## Abstract

We study a novel mechanism of pre-pulse generation in an optical parametric chirped-pulse amplification (OPCPA) system through an analytical approach together with numerical simulations. The acquired pre-pulses are initiated from the surface-reflection-induced modulation of the seed spectrum and occur as a consequence of high-order distortion of such modulated spectrum due to the instantaneous gain saturation effect. We demonstrate that the intensities of pre-pulses increase quadratically with the initial temporal modulation-depth of the stretched signal pulse as well as the conversion efficiency prior to substantial pump-depletion. Explicit formulas for estimating the contrast limit due to surface reflections are present. We also discuss the impact of group-velocity mismatch on the pre-pulse generation. The results of this work may deepen our cognition on the complexity of the pulse-contrast problem in OPCPA systems.

©2013 Optical Society of America

## 1. Introduction

Pulse-contrast of high-peak-power laser pulses has been a major concern in applications such
as high field physics [1]. Undesired laser light in the
form of pedestal or pre-pulses could trigger an early ionization of the target before the main
pulse arrives and alter the interaction properties detrimentally. Thus the pulse-contrast,
defined as the ratio of the peak intensity of the main pulse to the level of pedestal or
pre-pulse, is critical for a well-controlled laser-matter interaction experiment. Contrast of
10^{10} - 10^{11} is usually required for pre-plasma sensitive experiments
under the typical intensity of 10^{21} W/cm^{2}. Multi-petawatt laser systems in
the future will have even more demanding expectations for pulse-contrast (e.g., up to
10^{12} - 10^{14}), although pulse-contrast usually degrades with the increase
of operating power. Nowadays, contrast of 10^{9}- 10^{10} within tens of
picoseconds before the main pulse is almost the highest achievable level in high-power laser
systems either based on chirped-pulse amplification (CPA) or optical parametric chirped-pulse
amplification (OPCPA) techniques [1–8]. The mechanisms for pulse-contrast degradation have been
widely studied. Except the background noises such as amplified spontaneous emission (ASE) [1] or optical parametric fluorescence (OPF) [9,10], which are always
present in optical amplifiers, almost all of other proven contrast degradation mechanisms are
directly or indirectly associated with modulations of the spectral amplitude or spectral phase.
Spectral clipping from the acceptance of stretcher [2],
pump-noise transfer [11–13] and the presence of weak post-pulses [14,15] are typical sources of spectral amplitude
modulations. Incomplete compensation of the dispersion phase in the compressor [2], aberrations in the diffraction gratings in the stretcher
[16], and the Kerr nonlinearity [14,15,17–19] could readily lead to spectral
phase modulations. All of these effects will result in pulse-contrast degradation. Various
pulse-contrast enhancement schemes have already been investigated. Nonlinear-pulse-cleaning to
create a “cleaned” seed [8,20,21], double-CPA
architecture [22] and operation optimization of amplifier
[23,24] are
typically recommended for noise control in the part of laser driver. Beyond that, further
cleaning of the amplified high-power pulse based on certain peak-switching effect intrinsic to
intense laser interacting with matter has also been proposed. Such techniques include plasma
mirror (PM) [1] and second-harmonic generation of
high-intensity lasers [25].

It must be stressed that an OPCPA system, which presents itself as a perfect integration of optical parametric amplification (OPA) and CPA, has its special complexity in pulse-contrast control. This complexity mainly originates from the instantaneous nature of parametric amplification process, which implies that any temporal noise or variations on or among the interacting fields could distort the temporal amplitude of amplified signal. Furthermore, as the signal pulse is usually highly chirped, any temporal distortion can directly map to spectral domain and lead to pulse-contrast degradation. On the one hand, some of previous studies focus on the inherent advantage of OPCPA in contrast enhancement [26] as the instantaneous gain only exists within the time window defined by the short pump pulse. On the other hand, also due to the instantaneous parametric gain, it has been found that the picosecond-scale contrast of OPCPA is readily affected by almost all of the elements involved in the parametric process. Previous studies have shown that OPCPA contrast is susceptible to various pump noises, including pump intensity modulations [11–13], pump phase modulations [27] and the pump-seed jitter [10,29]. As for the ultra-broadband OPCPA, the contrast is also limited by the wavelength-dependent phase mismatch [28]. In this work, we look into the influence of a rarely studied but ubiquitous physical factor in OPCPA systems: the surface reflections. Post-pulse generation due to surface reflections is straightforward and has been demonstrated experimentally by Tavella et al. [29]. However, its impact on the pulse front is nontrivial and has not been explored so far.

In this paper, we first demonstrate that multiple pre-pulses can be generated in an OPCPA system due to the existence of surface reflections. In the presence of surface reflections, the chirped signal pulse will inevitably be modulated both in the temporal and spectral domains since the seed will interference with its reflected replica after stretching. We hence develop an analytical model to follow the evolution of the temporal and spectral amplitude of such modulated chirped signal pulse during parametric amplification process. We find that, the instantaneous response among the amplitude of such modulated signal, the amplitude of pump, and the amplitude of amplified signal will lead to rapid variations of parametric gain across the chirped seed pulse even in the case of perfect phase-matching and no pump noise, and result in high-order distortion of the spectrum of amplified signal. Consequently, the spectrum acquires several new components in spectral phase that can guide the energy of amplified signal to multiple pre- and post-pulses on the compressed output signal. Both of our analytical and numerical studies show that, prior to significant pump depletion, the maximum intensity of the induced pre-pulses are in direct proportion to the square of the temporal modulation depth of chirped signal before amplification as well as the square of the conversion efficiency. Despite that the intensities of pre-pulses get much more complicated law of growth when pump is substantially depleted, they continue to grow and can become comparable with or even stronger than that of post-pulses.

Note, quantum energy-level systems belonging to the non-instantaneous amplification process such as Ti:sapphire CPAs, are devoid of this pre-pulse generation mechanism. Only in nonlinear CPAs with very large B-integral (i.e., ΣB ≥ 10), pre-pulses could be significantly built up due to the Kerr nonlinearity since self-phase modulation (SPM) can induce nonlinear phase shift across the signal spectrum [14,15]. The Kerr-nonlinearity-initiated pre-pulse generation has also been identified as nonlinear temporal diffraction that is the time-domain analog of spatial nonlinear diffraction [18,19].

This paper is organized as follows. An approximate analytical model for understanding the pre-pulse generation behavior is introduced in Section 2. The numerical models concerning the surface reflections before and inside the OPCPA crystals are respectively built and the analyses of simulation results as well as the explicit formulas for calculation of contrast limit due to surface reflections are present in Section 3. Section 4 discusses the impact of the group-velocity mismatch (GVM) on the pre-pulse generation. Section 5 presents a summary of conclusions.

## 2. Analytical analyses

Optical amplification systems either based on CPA or OPCPA techniques inevitably suffer from post-pulses formation as the surface reflections of optics cannot be entirely eliminated. Generation of powerful pre-pulses due to the presence of post-pulses has been identified experimentally in Ti:sapphire regenerative and multi-pass amplifiers by Didenko et al. [14]. A theoretical analysis of pre-pulse generation in nonlinear CPA systems has also been thoroughly presented by Schimpf et al. [15]. It can be concluded that the Kerr nonlinearity is the cause for such pulse distortion. In OPCPA, however, the Kerr nonlinearity becomes trivial due to the short medium length (~1 cm) attributed by the high single-pass gain. In this contribution, by employing analysis methods analogous to that in Ref [15], we show that the instantaneous nonlinearity of parametric gain can also induce powerful pre-pulses generation when the amplification stages of OPCPA is seeded with pulses possessing weak post-pulses originating from surface reflections.

For the analytical derivations in this section, the signal and pump fields are represented
only as a function of time *t* (and equivalently optical frequency), without
spatial resolution. This assumption is reasonable for most cases in which the crystal length is
shorter than the diffraction length. The chirped pulse incident upon the amplification stage is
taken as

*r*and time-delay of

*t*.

_{δ}*ϕ*

^{(2)}is the second-order group-velocity dispersion (GVD) presented by the stretcher. The corresponding intensity profile can then be given by

*r*<< 1), we can drop the second term of Eq. (2) and rewrite it as

*p*is introduced to signify

*A*

_{0}((

*t-t*)/

_{δ}*ϕ*

^{(2)})/

*A*

_{0}(

*t*/

*ϕ*

^{(2)}). Then the modulation depth is explicit as $2p\sqrt{r}$ and the modulation period is 2π/(

*t*/

_{δ}*ϕ*

^{(2)}).

Using the small-signal approximation, the growth of signal intensity along the OPCPA crystal can be approximated by an exponential function [10,23]

*I*(

_{p}*t*),

*I*

_{s}_{0}(

*t*) and

*I*(

_{amp}*t*) are the intensity profiles of pump, input signal and output signal, respectively,

*L*is the crystal length. $\kappa =\sqrt{2{\omega}_{s}{\omega}_{i}{d}_{eff}^{2}/\left({n}_{i}{n}_{s}{n}_{p}{\epsilon}_{0}{c}^{3}\right)}$ is the gain coefficient, where

*n*are the refractive indices of the signal, idler, and pump fields respectively,

_{s,i,p}*ω*are angular frequencies of the signal and idler,

_{s,i}*d*is the effective nonlinear coefficient,

_{eff}*c*is the light speed in a vacuum and

*ε*

_{0}is the permittivity of vacuum. This simple solution is valid under the assumptions of constant pump, large parametric gain (i.e.,

*G*> 10) and perfect phase matching. We expect on a physical basis that, when the pump is depleted in small quantities, the solution to the parametric amplification process will still maintain the form of Eq. (4), except that

*I*(

_{p}*t*) will become a slowly varying function of

*L*. We hence take two steps to treat the parametric amplification in a thin crystal (making the pump depletion remain in small quantities) of length Δ

*L*pumped by a flattop pulse with intensity of

*I*

_{p}_{0}. We ignore the impact of pump depletion on the gain in the first half of crystal, while the time-dependent pump depletion Δ

*I*(

_{p}*t*) is then introduced by correcting the pump intensity for the second half of crystal as

*I*(

_{p}*t*) =

*I*

_{p}_{0}- Δ

*I*(

_{p}*t*). Thus the total gain in the crystal of Δ

*L*has the form

*I*(

_{p}*t*) as

*I*(

_{s}*t*) is the increase of signal intensity. For a chirped signal with (weak) temporal modulation, we can express

*I*

_{s}_{0}(

*t*) aswhere

*ρ*(

*t*) (∑

*ρ*(

*t*) = 0) represents the modulation superimposed on a uniform profile

*I*

_{0}. We also introduce the definition of the conversion efficiency

*η*:

*τ*and

_{p}*τ*are the temporal duration of pump and stretched signal, respectively. Combining Eqs. (5) through (8), we find that the amplitude of amplified signal is given by

_{s}*η*is low (i.e.,

*η*< 10%), we can further simplify Eq. (9) to

*I*

_{p}_{0}) and a time-independent coefficient, respectively. The term exp(-

*γη*) in Eq. (10) signifies the over-all gain saturation due to

*η*≠ 0 (i.e., pump depletion), while exp[-

*γηρ*(

*t*)] term indicates an instantaneous gain saturation, that is the instantaneous nonlinearity of parametric gain.

By substituting the modulation term *ρ*(t) in Eq. (3) into Eq. (10), we can obtain the temporal amplitude of the amplified signal as

*imt*Ω), where

_{δ}*m*could be any positive or negative integer. This expression bears a marked resemblance to the spectral amplitude of amplified signal in nonlinear CPA-systems, i.e., Eq. (12) in Ref [15]. In nonlinear CPAs, the distortion of spectral phase is straightforward as SPM effect introduces nonlinear phase shift directly. However, the derivations from Eq. (10) till Eq. (14) show that similar spectral phase distortion can also be induced by the instantaneous nonlinearity of parametric gain.

By compensating the GVD phase
exp(*iϕ*^{(2)}Ω^{2}/2) and conducting Fourier
transform of Eq. (14), we can obtain the temporal
amplitude of compressed signal as

*t*before and after the main pulse (at

_{δ}*T*= 0), but also indicates the intensity distributions of these pulses. In particular, when $2\gamma \eta p\sqrt{r}$< 1, the relative intensity of the most intense pre-pulse (i.e., 1st pre-pulse)

*I*/

_{pre-pulse}*I*

_{0}can be approximated by

*I*/

_{pre-pulse}*I*

_{0}, which could characterize the resulting decrease in pulse-contrast of OPCPA, is in direct proportion to the square of the conversion efficiency

*η*and the temporal modulation depth of the incident chirped-pulse$2p\sqrt{r}$.

In Fig. 1, the analytical calculations (using Eq.
(15)) of the compressed pulses from OPCPAs under different *η* and
*r* are present (as red solid line). With *η* increases
from 0.01% [Fig. 1(a)] to 0.1% [Fig. 1(b)] and then to 1% [Fig. 1(c)],
the relative intensity of the 1st pre-pulse (at *T* = −56 ps),
*I _{pre-pulse}*/

*I*

_{0}, increases from 10

^{−8}to 10

^{−6}and then to 10

^{−4}, showing a linear proportion between

*I*/

_{pre-pulse}*I*

_{0}and

*η*

^{2}. Similarly, when we keep

*η*( = 1%) the same and increase

*r*from 10

^{−3}[Fig. 1(d)] to 10

^{−2}[Fig. 1(e)] and then to 10

^{−1}[Fig. 1(f)], the resulting

*I*/

_{pre-pulse}*I*

_{0}increases from 10

^{−3}to 10

^{−2}and then to 10

^{−1}, showing a direct proportion between

*I*/

_{pre-pulse}*I*

_{0}and

*r*. These results lend support to the validity of the approximation formula Eq. (16). In addition to the direct analytical calculation, we also numerically calculated the compressed output of the signal in Eq. (12) employing Fast-Fourier transform. The numerical results (as black dashed line) are in good agreement with the analytical one for the intensities of pre-pulses. As for the intensities of post-pulses, the analytical results are lower than the numerical results when

*η*is low, as shown in Fig. 1(a) and Fig. 1(b). With the increase of

*η*, this difference gradually becomes negligible. The reason for such difference is that, for the OPCPA seeded with pulses possessing initial post-pulses, the post-pulses in the compressed output consist of two parts: the direct amplification of the initial post-pulses and the post-pulses induced by the nonlinearity of parametric gain. Thus, when

*η*is relatively low, the former part plays a major role, while it would be overtaken by the latter part when

*η*becomes high enough. The numerical calculations based on Eq. (12) have included both of the two parts, while the analytical results based on Eq. (15) have only considered the latter part. Since the pre-pulses are totally generated by the instantaneous nonlinearity, there is negligible difference between numerical and analytical results for the pre-pulse intensities. Note, all the analytical results in this section are only valid when pump depletion is not significant (i.e.,

*η*< 10%). To identify the pulse-contrast degradation characteristics when the OPCPA approaches the saturation regime of amplification, we have to resort to the full numerical simulations based on the nonlinear coupled-wave equations where the effect of pump-depletion integrated. This will be discussed in detail in the next section.

The dependence of *I _{pre-pulse}*/

*I*

_{0}on

*η*can be interpreted as the strengthening of the nonlinear coupling between signal and pump with the increase of pump depletion. Given that

*η*should be as large as possible for efficient energy extraction in some applications, we should deliberately minimize the initial amplitude modulation of chirped pulse for pre-pulse suppression. In practice, it can be realized by coating or wedging the transmissive optical elements in the laser path to minimize the surface reflectivity, as well as conducting OPCPA optimization to make the reflected replicas stay far away from the main pulse on the time axis.

The derivations in this section also reflect that, except for surface reflections, any kind of spectral amplitude or phase modulations that could lead to temporal amplitude modulation of the chirped pulse under amplification could result in detrimental pulse-contrast degradation in OPCPA systems due to the instantaneous parametric gain. And the higher the conversion efficiency is, the more the resulting pulse-contrast degrades.

## 3. Numerical modeling and simulation results

The analytical model built in Section 2, based on an extension of the traditional small-signal gain solution by introducing a pump-depletion factor, has predicted the contrast degradation characterized by pre-pulse generation. In the present section, full numerical simulations will be conducted to test the predictions and also extend the results into the saturation regime of amplification, where the above analytical solution is no longer valid. We classify the surface reflections in OPCPA into two categories. One is the reflections before the OPCPA crystal, which will provide the amplification process with a chirped signal carrying temporal and spectral modulations as mentioned before. The other is the reflections between the two surfaces of nonlinear crystal. In this case, the formation of post-pulses due to internal reflections, the interference between the original chirped signal and the time-delayed post-pulses, and the parametric amplification process occur in parallel, resulting in a more complicated behavior of pulse-contrast degradation.

#### 3.1 Numerical modeling

The parametric amplification process can be described by the coupled-wave equations having the form [10,31]

*A*are the complex envelopes of the signal, idler, and pump fields, respectively.

_{s,i,p}*n*and

_{s,i,p}*ω*are their refractive indices and angular frequencies. Δ

_{s,i,p}*k = k*is the wave-vector mismatch.

_{p}- k_{s}- k_{i}*v*,

_{gs}*v*and

_{gi}*v*are group-velocities defined as

_{gp}*v*. The reference frame is assumed to be moving at

_{g}= dω/dk*v*. Since crystal lengths are typically ~1 cm or shorter, high-order dispersion effects can be neglected. For definiteness, GVM effect is not included in this section and we will give a detailed discussion on its impact in Section 4. Besides, our simulations exclude the spatial transverse dimensions.

_{gp}#### 3.2 Surface reflections before the OPCPA crystal

To model the surface reflections before the OPCPA crystal, we consider an OPCPA seeded by a
femtosecond pulse followed by a post-pulse with an intensity ratio of *r* =
10^{−4} and a temporal separation of *t _{δ}* = 56
ps, as shown in Fig. 2(a). Both of the two pulses are transform-limited with center wavelength of 800 nm, spectral
width of 30 nm and temporal duration of 29 fs. They are stretched to 40 ps with peak intensity
of

*I*

_{s}_{0}= 5 kW/cm

^{2}(Fig. 2(b)) before being injected into a type-I β-barium borate (BBO) crystal. The pump pulse at 532 nm has a duration of 80 ps and peak intensity of 5 GW/cm

^{2}. It is set to be Gaussian and flattop distributed respectively to study the impact of pump profile. The inset in Fig. 2(b) shows the sinusoidal modulation on the trailing edge of stretched signal induced by the interference between the two pulses in Fig. 2(a) after stretching, just as predicted by Eq. (3).

Assuming an ideal compression (full compensation for the imposed GVD) and no further surface
reflections inside the crystal, we numerically simulate the temporal profile of compressed
output signal for an OPCPA system operating in different regimes of amplification. The results
are shown in Fig. 3. The output presents the same double-pulse structure [Fig. 3(a)] with the input if we set ∂*A _{p}* / ∂z
= 0 (constant pump). However, once we take account of pump depletion [Eq. (19)], the 1st pre-pulse preceding the main pulse
with a time interval of 56 ps appears with a relative intensity of ~10

^{−13}as shown in Fig. 3(b), even though

*η*is as low as 0.01%. When

*η*increases to 10%, the relative intensity of the 1st pre-pulse grows to ~10

^{−7}as shown in Fig. 3(c), agreeing well with the analytical prediction of

*I*/

_{pre-pulse}*I*

_{0}∝

*η*

^{2}in Eq. (16). Besides, a 2nd pre-pulse at 112 ps before the main pulse is also generated at

*η*= 10%. When we continue to increase the crystal length and push the OPCPA into the over-saturation regime, more side-pulses are formed and

*I*/

_{pre-pulse}*I*

_{0}increases to ~10

^{−5}, as depicted in Fig. 3(d).

Figure 4 shows the evolution of the relative intensities of side-pulses (normalized to the main
pulse intensity *I*_{0}) under a flattop [Fig. 4(a)] or Gaussian [Fig. 4(b)]
pump pulse, respectively. Despite of small difference in quantitative results, the evolutions
under Gaussian and flattop pump pulses share similar trends. The conversion efficiency is also
presented in the same figure to highlight the contrast level in different regimes of
amplification. The intensity of the 1st post-pulse shows little change throughout the
amplification process. In contrast, the newly generated side-pulses show sustaining rapid
growth. The growth of the 2nd pre-pulse lags a little behind that of the 1st pre- and the 2nd
post-pulse. In particular, the intensity of the 1st pre-pulse can increase to be comparable
with or even higher than that of the 1st post-pulse, when OPCPA operates in the over-saturation
regime.

In order to identify the dependency of the consequent contrast degradation on the parameters
of the initial post-pulse as well as the operating point of OPCPA, simulations were run for the
four sets of parameters: (i) *I _{s}*

_{0}= 5 kW/cm

^{2},

*r*= 10

^{−4}and

*t*= 56 ps; (ii)

_{δ}*I*

_{s}_{0}= 5 kW/cm

^{2},

*r*= 10

^{−3}and

*t*= 56 ps; (iii)

_{δ}*I*

_{s}_{0}= 5 kW/cm

^{2},

*r*= 10

^{−4}and

*t*= 48 ps and (iv)

_{δ}*I*

_{s}_{0}= 50 kW/cm

^{2},

*r*= 10

^{−4}and

*t*= 48 ps. Here, the pump pulse is assumed to be flattop distributed.

_{δ}By fitting the numerical results of
*I _{pre-pulse}*/

*I*

_{0}under varying crystal length from 1 to 10 mm, with an explicit formula having the form

*η*< 20% and the coefficient

*a*is set to be 0.4, as shown in Fig. 5. The parameter

*p*is related to

*t*through:

_{δ}*p*=

*A*

_{0}((

*t-t*)/

_{δ}*ϕ*

^{(2)})/

*A*

_{0}(

*t*/

*ϕ*

^{(2)}), just as defined in Section 2. For a Gaussian chirped signal pulse with duration of 40 ps,

*t*= 56 ps and 48 ps used in Fig. 5 correspond to

_{δ}*p*= 0.066 and 0.136, respectively.

Figure 5 and Eq. (20) illustrate that
*I _{pre-pulse}*/

*I*

_{0}increases quadratically with

*η*and $2p\sqrt{r}$when pump depletion is not significant, showing a very good quantitative agreement with our analytical predictions in Eq. (16). This provides further confidence in the accuracy of our analytical model in calculating the temporal and spectral distribution of amplified signal as well as the final pulse-contrast of OPCPA when the pump depletion cannot be ignored and when the chirped signal pulse carries temporal modulations, without the need of full numerical simulations. When

*η*> 20% or when OPCPA operates in the over-saturation regime,

*I*/

_{pre-pulse}*I*

_{0}can no longer be described by the simple formula of Eq. (16). However, it maintains growing, leading to a continual degradation of pulse-contrast as indicated by Fig. 5. The resulting pulse-contrast can even degrade to ~10

^{4}, which is absolutely unacceptable for many applications. The reason for the difference between the numerical and analytical results in the saturation regime of amplification is that

*η*would get saturated when significant pump depletion occurs. The saturation effect can be truly reflected by the full numerical simulations based on Eqs. (17), (18) and (19), yet it is overlooked by the exponential gain function of Eq. (4), which our analytical derivations are based on.

#### 3.3 Surface reflections inside the OPCPA crystal

Even though we seed the OPCPA crystal with a totally clean signal, the surface reflections
inside the crystal can still generate a time-delayed replica of chirped signal that overlays on
the trailing edge of the signal pulse as a post-pulse. To identify the contrast degradation
behavior in this case, we firstly simulate an OPCPA process in a 5 mm type-I β-BBO
pumped by a Gaussian pulse at 532 nm with a duration of 80 ps and peak intensity of 5
GW/cm^{2} and injected by a chirped pulse at 800 nm with a stretched duration of 40
ps, spectral width of 20 nm and peak intensity of 0.5 MW/cm^{2}. The surface
reflectivity of the β-BBO is assumed to be 0.1% (reflectivity for laser intensity),
aiming at highlighting the destructive effect of surface reflections, even when it is low in
the view of quantity.

The result of full numerical simulations shown in Fig.
6(a) demonstrates that all of the intensity profiles of pump, signal and idler acquire fast
quasi-sinusoidal modulations with modulation period of *T _{m}* = 0.08 ps
after amplification, even though the pump and signal is totally clean before amplification. The
modulation of pump has a complementary pattern with that of signal, while the modulation
pattern of idler is the same with that of signal. This can be interpreted as the Manley-Rowe
relations, which tells that in a lossless medium the rate at which the signal photons are
created is equal to the rate at which the idler photons are created and the rate at which the
pump photons are destroyed. The calculated spectra of signal before and after amplification are
plotted in Fig. 6(b), demonstrating the occurrence of a
quasi-sinusoidal modulation with period of 0.04 nm (i.e.,

*Ω*= 1.18 × 10

_{m}^{11}rad/s in the term of angular frequency) on the amplified spectrum. The consequent compressed output signal is depicted in Fig. 6(c). Despite that the injected seed possesses no post-pulses, the compressed output signal manifests as a main pulse accompanied by multiple side-pulses. The time separation between two adjacent pulses is 56 ps, which is just the ratio of twice the crystal length and the group velocity of signal as 2

*L*/

*v*(as

_{gs}*L*= 5 mm and

*v*= 1.78 × 10

_{gs}^{8}m/s in this case), claiming that these side-pulses are initiated from the surface reflections of signal within the crystal. We can also find that the modulation period of amplified fields in the temporal domain,

*T*, obeys

_{m}*T*= 2π/(

_{m}*t*/

_{δ}*ϕ*

^{(2)}) (as

*ϕ*

^{(2)}= 6.8 × 10

^{−25}s

^{2}/rad in this case), which agrees with the prediction of Eq. (3), and the modulation period in the spectral domain,

*Ω*, fits well with

_{m}*Ω*=

_{m}*T*/

_{m}*ϕ*

^{(2)}. The evolution of the intensities of acquired side-pulses within different amplification regimes is summarized in Fig. 6(d). The intensity of the 1st post-pulse is significantly higher than that of the 1st pre-pulse in the regime of small-signal amplification. Yet, the gap rapidly shrinks with the increase of conversion efficiency. Quantitatively, in the over-saturation regime, the generated pre-pulse can degrade the contrast of pulse front to ~10

^{4}.

To obtain an explicit formula like Eq. (20)
for *I _{pre-pulse}*/

*I*

_{0}in this case, we start from the difference between the contrast degradation processes initiated by the surface reflections before and inside the OPCPA crystal. For the “before-crystal” case, the modulation depth of chirped signal pulse before amplification is straightforward as 2

*p$\sqrt{r}$*, as indicated by Eq. (3). For the “inside-crystal” case, however, there is no initial post-pulse. Only after the injected signal undergoes single-pass amplification and double reflections by the inner surfaces of the crystal can a post-pulse be formed. Thus the intensity of such a post-pulse is expected to be proportional to the single-pass parametric gain

*G*and the square of the surface reflectivity

*R*. By replacing the

*r*in Eq. (16) by

*GR*

^{2}, we obtain a trial solution for

*I*/

_{pre-pulse}*I*

_{0}as

*b*is the constant coefficient for the trail solution. The parameter

*p*shares a similar definition of

*p*=

*A*

_{0}((

*t-t*)/

_{δ}*ϕ*

^{(2)})/

*A*

_{0}(

*t*/

*ϕ*

^{(2)}) with that in Eq. (4), where

*A*

_{0}is the temporal amplitude of the main signal pulse. The time-interval

*t*here equals the ratio of twice the crystal length and the group velocity of signal. To test the validity of Eq. (21), numerical simulations were run for the five sets of parameters: (i)

_{δ}*I*

_{s}_{0}= 10

^{−6}

*I*

_{p}_{0},

*R*= 0.1% and

*L*= 5 mm; (ii)

*I*

_{s}_{0}= 10

^{−6}

*I*

_{p}_{0},

*R*= 0.1% and

*L*= 4 mm; (iii)

*I*

_{s}_{0}= 10

^{−6}

*I*

_{p}_{0},

*R*= 0.1% and

*L*= 3 mm; (iv)

*I*

_{s}_{0}= 10

^{−6}

*I*

_{p}_{0},

*R*= 0.01% and

*L*= 5 mm and (v)

*I*

_{s}_{0}= 10

^{−5}

*I*

_{p}_{0},

*R*= 0.1% and

*L*= 5 mm. The simulation results, as shown in Fig. 7, are found to fit well with the trail solution of Eq. (21) when

*η*< 20% and the coefficient

*b*is set to be 0.4.

In addition, we find that the determination of
*I _{pre-pulse}*/

*I*

_{0}in this case can be divided into two relatively independent and interconnected steps, that is

*I*/

_{post-pulse}*I*

_{0}is the relative intensity of the 1st post-pulse against the main pulse. It is found to be proportional to

*GR*

^{2}but almost independent of

*η*, as shown in Fig. 8(a). The second term

*I*/

_{pre-pulse}*I*is the relative intensity of the 1st pre-pulse against 1st post-pulse. It is proportional to

_{post-pulse}*η*

^{2}but almost independent of

*G*and

*R*, as plotted in Fig. 8(b). This result is reasonable. As mentioned before, a post-pulse can be formed as the amplified reflection of the main pulse, with its intensity proportional to the parametric gain

*G*and the double reflection loss

*R*

^{2}. This post-pulse will be amplified and then guided to the position of the 1st post-pulse during compression. Although the nonlinear parametric gain can superimpose an additional 1st post-pulse to the compressed signal, according to Eq. (15), the intensity of this nonlinearity-induced post-pulse is typically lower than that of the post-pulse originating from surface reflections. That is the reason that

*I*/

_{post-pulse}*I*

_{0}is nearly proportional to

*GR*

^{2}and is little affected by

*η*. In contrast, since the 1st pre-pulse is totally induced by the nonlinearity of parametric amplification, its relative intensity is mainly determined by

*η*.

## 4. Effect of group-velocity mismatch

In this section, we concern the impact of group-velocity mismatch (GVM) on the pre-pulse
generation demonstrated above. Based on an intuitive though, since GVM can cause relative
movements among the interacting fields, the transfer of temporal modulations from chirped signal
pulse to the pump pulse may become less effective. At this point, we expect that the distortion
of signal spectrum will become less significant and the consequent pre-pulses will be weaker.
Although that the group velocities of signal and idler are typically matched to obtain a large
phase matching bandwidth, the GVM between pump and signal
(*δ _{sp}* = 1/

*v*- 1/

_{gs}*v*) is usually nonzero, we study the impact of

_{gp}*δ*, while assuming

_{sp}*δ*= 0 (

_{si}*δ*= 1/

_{si}*v*- 1/

_{gs}*v*).

_{gi}Full numerical simulations were primarily run for a set of parameters similar to that for
Fig. 3, except a shorter delay time of 10 ps rather than
56 ps for the initial post-pulse. This parameter is designed to make the modulation depth of
chirped signal pulse (2*p$\sqrt{r}$*) larger, so that the evolution of the modulations of
pump and signal during amplification could be identified more clearly. According to Eq. (3), the modulation period of chirped signal pulse
becomes *T _{m}* = 256 fs. With

*δ*varying from 0 to 1000 fs/mm, the consequent

_{sp}*I*/

_{pre-pulse}*I*

_{0}under a fixed conversion efficiency of 32% (saturated amplification) decreases from ~10

^{−5}to ~10

^{−7}, as demonstrated in Fig. 9(a). Simulations were also run for the chirped signal with modulation periods of 170 fs and 384 fs, obtained by changing the spectral width of seed to 45 nm and 20 nm respectively. Comparison of the three sets of results shows the shorter the modulation period, the effective for the GVM in impeding the growth of pre-pulses.

Figure 9(b) presents part of the intensity profiles of
pump and signal in the case of *δ _{sp}* = 0 and

*δ*= 1000 fs/mm, respectively. Compared with the case of

_{sp}*δ*= 0, the modulation depth of pump undergoes a significant reduction while that of the chirped signal pulse undergoes a slight increase after introducing

_{sp}*δ*= 1000 fs/mm. This is reasonable because the instantaneous pump-depletion in the case of

_{sp}*δ*= 0 could, on the one hand, induce a complementary modulation pattern on the pump pulse, yet on the other hand, partially compensate the intensity variation of the temporal profile of chirped signal as a negative feedback. From Fig. 8, we can conclude that an increase of

_{sp}*δ*is effective in impeding the growth of pre-pulses as it results in a decrease in the modulation depth of the pump pulse under the same modulation condition of chirped signal pulse. This result is consistent with our conjecture. Nevertheless, the advisable increase of

_{sp}*δ*is very limited, since

_{sp}*δ*can cause the signal to shift away from the peak of pump pulse, and other optical noises like OPF can then be preferentially amplified by the peak of pump pulse, resulting in a degradation of pulse-contrast in the form of pedestals with increasing level.

_{sp}## 5. Conclusion

We have shown that the surface reflections can lead to pre-pulses generation on the compressed
output signal, causing a degradation of the contrast of pulse front to be lower than
10^{6} in an OPCPA system, even when there is no pump noise, OPF or SPM at all. Such
the pre-pulse generation originates from the instantaneous nature of parametric gain and thus is
extremely hard to be eliminated. Although antireflection coatings and wedged optics have been
usually used to reduce the surface reflections, we claim the existence of surface reflection
will still be a potential bottleneck for the contrast improvement of OPCPA systems, especially
for the large aperture systems or systems with short inter-component spacing. The explicit
formulas derived in this paper can be used for direct calculation of the intensities of acquired
pre-pulses initiated by the surface reflections before or inside the OPCPA crystal, without one
having to resort the full numerical simulation. Besides, the analytic model built in this paper
revealing the instantaneous gain saturation effect provides another useful analytical tool for
analyzing the performance of OPCPA systems for real pump and signal pulses which are not
uniformly distributed in the temporal domain.

## Acknowledgments

This work was partially supported by the National Basic Research Program of China (973 Program) (Grant No. 2013CBA01505), and Natural Science Foundation of China (grant Nos. 61008017 and 11121504).

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