Supercontinuum white-light generation in optical fibers is a process that is known for its extreme sensitivity toward fluctuations of the input pulses, giving rise to a strong amplification of input noise. Such noise amplification has been recognized as a detrimental effect that prevents compression of the broad white-light spectra into a few-cycle pulse. Here, we show that the same effect can be exploited to amplify and recover faint modulation signals to an extent that seems impossible with any electronic method. We experimentally demonstrate the deterministic amplification of faint amplitude modulation signals by up to 60 dB. As we show from numerical simulations, this amplification process arises from the interaction dynamics between solitons and dispersive radiation in the fiber. The resulting all-optic signal restoration provides a new photonic building block that enables signal processing at virtually unlimited processing speeds.
© 2015 Optical Society of America
Optical supercontinuum (SC) generation has been widely applied in spectroscopy , optical imaging , precision metrology [3,4], and in measuring the carrier–envelope phase in attoscience . Despite its complexity, the physical mechanism of SC generation in optical fibers has been extensively investigated over the past 15 years with various types of pump sources [6–8] and fibers [9–11]. The dynamics are now very well understood , and arise from three key processes: group-velocity dispersion, Kerr nonlinearity, and stimulated Raman scattering. However, it has also been realized early on that the remarkable efficiency of fiber-based white-light sources comes with a caveat: a rapid loss of temporal and spectral coherence that can be particularly pronounced for pulses of 100 fs duration and above [12,13]. This inherent limitation of SC generation arises from nonlinear noise-amplification dynamics which cause the phase stability of the SC [14,15] to deteriorate. Consequently, amplitude or phase variations induced by, e.g., amplified spontaneous emission in the pump laser may already cause substantial shot-to-shot variations in the output spectra of the fiber [13,16,17]. Noise amplification strongly limits or even thwarts SC compression into a short few-cycle pulse  and is responsible for the emergence of extreme-value statistics or rogue solitons [16,19]. While noise amplification in supercontinuum generation is well known in particular for pulses with duration exceeding 100–150 fs [15,20–23], it has so far only been considered as a nuisance, and the actual potential for useful applications of the SC extreme sensitivity to noise has not been realized.
Here we propose and demonstrate that supercontinuum generation allows for all-optical regeneration of faint modulation signals that are obscured by a strong carrier. This regeneration effect enables substantial enhancement of the obtainable signal-to-noise ratio, which exceeds capabilities of all-electronic signal restoration capabilities by orders of magnitude. Moreover, our scheme may help to overcome limitations of unavoidable shot noise and Johnson noise in the optoelectronic conversion process [24,25]. Somewhat similar regeneration schemes have been previously demonstrated for telecommunications applications, but these address a different physical scenario of fully modulated signals that additionally experienced phase jitter [26,27]. Specifically, we show that the coupling mechanism between the long and short wavelength edges embedded within SC generation induces changes in tens of nanometers wide spectral slices of the output pulse energy which follow input energy variations according to with a very high power . This effect enables the enhancement of faint modulation signals relative to a strong carrier by up to 60 dB and comes with 25 dB signal-to-noise improvement. In order to reach such a high level of signal restoration, it is paramount to operate in a regime where a deterministic one-to-one correspondence between input signal and output signal is expected. This correspondence can be maintained up to pulse durations of 100 fs or more in the Ti:sapphire wavelength range. For longer durations, modulation instability can induce significant shot-to-shot fluctuations, which then prevent the controlled amplification of the signal.
The paper is organized as follows. We first discuss the procedure for a proof-of-principle experiment, demonstrating the recovery of a faint modulation signal in the presence of a strong carrier. We compare our experimental results with numerical simulations, using sinusoidally amplitude-modulated sequences of input pulses. These simulations confirm the nonlinear interaction of dispersive waves in the normal dispersion region and solitons in the anomalous dispersion region as the key processes that provide the large amplification of the modulation signal. To obtain an understanding of the technical limitations of our method, we then conduct a detailed noise analysis, which identifies detection shot noise as the limiting mechanism. Finally, we discuss the role of the coherence for our signal amplification process as well as for noise amplification, and close with conclusions on possible applications.
Our experimental setup uses a commercial Ti:sapphire laser with a 76 MHz repetition rate and specified pulse duration of 6.5 fs (Fig. 1). This duration ensures coherence between the input and output of the fiber, i.e., it guarantees a one-to-one correspondence between phase and amplitude at the input and output of the fiber. The beam was coupled into a 70 cm piece of polarization-maintaining microstructured fiber (Thorlabs NL-PM-750). Coupling efficiency into the fiber is to 30% using a single aspheric lens.
In order to illustrate the high sensitivity of the energy in specific spectral bands to the input power, we performed a simple initial experiment. Using a digital single-lens reflex camera, we directly recorded the diffuse projection of the diffracted output spectrum from the fiber while varying the input power (Fig. 1 and movie sequence in Visualization 1). The observed spectra that cover a range from 500 to 1000 nm essentially consist of a sequence of bright zones and dark fringes, and the position of the dark fringes varies in correlation with the measured total power. Inspecting the power in relatively narrow wavelength ranges, we find regions that are nearly perfectly correlated with the spectrally integrated power but also others which are in anticorrelation. Note that the video recording provides an overview of the dynamics of the spectrum but cannot be operated at the nanosecond exposure times that are necessary to resolve a MHz modulation. While the signal amplification is still rather modest in this initial experimental demonstration, the effect becomes much more prominent when switching to an avalanche photodiode array in the Fourier plane of a spectrograph, which enables much higher spectral resolution at the laser repetition rate but is restricted to a small number of channels.
In this second set of experiments, the laser beam was focused into an acousto-optic modulator, where the optical pulse train experienced a weak amplitude modulation of about 75 dBc at 4.6 MHz. The frequency of 4.6 MHz was limited by the available amplitude modulator but the setup only relies on passive nonlinear effects and is thus, in principle, scalable to arbitrary speeds. The group delay dispersion of the modulator was compensated by 48 reflections off chirped mirrors. These chirped mirrors were arranged in three pairs with dispersion oscillations canceling each other out in each pair. The duration of the pulses at the fiber input was measured to be 8 fs. The pulses were not transform-limited due to residual third-order dispersion, which was not compensated for. The generated SC spectrum is spectrally dispersed in a Czerny–Turner grating spectrograph with an avalanche photodiode array (APD, Hamamatsu S8550) located in its output plane. The spectrograph focal distance was 175 mm, and the diffraction grating resulted in dispersion in the visible wavelength range. Given the width and pitch of the individual APDs, we computed a bandwidth of 35 nm for the individual wavelength channels. Moreover, the array offers a total of eight independent wavelength channels which we could not fully exploit as we only had four oscilloscope channels available, one of which being reserved for digitization of the input transient. Each of the three SC measurement channels in the APD array was amplified with a separate transimpedance amplifier (Maxim MAX3665, 470 MHz bandwidth) with 20 kΩ gain. Detector noise-equivalent power was with a 220 V reverse bias voltage on the APD. The spectral response range of the photodiode covers 300–1000 nm, with a peak quantum efficiency of 85% at 650 nm.
Signals were recorded by a digital storage oscilloscope with sampling rate and a 1 GHz analog bandwidth, using an 8 bit analog-to-digital converter. As the laser repetition rate was about 64 times lower than the sampling rate, this scenario allows for a resolution improvement of 3 bits, which translates into a noise floor of , effectively removing any dynamic range limitation of the oscilloscope. Aliasing artifacts were additionally suppressed by the use of analog 300 MHz lowpass filters. Given the choice of modulation frequency and electronic processing, the signal-to-noise ratio is expected to be dominated only by the number of detected photons per pulse, the resulting shot noise, and by Johnson noise whereas other technical laser noise contributions are not expected to play any role here.
A small part of the modulated beam before the microstructured fiber was reflected with a thin glass plate into an avalanche photodiode in order to have a reference signal for the induced modulation. Detection of this signal was shot-noise limited. The reference and selected three bands of the supercontinuum pulse train were recorded simultaneously with the oscilloscope. Recorded pulse trains contained a few thousand to tens of thousands of modulation cycles. The detection was conducted both in the normal dispersion region (Fig. 2) as well as in the anomalous dispersion region of the microstructured fiber that had a specified zero-dispersion wavelength at 750 nm. Both dispersion regions showed areas of the spectrum where the modulated signal seemed unaffected by the supercontinuum process, but then also areas where the signal was greatly enhanced. Here we restrict ourselves to the normal dispersion region, given that it showed higher signal amplification.
Experimental conditions ensured stability of the spectral shape and pulse energy of the laser output. However, our setup did not include any means for stabilizing the power launched into the fiber. Long-term measurements indicate a drop of 10% in the timescale of one hour. In the experiments, we allowed the launched power to slowly drift and recorded a large number of individual time series of three APD channels and the reference detector. We noted average signal amplification values as well as their peak values.
Applying a faint 75 dBc (electrical) amplitude modulation with 4.6 MHz frequency to the pulses injected into the fiber, we find input power ranges where the modulation on a single output channel is enhanced by more than 60 dB compared to an integrated measurement before the fiber [Figs. 2(b)–2(i)]. While the individual wavelength channels at the fiber output show a clearly visible 4.6 MHz modulation, the spectrally integrated signal before the fiber only reveals presence of the modulation in the Fourier domain. Moreover, the central wavelength channel at 670 nm is clearly anticorrelated with the two other channels at 700 and 620 nm. Adding the measured signals of the correlated channels and subtracting the anticorrelated one results in a modulated signal that even shows an signal enhancement. As the detected intensity levels in the spectrally resolved output are about three orders of magnitude smaller than in the input, we also see an increase of the noise background by approximately 30 dB. However, this increased noise level still enables a 30 dB signal-to-noise improvement in the combined signal. We observe similar results in the anomalous dispersion region, yet with a lower signal amplification of .
3. THEORY AND NUMERICAL SIMULATIONS
In our experiments, we exploit the sensitivity to the input peak power of SC generation with short pump pulses near the zero-dispersion wavelength of a highly nonlinear fiber. Prior to analyzing this process in detailed numerical simulations, let us first outline the mechanisms that lead to this pronounced input power dependence. The SC generation process consists of an initial stage of higher-order soliton compression followed by fission into fundamental solitons, with dispersive waves generated due to the presence of higher-order dispersion and stimulated Raman scattering . Near-identical group velocities of soliton and dispersive waves provided, the large nonlinear potential represented by the soliton amplitude leads to a situation where dispersive waves are effectively trapped by the soliton . Such soliton-dispersive wave dynamics have also been shown to be associated with transfer of energy to a frequency-shifted wave through a four-wave mixing process [29,30].
If the power of the injected pump pulses is modulated by a very weak signal, the resulting tiny variations in the power of the injected pulse transfer into wavelength jitter for the solitons ejected during the fission process. The power-dependent soliton jitter, although very modest (on the order of a few nm), dramatically affects the phase-matching condition for the four-wave mixing interaction with the DW, which, in turn, results in significant variations in the wavelength and energy of the frequency-shifted waves . Furthermore, the dispersive waves and solitons may mutually affect each other by power-dependent wavelength shifts , which further increase the sensitivity of the power-dependent shifting mechanism. These combined effects then translate into a very large amplification of the initial modulating signal when a narrow spectral bandwidth in the normal dispersion is filtered out from the supercontinuum. Note that this scenario is only possible for a SC generated under prevalent soliton dynamics and in a regime where a one-to-one correspondence between the input and output of the fiber holds.
We next analyze the experimental situation by conducting numerical simulations of the nonlinear fiber propagation, as shown in Fig. 3. These simulations employ the nonlinear Schrödinger equation with a standard split-step Fourier algorithm:12] as required for accurate modeling of broadband pulse propagation in optical fibers . The parameters of the simulations correspond to those of the experiments. Specifically, we inject 8 fs pulses into 60 cm of a photonic crystal fiber. The fact that the fiber was polarization maintaining greatly simplifies numerical modeling. We chose a zero-dispersion wavelength at 760 nm, i.e., slightly higher than specified as this value agrees best with previous comparisons between experiment and theory. This choice results in a SC that extends from 550 to 1100 nm as shown in Fig. 3. Note that, because the simulations do not include attenuation along the fiber, the length was adjusted to match the bandwidth of the experimental spectrum. The input pulses were modeled to include imperfections of the chirped mirrors present in the experimental setup, namely dispersion oscillations . Amplitude and frequency of these oscillations were fitted to autocorrelation measurements to correctly account for a pedestal-like background structure in the launched pulses. Broadband background noise in the form of one photon per frequency bin was added to the input field. Moreover, we included the effect of Raman noise and added it to the propagating field as well, although this was found negligible for the resulting RF modulation amplification.
We simulated the SC spectra generated by a pulse train of 4000 consecutive pulses with modulated peak power at a 4 MHz frequency (corresponding to a resolution bandwidth of 40 kHz in the RF spectrum) and with amplitude modulation contrast of (74 dBc electrical) [see Figs. 3(c) and 3(d), top panels]. The results plotted in Fig. 3(a) show how the very weak initial modulation of the input peak power is amplified by nearly 50 dB when detecting the energy fluctuations in wavelength channels with 10 nm bandwidth in the normal dispersion region [Figs. 3(c) and 3(d)], in very good agreement with the experiments. The mutual correlation between the different wavelength channels is also correctly reproduced.
In order to confirm the amplification mechanism, we plot the spectrogram representation of the SC generated by a single pulse in Fig. 4(a). We can clearly see how the spectral bands in the normal dispersion region, where we observe large amplification of the initial modulation, specifically correspond to dispersive waves that are trapped by solitons as highlighted by the dashed lines. The amplitude of the periodic power-dependent spectral jitter of these solitons resulting from the weak initial modulation is on the order of a few nm as shown in Fig. 4(b). This spectral jitter, in turn, induces a change in the trapped dispersive wave that may reach or exceed its bandwidth, as shown in Fig. 4(c). It is this dramatic change in the wavelength of the dispersive wave which is captured (or not) by the spectral filters and results in very large amplification of the initial modulation. Additional numerical simulations show that large amplification factors are obtained when the SC dynamics are self-seeded from the input pulse spectral components, which has been dubbed as the coherent regime previously . This regime can be maintained for pulses up to 100 fs duration. Shorter pulses are, in principle, more favorable due to the smaller number of solitons ejected during the fission process, which serves to further increase the power-dependent jitter.
4. NOISE ANALYSIS
Our experiments indicate that the modulation signal is elevated by up to 60 dB relative to the carrier. We also observe a less pronounced increase of the noise floor, which still enables a substantial improvement of the signal-to-noise ratio. A physically intriguing question is the origin of this noise floor, whether it relates to technical limitations of the photodetectors or whether it has a quantum noise origin. In order to clarify this aspect, we carefully simulated all possible noise mechanisms, as is explained in the following paragraphs.
Photodetection underlies a series of noise mechanisms that define the attainable signal-to-noise ratio. Most prominently, these mechanisms include Johnson noise  and shot noise . The latter is caused by the fact that measurements of optical pulse energies are quantized, with the detected photon number following Poissonian statistics. Here it often proves limiting that only a small fraction of the available laser power can be converted into an electric current without destroying the photodetector. The uncertainty of an individual pulse energy measurement is given by , resulting in the best obtainable signal-to-noise ratio (SNR) of in the absence of Johnson noise and other technical noise sources. The obtainable SNR therefore increases with the square root of the detected photons. Johnson (or thermal) noise follows Gaussian statistics that give rise to voltage fluctuations of per root Hertz across the feedback resistor of the transimpedance amplifier . Other than shot noise, the Johnson noise contribution is independent of signal levels. Here, is Boltzmann’s constant and was assumed as the temperature. In combination, both these mechanisms limit our capabilities to detect a faint modulation signal. Figure 5 shows how shot noise and Johnson noise affect the detection before and after the fiber in our signal-amplification scheme. These simulations have been carefully adjusted to provide the exact same noise floor as was measured experimentally.
Our detection scheme relies on massive oversampling  of the signals acquired at rate with 8 bit digitization. Numerical processing of the data allows for a maximum dynamic range increase by . The resulting effective 11 bit digitization, shown as a gray curve in Fig. 5, pushes dynamic range limitations to beyond 105 dBc, i.e., substantially below the other noise sources in the photoelectric conversion. For completeness, we additionally included quantum noise estimations from the numerical simulations discussed in the previous section. These are shown as green lines in Fig. 5. The discrepancy between detection shot noise and amplified spontaneous emission in Fig. 5(a) arises because only a small fraction of the available laser power can actually be converted in a photodiode without saturation or damage. The detection of this noise signature would require total photoelectric conversion near the watt level. Additionally, a dynamic range increase by an additional 4 bits is required, which seems out of reach for currently available fast photodiodes and high-speed analog-to-digital converters. Our estimates indicate that the obtainable signal-to-noise ratio of the input signal is limited by saturation of the photodiode and the resulting shot-noise limit. In contrast, power levels after the fiber are much lower, such that the detection of the modulation signal is not shot-noise limited but rather it is the Johnson noise that dominates.
Our experimental results and numerical simulations have made it clear that the highly nonlinear soliton fission scenario results in spectra that may exhibit one or several pronounced dark fringes, particularly in the normal dispersion region. Variation of the input power leads to a spectral shift of these fringes. The strongest signal enhancement is observed for a detected spectral bandwidth smaller than the width of a dark fringe. In an ideal situation, the amplitude modulation of the input signal is adjusted to lead to a spectral shift of the dark fringe that exactly changes the detected power from near zero to the maximum possible. In our numerical simulations, we find signal enhancements of 50 dB, close to the best experimental observations yet at narrower filter bandwidth than in the experiments. Even in this ideal situation of maximum signal amplification, spurious harmonics may appear.
It appears useful to analyze the behavior from a more general point of view, using the concept of a transfer function between input and output pulse energy and , respectively. This is illustrated in Fig. 6, with a simulated input signal [Figs. 6(a) and 6(b)] being amplified via a linear transfer function [Figs. 6(c) and 6(d)] and under weakly nonlinear transfer [Figs. 6(e) and 6(f)]. In the latter case, harmonics of the modulation frequency appear in the output and, for multifrequency driver signals, mixing products will appear. However, it can also be seen that the presence of these spurious harmonics does not strongly reduce the amplification of the modulation signal itself. This situation changes when the amplification process is completely overdriven [Figs. 6(g) and 6(h)]. Now the input modulation causes a spectral shift that is much larger than the spacing between the dark fringes, with a sinusoidal transfer function resulting in a multitude of harmonics that appear with comparable amplitude. Nevertheless, even this disadvantageous situation still enables some reduced signal amplification. All processes discussed so far have ensured coherence between input and output signal, with their explicitly assumed unambiguous functional dependence via the transfer function.
Finally, the concept of a transfer function may break down and stochastic processes take over. This breakdown of the transfer function concept shows up as a white noise floor in the Fourier representation [Fig. 6(j)]. One possible source for such a contribution is modulation instability or the spontaneous Raman effect in fibers. Stochastic processes further reduce signal amplification to the verge of uselessness, whereas noise amplification is significant. These examples show that noise amplification may be easy to observe, yet typically such behavior cannot be exploited for the coherent amplification of an input signal.
At first glance, our findings may not appear overly surprising given that similar noise amplification has been discussed before with lower amplification factors up to 30 dB [14,15]. The latter experiments actually related relative intensity noise of the SC to pump fluctuations, yet without investigating the phase relationship or coherence between input and output. Such amplification of input fluctuations will appear regardless of a deterministic nature of the underlying process. For pump pulse durations , our numerical simulations indicate that noise amplification quickly degrades spectral and temporal coherence. In the absence of coherence between the input and output, noise amplification becomes significant while the modulation amplification factor that can be obtained may be reduced significantly. The demonstrated amplification of up to 60 dB therefore strongly contrasts with the previously observed noise-amplification scenarios. For a fixed wavelength interval in the output of the fiber, such favorably high amplification factors can only be seen for a narrow input power range. Nevertheless, our simulations indicate that signal amplification at 10–20 dB lower factors is actually a fairly robust phenomenon in a 100–200 nm wide dispersive wave region that copropagates with solitons in the anomalous-dispersion region. Outside this range, signal-amplification factors are substantially lower. Our scheme can also tolerate larger changes in the input power and still amplify the modulated signal by 60 dB, but in this case such high gain value may be obtained in a different wavelength band.
Moreover, to fully appreciate the observed improvements of the signal-to-noise ratios by up to 30 dB, it is useful to compute the photon fluences necessary to simply obtain the same enhancement via increase of the optical input power. In the case of shot-noise-limited detection, a thousandfold increase of the input power would be necessary to measure a signal 40 dB above the noise floor which, in turn, would require 110 dB dynamic range as well as the total photoelectric conversion of about 100 mW of Ti:sapphire light. Such parameters pose a serious challenge for the dynamic range of available electronic test equipment  and are at the limit of high-power photodetector designs [36,37]. Although all-optical regeneration for optical telecommunications can be conveniently achieved with simpler approaches, our scheme offers an important alternative whenever small modulations need to be recovered in a strong pulsed laser beam. In such a situation, we may be able to push detection limits by one or two orders of magnitude in intensity, which would greatly alleviate extremely sensitive pump-probe spectroscopy studies . Other possible applications include high-sensitivity interferometry and sensitivity enhancement in the measurement of small induced refractive index changes.
Suomen Akatemia (Academy of Finland) (128844, 130099, 132279).
The authors acknowledge fruitful discussions with John Dudley and Miro Erkintalo.
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