We theoretically study the pulse-contrast degradation of optical parametric chirped pulse amplification (OPCPA) system pumped by a temporally phase-modulated laser. We reveal two physical mechanisms that reduce the contrast in such a parametric process: the effect of phase transfer caused by group-velocity mismatch (GVM) and the effect of FM-to-AM conversion caused by group-velocity dispersion (GVD). Degradation of the contrast is quantified via numerical simulations. To reduce the effect of phase transfer, a pair of OPCPAs seeded by the signal and idler respectively, is studied for effectively improving the contrast in such a case with pump phase modulation. The results presented in this paper are of particular importance for extremely intense laser applications requiring high contrast.
©2010 Optical Society of America
The development of high-intensity chirped-pulse amplification (CPA) techniques creates exciting opportunities for experimental investigations in the relativistic regime of laser-matter interactions . One of the major challenges is the capability of providing sufficiently high pre-pulse contrast required by preplasma-free interaction [2,3]. The pulse contrast, defined by the ratio between the pre-pulse or pedestal peak-intensity to that of the main pulse, is therefore one of the most important parameters of a high-intensity laser system. At typical interaction intensity (I) of ~1020 W/cm2, one would require a pre-pulse contrast better than (i.e., lower than) 10−10 within the range a few picosecond beyond the pulse peak to ensure that ionization does not affect the target foil. Pulse contrast can be reduced significantly during the generation, propagation and amplification of the pulse, and the pulse-contrast degradation manifests itself as isolated prepulses or a slowly varying pedestal. In order to solve this bottleneck of pulse contrast in high-intensity CPA laser systems, the prerequisite is to reveal the physical mechanisms of pulse-contrast degradation. It is known that isolated prepulses occurring at picosecond time scale may stem from incomplete compensation of high-order dispersions, aperture-limited spectral clipping, spectral distortion during amplifications, or self-phase modulation [4–7]. Incoherent laser or parametric fluorescence can significantly affect the pulse contrast, which leads to a broad pedestal on the recompressed pulse . This contrast degradation is fundamental since fluorescence is always accompanied with the amplification of laser pulses, which may be greatly isolated between different amplification stages by using double CPA design .
The requirement on pulse-contrast level depends on the pulse peak-intensity. For instance, it requires a pulse-contrast better than 10−15 for the planned Extreme Light Infrastructure (at I ~1025 W/cm2) . For such an extreme requirement, even very weak limiting factors must be taken into account in analyzing pulse-contrast degradation. It is now clear that the cross-phase modulation in CPA plays an important role in degrading the pulse contrast: the postpulses may result in the generation of prepulses , and also the prepulses caused by spectral phase ripples may be enhanced . Because of the prospects in obtaining high-power few-cycle laser pulses, the contrast of optical parametric chirped-pulse amplifiers (OPCPA) is of great interest. Due to the instantaneous nature in OPCPA processes, intensity modulation of the pump pulse will be directly coupled into the spectrum of the chirped signal, which is fundamentally different from the conventional quantum energy-level laser amplifiers. The impact of temporal intensity-modulation on the contrast in an OPCPA system was first identified by Forget et al. , followed by numerically studies [13,14]. With various frequencies of temporal intensity modulations, the pump-induced contrast degradation manifests itself as a temporal pedestal with duration of ~10-ps that depends on the coherence time of amplified spontaneous emission (ASE). Though less appreciated, phase modulation of the pump pulse may also be converted to the chirped signal in a practical parametric process due to the group-velocity mismatch (GVMi-p) between the pump and idler [15,16]. Such a phase transfer may also lead to a degradation of temporal contrast for the recompressed signal pulse in an OPCPA system, which has not been well addressed in the literatures. It should be pointed out that a practical pump pulse is always accompanied with a certain extent of phase noise due to non-ideal lasing mode and/or ASE. In the case of high-energy pump pulse, particularly, the laser pulse is intentionally modulated in phase to suppress the effect of stimulated Brillouin scattering (SBS) .
Our research in this paper is motivated by developing large-scale OPCPA with high pulse-energy well over tens of Joule [18–20]. In these high-energy OPCPAs, high-energy Nd:glass lasers were adopted as the pumping source, which were typically modulated in phase with the frequency of ~10-100 GHz if their pulse energies are ~KJ or high . It is necessary to study the impact of pump phase modulation on pulse contrast and to develop approaches to solve the problem. In this paper, we focus our study on the pulse-contrast degradation due to pump phase-modulation in an OPCPA system while ignore the well-known factors such as parametric fluorescence and pump intensity-modulation.
As a main result of this paper, our numerical simulations demonstrate that temporal phase-modulation of the pump pulse will be partly transferred to the phase of the chirped signal in a typical parametric process with non-zero GVMi-p, and therefore lead to a pulse-contrast degradation. Since phase modulation (hence frequency modulation, FM) may be converted to the amplitude modulation (AM) of a laser pulse , in this paper we also study the impact of group-velocity dispersion (GVD) on the pulse contrast of an OPCPA system. We note that the FM-to-AM conversion might be avoided by using multi-pass phase modulator in a fiber loop [23,24], however, current high-energy Nd:glass laser system still adopts single-pass modulator in which the FM-to-AM conversion in OPCPA will appear. Although the effect of GVD of a practical nonlinear crystal on nanosecond pump pulses is negligible, the FM-to-AM conversion due to GVD can be significant, which in turn leads to the pulse-contrast degradation. To determine the pulse contrast of an OPCPA system with pump phase-modulation, it is necessary to include both the effects of phase transfer and FM-to-AM conversion (i.e., GVD). A pair of OPCPAs seeded by the signal and idler respectively, the so-called hybrid seeded OPCPA (HS-OPCPA) [25,26], is studied for effectively improving the contrast in such a case. The paper is organized as follows. In Section 2, we present the numerical model to deal with an OPCPA system. For the purpose of offering a physical explanation of the phase-modulation induced pulse-contrast degradation, Section 3 discusses the mechanisms of phase transfer and FM-to-AM conversion, respectively. Section 4 presents the numerical results to quantify the pulse-contrast caused by the effects of phase transfer and FM-to-AM conversion. In Section 5, we study a pair of OPCPAs for the purpose of improving pulse contrast. Finally, conclusions are given in Section 6.
2. Numerical model
In this paper, the OPCPA pumped by a temporally phase-modulated pulse is treated by the nonlinear coupled-wave equations in time domain, which implies that all the transverse effects in spatial domain are ignored. To the first order in the quadratic susceptibility χ(2) and when all derivatives of the linear refractive index beyond the second-order are neglected, the equations that govern the envelopes E p, E s and E i of the pump, signal and idler pulses, respectively, are
The source term for quantum noise is not included in the equations, thus the effect of parametric fluorescence will not appear in our simulations. The time variable t is normalized to the Fourier-transform-limit-corresponding duration (τs) of the signal, and Δk = k s + k i −k p is the phase mismatching at the central frequencies. The nonlinear length is defined by as a measure of the pump intensity, where is the input pump field. () is the GVM length of the idler (signal) with respect to the pump pulse. The ratio of L ip to the crystal length L indicates the practical GVM magnitude of the parametric amplification process, and the smaller L ip is, the larger practical GVMi-p will be. Since most OPCPA systems work near at wavelength degeneracy, we assume equal group-velocities of the signal and idler (i.e., GVMs-p = GVMi-p and L sp = L ip) in this paper. The dispersion effect is characterized by the dispersion length, in which is the group-velocity dispersion of the j-th wave. Standard split-step method and Runge-Kutta algorithm are adopted to simulate the parametric processes in this paper. Although the above parameters are normalized and may also be varied for investigation purposes, we should point out that they are based on a typical OPCPA system at wavelength of ~1-μm using KDP or β-BBO crystal and a green pumping laser .
3. Physical mechanisms reducing the pulse contrast
To characterize the impact of pump phase-modulation, we assume a 1-ns Gaussian pump pulse with a sinusoidal modulation in phase (Fig. 1(a) ): , , where the parameters m and f m correspond to the modulation index and frequency (speed rate), respectively. Due to phase modulation, the spectrum of the pump pulse consists of several equal-spaced narrow peaks, as shown in Fig. 1(b). An initial Fourier-transform-limited 100-fs Gaussian signal pulse (Fig. 1(c)) is stretched to 1-ns with a parabolic phase profile (Fig. 1(d)). In the process of parametric amplification, due to the effect of phase transfer caused by GVM or the effect of FM-to-AM conversion caused by GVD, the temporal phase-modulation of pump pulse may cause the chirped-signal modulated in both the phase and amplitude, and in turn will result in pulse-contrast degradation of the recompressed signal pulse. In the following, we will discuss the mechanisms that reduce the pulse contrast by considering the effects of GVMi-p, group-velocity dispersion at the idler (GVDi), and group-velocity dispersion at the pump (GVDp), respectively.
3.1 Phase transfer caused by GVM
In the process of parametric amplification, phase-matching condition (Δk = 0) leads to a well-known phase relation, ϕ p = ϕ s + ϕ i + π / 2. The generated idler wave bears the phase difference between the pump and signal waves, while the phase of amplified signal will be independent on the pump phase and maintains its initial phase . This phase relation, however, only holds when all the group velocities of interacting waves are matched to each other. In a practical OPCPA system, there always exists GVM between the pump and idler (signal). This nonvanished GVMi-p has little impact on the bandwidth of parametric gain, and will not affect the feature of chirped signal pulse if the pump pulse is ideal in phase (i.e., without phase modulation or noise). The situation will be very different if pump phase is not uniform in time. The exist of GVMi-p makes the phase profiles of the pump and idler relatively shift in time, thus their phase difference will no longer be constant, which will in turn affect the signal phase. This effect was termed as the so-called phase transfer [15,16]. As a result of the phase transfer due to GVMi-p, the amplified signal pulse partly bears the temporal phase-modulation of the pump (Fig. 2(a) ) that is imposed on the initial parabolic phase of the chirped pulse, which naturally induces spectral modulation itself, as shown in Fig. 2(b).
In addition, we note that the temporal phase modulation of signal pulse is linearly varied with time globally (inset in Fig. 2(a)). This is equivalent to a shift of the center wavelength. During the parametric process in OPCPA, the idler pulse is also chirped with an opposite sign to that of the chirped signal pulse. Signal wavelength shifting is a result of nonlinear interaction of chirped pulses under the condition of GVM , which may be understood easily by regarding the amplification of the signal pulse as a process of difference frequency generation (DFG) between the pump and signal. At a certain time t apart from the pulse center, the DFG between the pump and idler generates the signal component with an instantaneous frequency ωs = ωp − ωi(t) at the crystal entrance. As their propagations to a position (z) within the crystal, the instantaneous frequency of locally generated signal will be changed to a different value ωs = ωp − ωi(t + z × GVMi-p) due to the temporal walk-off between the pump and idler. After the amplification passing through the whole crystal, a shifted signal spectrum will be resulted since longer crystal (and also larger z) favors signal amplification.
In comparison, the amplitude-modulation on the chirped signal pulse is not observable (Fig. 2(a)). Since the time period of the modulation is much longer than the overall temporal walk-off, the magnitude of transferred signal phase-modulation linearly grows with the increase of GVMi-p (the decrease of L ip), also does its induced spectral modulation of the amplified chirped signal. For example as shown in Fig. 2(a), the magnitude of the signal phase-modulation (about 0.2 rad) with GVMi-p = 20 fs/mm (L ip = 0.5L, red curve) is twice as large as that (about 0.1 rad) with GVMi-p = 10 fs/mm (L ip = L, black curve). The spectral modulation changes with GVMi-p in a similar way (Fig. 2(b)). In the case of small-signal amplification, the pump pulse and its phase almost maintain their initial values (Fig. 2(c)).
To compare different regimes of parametric amplification, Fig. 3 presents the results for the case of saturated amplification (conversion efficiency η ~28%). Although the pump depletion may saturate the peaks of the signal pulse and its spectrum, the induced signal phase modulation and spectral modulation are quantitatively the same as those in the small-signal regime (Fig. 3 (a) and (b)). Meanwhile, the pump phase may also keep its initial value until back conversion from the signal to the pump occurs. Figure 3 (c) displays the situation that the pump pulse is strongly depleted in the center part corresponding to our simulation condition.
3.2 Phase transfer caused by GVD at idler
Similar to the reason of GVMi-p, the relative difference between dispersions of the pump and idler pulses in the nonlinear crystal may also cause the phase transfer from the pump to signal, and in turn induce the spectral modulations of the chirped signal pulses both on phase and amplitude. On the other hand, dispersion at the signal wave (GVDs) alone has little impact on the phase transfer and contrast degradation, thus the effect of GVDs will not be discussed in this paper. Since GVDp may also cause the effect of FM-to-AM conversion, we will separately discuss the impacts of GVDi and GVDp.
Figure 4 plots the pulse, spectrum and phase modulation of the amplified chirped signal with the effect of GVDi in the small-signal regime. As expected, the amplified signal pulse partly bears the temporal phase-modulation of the pump (Fig. 4(a)), which also induces spectral modulation itself, as shown in Fig. 4(b). Different from the case with GVM, the magnitudes of the signal phase modulations in both time and frequency domains are not uniformly distributed, which are weaker in the center parts of the chirped pulse and its spectrum. By increasing GVDi, both the magnitudes of signal phase-modulation and spectral modulation also linearly increase. Compared to the case of GVMi-p in terms of equal GVM and dispersion lengths, GVDi has a relatively smaller impact on phase transfer and spectral modulation of the chirped signal. As shown in Fig. 4(a), the effect of GVDi has little impact on the pump pulse in the small-signal regime. The pump pulse is smooth in amplitude and maintains its initial phase, which is very different from the case with GVDp as will be discussed later in Subsection 3.3.
In the saturated regime as illustrated in Fig. 5(a) , some degrees of amplitude modulation at the pump pulse, though not severe, can be observed due to the strong coupling among the three interacting waves. As a result, the spectral modulation in the saturated regime (Fig. 5(b)) is significantly larger than that in the small-signal regime.
3.3 FM-to-AM conversion caused by GVD at pump
Similar to the case with GVDi, GVDp will also cause the phase transfer from the pump to the signal, with uniformly distributed phase-modulation in time (inset, Fig. 6(a) ). Compared with the effects of GVMi-p and GVDi, the influence of GVDp on the pump-to-signal phase transfer and spectral modulation of the chirped signal is the smallest under the similar conditions. The magnitude of the signal phase-modulation is only ~0.028 rad (red curve in Fig. 6(a)) with L 2p = 0.5L, which is nearly one order of magnitude smaller than that with GVMi-p (~0.2 rad, red curve in Fig. 2(a)) and GVDi (~0.03 to 0.16 rad, red curve in Fig. 4(a)).
Due to FM-to-AM conversion at the pump pulse, the situation with GVDp will be more complicated. Although the effect of GVD of a practical nonlinear crystal is negligible for a nanosecond pump pulse, the FM-to-AM conversion due to GVD can be significant. As clearly shown in Fig. 6(a), a certain degree (~5%) of intensity modulation on the pump pulse will be developed by GVDp with moderate values (the dispersion length L 2p comparable to the crystal length L). The impact of GVDp on FM-to-AM conversion is determined by the spectral width of the pump pulse, and is not affected by the duration of the pump pulse. Since the instantaneous nature in a parametric process, the amplitude modulation will be imprinted on to the signal pulse (Fig. 6 (a)), whose magnitude is much larger (~5 times) than that of the pump pulse due to its exponential dependence of field amplitude of pump pulse in the regime of small-signal gain.
Both the phase and amplitude modulations of the chirped signal will contribute to the spectral modulation, in which the amplitude modulation makes a larger contribution. The two sets of spectral modulation are anti in phase, which therefore leads to a periodically structure in the signal spectrum (Fig. 6(b)). By increasing GVDp, both the pump pulse modulation and signal phase modulation will increase, and so that the amplitude and phase modulations of the signal in the frequency domain will also increase (Fig. 6(b)). These amplitude and phase modulation in frequency domain will degrade signal contrast significantly. Taking the temporal phase modulation of speed 40 GHz, index 10 and GVDp = 100 fs2/mm in β-BBO crystal, for example, the signal contrast will be degraded to about 10−6 after the parametric amplification in a 20 mm thick crystal.
When the pump is depleted in the saturated regime, the signal pulse modulation in time domain will be greatly suppressed (Fig. 7(a) ). Accordingly, the spectral modulation of the signal pulse will be smaller (Fig. 7(b)). Comparing Fig. 6 and Fig. 7, the signal phase modulations in time domain are almost the same, while the spectral phase in the saturated regime will be much smaller than that in the small-signal regime. This highly suggests that the amplitude modulation of chirped pulse will significantly affect the spectral phase, and also identifies the two contributions in the spectral modulation (i.e., the amplitude and phase modulations in time domain).
4. Contrast degradation
As discussed above, both the effects of GVM and GVD will cause the amplitude and phase modulations of the signal pulse in frequency domain, which will in turn degrade contrast of the recompressed signal pulse. The intensity distributions of the recompressed signal pulse with GVMi-p, GVDi and GVDp are plotted in Fig. 8 , respectively. We can see that there are several pre- and post-pulses appearing around the main pulse, and the pulse contrast is significantly degraded depending on the modulation index and frequency of the pump phase and the magnitudes of GVMi-p, GVDi and GVDp. In all the cases as shown in Fig. 8, the time interval between the pulse peaks is determined by the modulation frequency of the pump pulse and the chirp parameter of the signal, and is not affected by GVMi-p, GVDi and GVDp. The higher modulation frequency is, the larger time interval will be. As suggested by Fig. 8, contrast degradation is more serious in the case with GVM effect.
Contrast degradation is a result of combined effects of pump phase-modulation and GVM or GVD. It is important to quantitatively study the dependence of the contrast with the phase-modulation parameters, GVM and GVD. Figure 9 summarized the simulation results for the intensity of the first side-pulse relative to the main peak (i.e. pulse contrast) under different conditions. As shown in Fig. 9(a), when the modulation index (m) increases by a factor of n, the pulse contrast (S) will be degraded by a factor of n 2 if GVMi-p, GVDi and GVDp are all fixed. With fm = 20 GHz and m = 5, for example, the pulse contrast is 1.98 × 10−4 for GVMi-p, 7.47 × 10−5 for GVDi, and 3.89 × 10−6 for GVDp, respectively. By increasing the modulation index to m = 5 × 2 and fixing the modulation frequency at fm = 20 GHz, the contrast in each case will be decreased by 4 times to a value of S = 7.93 × 10−4 ≈4 × 1.98 × 10−4 for GVMi-p, S = 2.98 × 10−4 ≈4 × 7.47 × 10−5 for GVDi, and S = 1.55 × 10−5 ≈4 × 3.89 × 10−6 for GVDp, respectively. The relationship between the contrast and the modulation frequency fm is very similar for fixed GVMi-p and GVDi, and the change of modulation frequency fm to n fm will lead to a degradation of the contrast from S to n 2 S at fixed GVMi-p or GVDi. As shown in Fig. 9(b), however, the relationship between the contrast and the modulation frequency fm will be very different for fixed GVDp, and the change of modulation frequency fm to n fm will lead to a degradation of contrast from S to n4 S at fixed GVDp.
At fixed modulation frequency (fm = 20 GHz) and index (m = 10), the dependences of contrast with GVMi-p, GVDi, and GVDp are illustrated in Fig. 9 (c). We can see that with the increase of GVMi-p, GVDi, or GVDp (decrease of L ip, L 2i, or L 2p), the pulse contrasts decease monotonically. Since the criteria for designing femtosecond OPAs or OPCPAs is generally in terms of GVM and dispersion lengths (i.e., these lengths are larger than the crystal length), it might be more reasonable to compare the effects of GVM and GVD with equal characteristic lengths. As indicated by Fig. 9, GVMi-p-caused phase-transfer is the most severe factor in degrading the pulse contrast, which is about 3 times stronger than that of GVDi, and nearly two orders of magnitude more serious than that of GVDp-caused FM-to-AM conversion. Since GVMi-p and material dispersions are inevitable in OPCPAs, pump phase modulation can lead to a significant degradation of the pulse contrast, which should be taken into account in designing high contrast OPCPA systems.
5. Contrast improvement
Pulse contrast is one of the main bottlenecks limiting the development of high-intensity laser systems. As discussed in Section 4, GVMi-p-caused phase-transfer is the most severe influencing factor compared with the GVDp caused FM-to-AM conversion. Therefore we employ an HS-OPCPA configuration (Fig. 10 ) to reduce the pump-to-signal phase transfer and to improve the pulse-contrast. The HS-OPCPA consists of two OPA crystals, in which the first crystal is seeded by the signal while the second crystal is seeded by the generated idler from the first crystal. Thus, the signal phase modulation experienced in the first OPA crystal due GVMi-p can be well removed in the second OPA crystal.
As shown in Fig. 11 , due to GVMi-p, the pump phase-modulation is imprinted onto the chirped signal (black curve in Fig. 11(a)), and induces spectral modulation of the signal itself (black curve in Fig. 11(b)), which leads to the pulse-contrast degradation. These signal phase and spectral modulations will be enhanced (red curve in Fig. 11 (a) and (b)), and the pulse contrast will be further degraded in the second OPCPA stage if it is seeded by the amplified chirped signal from the first stage. By contrast, the situation will be completely different if the idler seeding is adopted in the second stage, the pump-to-signal phase transfer can be reduced drastically and the phase of the final output signal can almost be resumed to its initial value (blue curve in Fig. 11(a)), also does its induced spectral modulation of the amplified chirped signal (blue curve in Fig. 11(b)).
Figure 12 summarizes the results of the pulse-contrast using HS-OPCPA. Due to the enhancement of the phase-transfer in the second OPCPA with signal seeding, the pulse-contrast will be further degraded by several times in magnitude (Fig. 12 (a) and (b)). By using the HS-OPCPA, on the other hand, the contrast of the compressed signal pulse from the second stage can be greatly improved by about four orders of magnitude (Fig. 12(c)).
In summary, we have studied the pulse contrast of an OPCPA system pumped by a temporally phase modulated laser. Based on the numerical study on such a parametric process, we have revealed and discussed two novel physical mechanisms reducing the pulse contrast: the effect of phase transfer caused by GVM or GVD and the effect of FM-to-AM conversion caused by GVD. Degradation of the pulse contrast has been quantified via numerical simulations. We have shown that GVM-caused phase-transfer is the most severe factor in degrading the pulse contrast, which may limit the contrast to only ~10−4 to 10−5 depending on the pump phase modulation. By using the HS-OPCPA, the GVM-caused phase-transfer can be well reduced and the pulse-contrast can be effectively improved. Our study should be useful for understanding the contrast limitation in OPCPA systems. The impact of contrast degradation due to pump phase modulation should be considered in the design of high-contrast OPCPA systems.
This work was partially supported by the Natural Science Foundation of China (NSFC) grants 10776005, 60890202, and 60725418, and the National Basic Research Program of China 973 Program grant 2007CB815104.
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