talbot amplification for ASE-like noisy fluctuations. (a) Red squares show experimental data for the coefficient of variance (CV) of a passively amplified noisy pulse (OSNR=10) for a given passive amplification factor m versus the inverse of the square root of the amplification factor m with no scope averaging; solid blue circles show experimental data for the CV of the same noisy input pulse (OSNR=10) for N scope averages and no passive amplification versus the inverse of the square root of the number of scope averages N; the dashed green line shows the expected (N)1/2 dependence of averaging governed by Poisson statistics. (b) Experimental sampling oscilloscope trace for a pulse without passive amplification using scope averages, N=15. (c) Experimental sampling oscilloscope trace of the same pulse train as in b, but using passive Talbot amplification with m=15 and no scope averaging. Full size image

Finally, Fig. 9 shows a side-by-side comparison of the oscilloscope time trace of a Talbot-amplified pulse train versus an actively amplified pulse train using an EDFA. Both amplification techniques amplified a very noisy input pulse train with an input OSNR=5, and both had a gain of 15. The noisy input is shown in the top trace, and after active amplification, bottom trace, the pulse is significantly degraded. Noise is injected onto the already-noisy train, and both the noise floor and pulse peak are amplified about the same amount, negatively affecting the pulse quality. However, in the case of passive amplification, no noise is injected, the ER is enhanced by a factor approaching m—the peak is amplified while the noise floor is not, and the point-to-point fluctuations from ASE noise are reduced by a real-time optical average in which 15 averages have been taken. This can also be seen in the Supplementary Movie, which shows passive amplification for m=15 for a signal that is approximately the same level as the noise floor. Both scope traces in Fig. 9 show zero averaging in the scope representation, but the Talbot-amplified pulse has notably better extinction and less noise fluctuation. This signal recovery resembles the strategies used in optical spread-spectrum methods to hide an optical pulse in noise and recover it from the noisy background through the use of a pseudo-random spectral phase mask (transmitter) and its conjugate (receptor)27. However, for Talbot passive amplification the analogous ‘phase masks’ are in time and frequency, respectively, instead of frequency alone, effectively implementing a coherent addition of multiple consecutive waveform copies.