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Abstract
The pulse contrast of state-of-the-art petawatt lasers is limited by coherent noise. This paper reports on a new family of noise, termed nonlinear beat noise, which is generated by the nonlinear mixing of two kinds of coherent noise in optical parametric chirped-pulse amplification (OPCPA). We theoretically study the various nonlinear beat noises and reveal their intensity evolutions in an OPCPA amplifier. The results suggest that nonlinear beat noise will be destructive to the future hundred-petawatt lasers.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The fundamental limits on laser applications are often governed by the noise in optical amplifiers [1]. In the field of ultra-intense lasers, pulse contrast, as a measure of noise, is critically important in strong-field and solid-target interactions for particle accelerations [2] and the generation of giant attosecond pulses [3]. So far, laser facilities with peak powers exceeding one petawatt (PW) have been available at laboratories around the world [4–7]. These multi-petawatt lasers are aiming at a pulse-contrast parameter of 10−11 to 10−12 for launching well-controlled strong-field experiments at the laser intensities of ~1021 W/cm2 or higher. The high contrast is achieved via the architecture of double chirped-pulse amplification with an intermediated pulse-cleaning device [6–9], such that the incoherent laser and parametric fluorescence can be minimized. However, the coherent noise in these systems remains uncontrolled and manifests as a major limitation to the accessible contrast [10–12]. Taking the recently-reported 4.2 PW laser (GIST, Korea) [6] as an example, a high incoherent contrast of 3 × 10−12 has been demonstrated at 100 ps before the main pulse, but its coherent contrast, located within 100 ps from the main pulse, has been still limited to 10−9 to 10−10. This status has stimulated intense research activities in coherent noise. The typical coherent noise is induced by the surface scattering or spectral clipping inside the pulse stretcher and compressor [13,14]. In contrast to incoherent noise, coherent noise has its intensity linearly proportional to the main pulse, and manifests itself as picosecond pedestals at both fronts of the main pulse. On the other hand, the amplification process may also induce coherent noise nonlinearly [15]. In optical parametric chirped-pulse amplification (OPCPA), the noise at pump pulses will directly imprint onto the signal spectrum, and displays as a kind of coherent noise after pulse compression [16–19]. In a similar manner, the stray light of the chirped signal pulse in an OPCPA crystal will also distort the signal spectrum nonlinearly, and induce another kind of coherent noise in the form of pre-pulses [20]. These two kinds of coherent noise in OPCPA amplifiers are, in principle, related to the nonlinear mixing of incident noise (e.g., pump-pulse distortions or seed-pulse replicas) with the chirped signal pulse.
The contrast requirement will climb to ~10−13 for the planned 200 PW lasers (e.g., the Extreme Light Infrastructure) aiming at the extreme intensity of 1024 W/cm2 or higher [21]. It is therefore necessary to explore new mechanisms of coherent noise in the scale of 10−13 that is much more demanding than the requirement (10−10) of current PW lasers. In view of the tremendous noise (possibly as high as megawatt) in 200 PW lasers, the nonlinear mixing of noise with noise will be strong enough to produce new kinds of noise. Here we present, to the best of our knowledge, a first study on the nonlinear mixing of coherent noise themselves in OPCPA. When two common kinds of coherent noise simultaneously present in a parametric amplifier, they will interfere with each other and mix nonlinearly. As a result, new noise components at the beat-modulation frequencies are generated, which is termed as nonlinear beat noise and will be theoretically characterized in this paper.
2. Noise mechanism and nonlinear evolutions
2.1 Nonlinear mixing of pump-pulse modulations
We first deduce an analytical formalism for the production of nonlinear beat noise from pump-pulse modulations. Under the ideal OPCPA condition that neglects the media dispersion and transverse diffraction effects, the instantaneous parametric gain G(t) is determined by the pump intensity Ip(t) in the pump non-depletion regime as G(t) = cosh2[v], where z is the crystal length and Γ2 = 2ωsωideff2/(nsninpε0c3) [22]. Two sinusoidal modulations (with the modulation rates of Ωp1, Ωp2 and amplitudes of rp1, rp2, respectively) are introduced to the pump pulse, which leads to Ip(t) = Ip0(t) × [1 + rp1cos(Ωp1t) + rp2cos(Ωp2t)]. The amplitude envelope of the amplified chirped-pulse As(z, t) links directly with the time-varying parametric gain G(t) as
where As0(t) is the temporal amplitude of signal before amplification. When the gain is reasonably large (i.e.,Γz >>1), Eq. (1) can be approximately rewritten aswhere m and n are arbitrary integers. Aamp(t) = As0(t) × cosh() represents the amplitude envelope of signal pulse after amplification with a clean pump, and G0 = cosh2 [] is the corresponding parametric gain. Equation (2) clearly shows that the amplified signal pulse undergoes a discrete sum of sinusoidal modulations with the modulation ratesjointly determined by Ωp1 and Ωp2. These modulation terms indicate the nonlinear mixing of the two pump modulations at Ωp1 and Ωp2, which we call the nonlinear beat noise. Equations (1)-(3) reveal the mechanism for generating nonlinear beat noise: the nonlinear pump-intensity dependence of parametric gain [Eq. (1)] allows the pump-pulse modulations to mix with each other, and also enables such nonlinear mixing to fall into the high-order harmonic-modulation regime [Eq. (2)], which induces high-order signal modulations at the beat-modulation rates given by Eq. (3). The temporal intensity of the amplified signal pulse after compression can be also obtained aswhere C is the chirp rate (C = Δωs/τs, with Δωs and τs as the frequency bandwidth and temporal duration of the chirped signal pulse, respectively). As described by Eq. (4), the compressed pulse consists of a main pulse I0(t) centered at t = 0 and a series of noise spikes at t = (mΩp1−nΩp2)/C (termed as (|m| + |n|)th order spike in what follows) that are converted from the nonlinear beat noise described in Eq. (2). The intensities of these noise spikes are linearly proportional to the main pulse intensity I0, which is the major attributor of coherent noise.To prove the validity of Eq. (4) and further characterize the nonlinear beat noise in the regime of pump depletion, we carried out numerical simulations based on the nonlinear coupled-wave equations that describe the parametric amplification [23]. We consider an OPCPA system consisting of a type-I β-BBO crystal pumped by a 532-nm, 100-ps Gaussian pulse with an adjustable intensity up to 4 GW/cm2. The seed pulse at 800 nm has a bandwidth of ∆ωs = 90 THz, a chirped-pulse duration of τs = 60 ps and an intensity of Is0 = 40 W/cm2. Figures 1(a)-1(c) display the temporal profiles of the compressed signal pulse after amplification under three different situations of pump-pulse modulations. In the case with a single sinusoidal modulation of Ωp1 = 10.5 THz (Ωp2 = 15.0 THz), a series of noise spikes with an equal time interval of tp1 = Ωp1/C = 7 ps (tp2 = Ωp2/C = 10 ps) are induced at both fronts of the main pulse, as plotted in Fig. 1(a) [Fig. 1(b)]. These coherent-noise spikes are resulted from the nonlinear mixing of each pump modulation with the main pulse, which has been previously reported [18]. In the case that the two modulations exist together on the pump pulse, more noise spikes, except for those induced by each pump modulation, can be observed in the OPCPA output, as highlighted by the arrows in Fig. 1(c). These spikes at t = −3 ps, −4 ps, −13 ps, −17 ps, −23 ps, −24 ps, −27 ps, −34 ps and −37 ps are just the nonlinear beat noise induced by the nonlinear mixing of the two pump modulations, whose temporal locations satisfy the relation of t = mtp1− ntp2 (an equivalence of Eq. (3)).
Fig. 1 Temporal profiles of the OPCPA output under the pump-noise situations of (a) a single sinusoidal modulation with modulation rate Ωp1 = 10.5 THz and amplitude rp1 = 10−3; (b) a single sinusoidal modulation with Ωp2 = 15.0 THz and rp2 = 10−2; and (c) both of these two modulations, respectively. The OPCPA operates in the strong pump-depletion regime (ηp = 45%) with a pump intensity of Ip0 = 4 GW/cm2. The red, blue and green arrows in (c) indicate the 2nd, 3rd and 4th order spikes of the nonlinear beat noise. (d) Intensity evolutions of the 2nd (red, 3rd (blue) and 4th (green) order nonlinear-beat-noise spikes at t = −17 ps, −27 ps and −34 ps respectively in (c), versus the parametric gain G0 and pump-depletion ratio ηp (gray line). The solid and dashed lines represent the results obtained with numerical simulations and Eq. (4), respectively.
Figure 1(d) shows the intensity evolutions of the three nonlinear-beat-noise spikes at t = −17 ps (2nd order spike, i.e., ‘m = −1, n = 1’ in t = mtp1− ntp2), −27 ps (3rd order spike with ‘m = −1, n = 2’) and −34 ps (4th order spike with ‘m = −2, n = 2’) versus the parametric gain G0 and pump-depletion ratio ηp. The results obtained with the analytical formula Eq. (4) match well with the numerical simulations. These intensity curves clearly show that the nonlinear beat noise undergoes a drastic growth at the initial stage of amplification (i.e., G0 < 10), and gradually levels off in the further amplification. As in our case, the nonlinear mixing of the two weak modulations (with the modulation amplitudes of rp1 = 10−2 and rp2 = 10−3) on the pump pulse immediately induces the 2nd order nonlinear-beat-noise spike with an intensity of 10−15 at the crystal entrance (G0 ~1). This noise intensity then quickly grows by 5 orders of magnitude to 1 × 10−10 with a parametric gain of G0 ~103. After that, it maintains around 10−10 in the high-gain (up to G0 ~109) and strong pump-depletion (up to ηp ~45%) regime. In comparison, the 3rd order spike begins with a much lower intensity of ~10−25 and grows by 10 orders of magnitude to ~10−15 in the amplification process. These results suggest that the higher-order nonlinear-beat-noise spikes (with locations far from the main pulse) grow with the signal amplification much faster than lower-order ones (with locations near the main pulse).
2.2 Nonlinear mixing of signal post-pulses
The post-pulses at the signal introduced by optical surface reflections result in the coherent noise in form of pre-pulses in OPCPA [20]. The nonlinear mixing of these pre- and post-pulses in a parametric amplifier will induce the second kind of nonlinear beat noise. Let us consider an OPCPA case with a clean pump and a seed signal followed by two post-pulses. As plotted in Fig. 2(a), the two post-pulses are assumed to separate with the main pulse by ts1 = 7 ps and ts2 = 10 ps with a relative intensity of rs1 = 10−5 and rs2 = 10−3, respectively. The signal leading edge is clean before amplification. Figures 2(b) and 2(c) display the temporal profiles of amplified signal pulses after compression, where a series of pre-pulses are induced. These pre-pulses can be classified into three groups in terms of their temporal locations: the pre-pulses at t = mts1 (as indicated with open triangles), the pre-pulses at t = nts2 (indicated with open cycles) and the pre-pulses at t = mts1 − nts2 (indicated with arrows), with m and n as arbitrary non-zero integers. The first two groups of pre-pulses are the coherent noise induced by nonlinear mixing of each post-pulse with the main signal pulse, while the third group of spikes is induced by the nonlinear mixing of the two post-pulses. When the OPCPA is run at low pump-depletion with ηp = 1%, only the 1st (t = −3 ps) and 2nd (t = −17 ps) order nonlinear-beat-noise spikes are observed at the leading edge of the main pulse, as indicated by the olive and red arrows in Fig. 2(b), respectively. But with ηp increasing to 45%, the noise spikes grow substantially with the presence of 3rd and 4th order spikes, as indicated with blue and green arrows in Fig. 2(c), respectively. This suggests that pump-depletion plays an important role in the growth of this kind of nonlinear beat noise.
Fig. 2 (a) Temporal profile of a seed pulse followed by two post-pulses at ts1 = 7 ps and ts2 = 10 ps with relative intensities of rs1 = 10−5 and rs2 = 10−3, respectively. (b), (c) Temporal profile of this signal pulse after amplification with a clean pump pulse (Ip0 = 4 GW/cm2) at low pump-depletion (ηp = 1%) and strong pump-depletion (ηp = 45%) level, respectively. The open-triangles (open-circles) mark the pre-pulses induced by the nonlinear mixing of the post-pulse at ts1 (ts2) with the main signal pulse. The arrows indicate the nonlinear beat noise spikes induced by the nonlinear mixing of the two post-pulses. The olive, red, blue, green and orange arrows correspond to the 1st, 2nd, 3rd and 4th order spikes, respectively. (d) Intensity evolution of the 2nd, 3rd, and 4th order nonlinear-beat-noise spike at t = −17 ps, −27 ps and −37 ps, versus the parametric gain and pump-depletion (gray line). The solid and dashed lines represent the results calculated with numerical simulations and analytical formula Eq. (5), respectively.
According to the intensity evolutions of the 2nd, 3rd and 4th order nonlinear-beat-noise spikes at t = −17 ps, −27 ps, and −37 ps obtained with numerical simulations [Fig. 2(d)], an empirical formula for the intensity of (|m| + |n|)th order spike located at t = mts1 − nts2 can be obtained as
which is proportional to the 2(|m| + |n|)th power of the pump-depletion ratio ηp. This formula fits well with the numerical simulations prior to significant pump-depletion (ηp ~30%). Figure 2(d) indicates that this kind of nonlinear-beat-noise spikes induced by signal post-pulses is negligible in the initial stage of amplification (ηp < 10−2), but rises steeply when the pump-depletion ηp gets considerable. This growth behavior is almost the opposite of the first kind of nonlinear beat noise illustrated in Fig. 1(d). In particular, when the amplifier operates in the strong pump-depletion regime, this kind of nonlinear beat noise becomes more pronounced than the first kind one. As in our OPCPA case, with ηp = 45%, the 2nd, 3rd and 4th order nonlinear-beat-noise spikes induced by post-pulses have the intensities of 10−9, 10−13 and 10−16, which are far greater than their counterparts in Fig. 1(d) as 10−10, 10−15 and 10−21, respectively. Equation (5) also suggests that higher-order noise spikes grow with pump depletion much faster than the lower-order ones, which explains the fast increase of spike numbers in the strong pump-depletion regime [Fig. 2(c)].Such a distinct difference between these two kinds of nonlinear beat noise can be interpreted from the nonlinear characteristics of parametric amplification. The first kind of nonlinear beat noise, induced by two pump modulations, is a direct result of the signal spectrum modulation. In the small-signal amplification regime (i.e., the pump non-depletion regime), the nonlinear beat noise appears immediately and its intensity growth is governed by the intrinsic nonlinearity of small-signal gain with pump intensity. As indicated in Eqs. (1)-(3), it is the nonlinear dependence of small-signal gain G(t) on the pump intensity Ip(t) that leads to the mixing of pump modulations, thus this kind of nonlinear beat noise grows drastically in this regime. In the strong pump-depletion regime, the induced signal spectrum modulation will be saturated more severely than the signal itself, which causes a significantly slower noise growth, as illustrated in Fig. 1(d). Conversely, the second kind of nonlinear beat noise does not appear in the small-signal amplification regime where the post-pulses as well as the main signal are amplified independently and no nonlinear mixing will occur between each other. However, pump depletion will introduce a strong nonlinear coupling between the pump and signal, which in turn leads to the nonlinear mixing of two post-pulses and the rapid growth of nonlinear beat noise in the strong pump-depletion regime, as illustrated in Fig. 2(d).
2.3 Nonlinear mixing of a pump modulation with a signal post-pulse
Except for the nonlinear mixing of pump noise and nonlinear mixing of signal noise, nonlinear mixing also occurs between the pump noise and signal noise. We next characterize the third kind of nonlinear beat noise in OPCPA induced by the mixing of pump-pulse modulation with signal post-pulse. We consider an OPCPA case with a pump pulse that carries an intensity modulation (with modulation rate Ωp = 10.5 THz and amplitude rp = 10−3) and a seed pulse followed by a post-pulse (temporal interval ts = 10 ps and relative intensity rs = 10−3), as plotted in Fig. 3(a). Figures 3(b) and 3(c) display the temporal profiles of the compressed signal pulses after amplification at a low pump-depletion level of ηp = 1% and a strong pump-depletion level ηp = 45%, respectively. The arrows in Figs. 3(b) and 3(c) indicate the nonlinear-beat-noise spikes produced in the amplification process, whose temporal locations satisfy t = mΩp/C − nts (with m, n as non-zero integers). Only 2nd order spikes (t = −3 ps, −4 ps, −17 ps) are observed with ηp = 1% [Fig. 3(b)], while the 3rd (t = −13 ps, −24 ps, −27 ps) and 4th (t = −23 ps, −37 ps) order spikes appear [Fig. 3(c)] when ηp increases to 45%, indicating that this kind of nonlinear beat noise grows quickly with pump depletion.
Fig. 3 (a) Temporal profile of a seed pulse followed by a post-pulse at ts = 10 ps with a relative intensity of rs = 10−3. (b),(c) Temporal profiles of this signal pulse after amplification by a pump pulse with an intensity modulation of Ωp = 10.5 THz, rp = 10−3 at low pump-depletion (ηp = 1%) and strong pump-depletion (ηp = 45%) level, respectively. The red, blue and green arrows indicate the 2nd, 3rd and 4th order nonlinear-beat-noise spikes. (d) Intensity evolution of the 2nd, 3rd and 4th order nonlinear-beat-noise spike at t = −17 ps, −27 ps and −37 ps versus the parametric gain and pump-depletion (gray line). The solid and dashed lines represent the results from numerical simulations and Eq. (6), respectively.
Based on the experiences from Eqs. (4) and (5), we can directly write an empirical formula for the intensity of noise spike at t = mΩp/C -nts as
which depends on both the gain and pump depletion. In Fig. 3(d), the intensity evolutions of the 2nd, 3rd and 4th nonlinear-beat-noise spikes calculated with Eq. (6) and numerical simulations are both plotted, which illustrates a good agreement of this empirical formula with numerical simulations in not only the pump non-depletion but also the strong pump-depletion regime.2.4 Nonlinear mixing of pump-ASE with pump modulation
We note that all kinds of coherent noise can induce nonlinear beat noise in an OPCPA amplifier. In the typical OPCPA output, there is always a coherent-noise-pedestal in ten picosecond time scale, which has been verified to be imparted from the ASE noise in the pump laser [16,17]. This ASE noise, despite its random phase, can also mix nonlinearly with the pump-pulse modulation and induces the fourth kind of nonlinear beat noise. Assuming the pump laser in our OPCPA system contains the ASE noise of 1 nm in a bandwidth and 10−4 in relative intensity, a coherent-noise-pedestal in the time range of ± 10 ps will be induced in the amplified signal pulse, as illustrated in Fig. 4(a). However, when the pump pulse additionally carries an intensity modulation with a modulation rate Ωp = 15.0 THz and amplitude rp = 10−2, several replicas of the coherent-noise-pedestal would be induced at the temporal locations (i.e., t = mΩp/C) set by the pump-modulation rate, as shown in Fig. 4(c). As a result, the total coherent noise pedestal is expanded dramatically to a time range of ± 30 ps. These replicas of coherent-noise-pedestal are just the nonlinear beat noise induced by the mixing of pump ASE with the pump modulation.
Fig. 4 Temporal profiles of OPCPA output under the pump-noise situations of (a) the pump laser contains the ASE noise with a bandwidth of 1 nm and an intensity of 10−4 with respect to the pump intensity); (b) the pump laser contains a sinusoidal intensity modulation with Ωp = 15.0 THz and rp = 10−2; and (c) the pump laser contains both the ASE and intensity modulation. The OPCPAs all operate in the strong pump-depletion regime with ηp = 45%. (d) Intensities of the noise spike (solid-lines) and pedestal (dotted-dash lines) located at t = −10 ps (red), −20 ps (blue) and −30 ps (green) versus the parametric gain and pump-depletion (gray line). The intensity of the main coherent-noise-pedestal at t = 0 is also plotted, as the black dotted-dash line.
Figure 4(d) shows the intensity evolutions of pump-ASE-initiated coherent-noise-pedestal at t = 0 as well as the replicas of this pedestal. The replicas exhibit a similar growth behavior with the main coherent-noise-pedestal, which all establishes immediately at the beginning of amplification (G0 < 10) and the noise intensities are quite insensitive to the pump depletion. The intensities of the noise spikes and pedestals at t = −10 ps, −20 ps and −30 ps are plotted in the same figure for comparison. The results show that the intensity of each noise-pedestal replica is linearly proportional to the intensities of coherent-noise spikes at the same temporal location. These results further manifest that the coherent-noise-pedestal replicas are induced by the mixing of ASE with the intensity modulation noise in pump laser.
2.5 Nonlinear mixing of pump-ASE with signal post-pulse
The pump ASE can also mix nonlinearly with the post-pulse at the signal and induces the fifth kind of nonlinear beat noise. As illustrated in Figs. 5(a)-5(c), this nonlinear mixing of noise also leads to the production of several replicas of the ASE pedestal and thereby a significant expansion of the total noise pedestal. In the simulations, we have assumed a seed pulse followed by a post-pulse at ts = 10 ps with a relative intensity of rs = 10−3. When the OPCPA operates at the pump-deletion level (ηp = 45%) as the same with that for Fig. 4(c), the nonlinear mixing of pump ASE with this signal post-pulse expands the total noise pedestal to a temporal range of ± 30 ps, as shown in Fig. 5(c). Figure 5(d) summarizes the intensity evolutions of the main coherent-noise-pedestal at t = 0, the replicas of this pedestal induced at t = −10 ps, −20 ps and −30 ps, as well as the coherent-noise spikes in this case. It is clear that this set of coherent-noise-pedestal replicas induced by the mixing of pump-ASE with signal post-pulse have their intensities grow quickly with the pump depletion, which is entirely different from the coherent-noise-pedestal replicas characterized in Fig. 4(c). It is because these pedestal replicas, as nonlinear beat noise, follow the growth behavior of not only the pump-ASE transfer, but also those coherent noise spikes at t = mts induced by the post-pulse. As plotted by the solid lines in Fig. 5(d), these noise spikes grow quickly with the pump depletion, and the intensity of each coherent-noise-pedestal replica is linearly proportional to the intensity of the corresponding spike (at the same temporal location).
Fig. 5 Temporal profiles of the compressed signal pulses after amplification (ηp = 45%) under the noise situations of (a) the pump pulse contains ASE noise (1 nm in bandwidth and 10−4 in relative intensity, the seed pulse is ideal clean; (b) the pump laser is clean while the seed pulse contains a post-pulse at ts = 10 ps with a relative intensity of rs = 10−3; and (c) the pump-intensity modulation and signal post-pulse exist together. (d) Intensity evolutions of coherent-noise spikes (solid lines) and replicas of coherent noise-pedestal (dotted-dash lines) located at t = −10ps (red), −20ps (blue) and −30ps (green), versus the parametric gain and pump-depletion (gray line). The intensity of the main coherent-noise-pedestal at t = 0 is also plotted (black).
3. Conclusions
We have demonstrated that nonlinear mixing of two kinds of coherent noise and the production of abundant nonlinear beat noise are inevitable in an OPCPA amplifier. A comprehensive characterization of this new family of noise has been developed in this paper. In general, the nonlinear beat noise can significantly expand the temporal range of the noise pedestal; and the magnitude of these noise is typically in the range of 10−10 to 10−15, implying that nonlinear beat noise will be an important limitation to the pulse contrast of future hundred-petawatt lasers. The nonlinear beat noise induced by pump noise (i.e., intensity modulations and ASE) establishes quickly in the small-signal amplification regime, while the nonlinear beat noise induced by signal noise (i.e., post-pulses) grows drastically in the pump-depletion regime. As these two groups of nonlinear beat noise exists simultaneously in practical OPCPA systems, it is difficult to circumvent all the nonlinear beat noise by optimizing the laser system design. Furthermore, the intensity of nonlinear beat noise is inherently proportional to that of the main pulse, implying it also cannot be effectively suppressed by the approach of double chirped-pulse amplification. It seems that minimizing the coherent noise at the source is a prerequisite to suppress the nonlinear beat noise in OPCPA systems.
Funding
National Natural Science Foundation of China (61727820); China Postdoctoral Science Foundation (2016M601577, 2017T100294).
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