## Abstract

Experiment and modeling show that the refractive index nonlinearity can significantly degrade the contrast of a chirped-pulse amplifier seeded with a pulse and a single postpulse. Multiple powerful non-equidistant pre- and postpulses are generated. For a Gaussian pulse and a hat-top beam, an incident postpulse of energy *W* results in a prepulse of energy 0.58*B*
^{2}
*W*, where *B* is the nonlinear phase (*B*-integral) of the main pulse. Calculations show that level of satellites due to gain saturation is negligibly small. Experimental results for Ti:Sapphire regenerative and multipass amplifiers and prepulse generation in fused silica agree well with the theory.

©2008 Optical Society of America

## 1. Introduction

Lasers based on chirped-pulse amplification (CPA) technique [1] can generate powers up to petawatt level [2,3] and create intensities up to 10^{22} W/cm^{2} [4]. One of the problems in CPA is to achieve high contrast ratio that becomes crucial in experiments on high-field laser-matter interactions. ASE and prepulses with intensities about 10^{12} W/cm^{2} can generate unwanted low-density plasma in advance of the main laser pulse and thus significantly influence laser-plasma interactions [5]. Up to date several sophisticated techniques have been proposed for pulse cleaning: using plasma mirrors [6], saturable absorbers [7], nonlinear rotation of the polarization ellipse in gases [8] and cross-polarized wave (XPW) generation [9]. The last method allows generation of high-energy pulses with contrast ratio peak-to-pedestal of 10^{11} [10].

It is well known that in many applications the time structure after the peak, like relatively weak post-pulses, can often be ignored. As a consequence, corresponding elements and designs, e.g. the plane-parallel optics, are not uncommon in both commercial and home-made CPA systems. However, any active medium has its nonlinear response due to saturation and nonlinear index of refraction. Hence, the amplified chirped pulse might have the spectral phase slightly shifted which in turn may result in prepulses after compression (Fig. 1).

The aim of this paper is to investigate this combined effect of the nonlinearity and post-pulses on temporal contrast of a CPA amplifier.

We will analyse here two possible sources of nonlinear prepulse generation and develop a simple quantitative model for the calculations. Data on satellite generation in multipass (MPA) and regenerative (RA) Ti:Sapphire (TiS) amplifiers using high dynamic range (>10^{9}) third-order autocorrelator (TOAC) has been obtained. Prepulse generation in RA and in fused silica is compared to the theory.

## 2. Theory

The mechanism of nonlinear generation of the satellites in a chirped-pulse amplifier is based on the following points:

- interference of the main chirped pulse and its delayed replica results in a stretched pulse sinusoidally modulated in intensity,
- the modulated light field changes an optical property of a nonlinear active medium,
- the time-dependent optical property modulates the propagating fields,
- the temporal modulation of the stretched pulse is equivalent to the modulation of the field spectrum,
- the compressor converts this sinusoidal modulation of the spectrum to the new satellites, since in a typical femtosecond CPA system the compressor (and, of course, the stretcher) in fact performs the Fourier transform of the incident field.

In other words, formally, a sinusoidal modulation in the Fourier original is equivalent to the satellites in the coupled Fourier image, and any nonlinear distortion of this initially sinusoidal modulation will produce new satellites absent in the original.

Direct analogies can be seen between this process and the higher-order diffraction on periodic patterns in optics, generation of the satellites under incomplete mode locking in lasers, and harmonic distortion in sound amplifiers.

In a CPA system, the effect of the refractive index nonlinearity is straightforward, it acts only as a pure phase modulator. However, the amplification in a saturated medium will modulate both the phase and the intensity of the field, because the radiation, which is near-resonant to the transition, changes both its intensity and phase. This is due to the fact that the rate of the depletion of the inversion is proportional to the intensity and hence the inversion itself follows the intensity modulation. In its turn, this modulation of the inversion modulates the amplified field through the time-dependent real and imaginary parts of susceptibility.

In what follows we will calculate and compare the magnitudes of the satellites which these two nonlinear mechanisms can generate.

## 2.1 Satellites produced by the nonlinearity of refractive index

An ideal stretcher (compressor) response to Dirac’s delta a *E*(*t*)=*δ*(*t - t _{0}*) is a delayed chirped pulse with bandwidth

*δω*centered at

*ω*

_{0}:

where *ω _{s}*=

*ω*

_{0}-

*δω*/2,

*t*is the response delay, τ

_{d}*=*

_{s}*δω/α*is the response time constant, and $\alpha ={\frac{{d}^{2}\phi}{d{t}^{2}}\mid}_{\omega ={\omega}_{0}}=-\frac{{\left({\frac{{d}^{2}\phi}{d{\omega}^{2}}\mid}_{\omega ={\omega}_{0}}\right)}^{-1}}{2}$ . We will omit

*t*for brevity further. Then the stretched pulse

_{d}is the response to the pulse *E*(*t*)=*E*
_{0}
*f*(*t-t*
_{0}) with pulsewidth *t _{p}*=∫

^{∞}

_{-∞}|

*f*(

*t*)|

^{2}

*dt*and bandwidth Δ

*ω*=

_{p}*a/t*, max|

_{p}*f*(

*t*)|=1. Here Δ

*ω*=(max|

_{p}*F*(

*ω*)|)

^{-2}∫

^{∞}

_{-∞}|

*F*(

*ω*)|

^{2}

*dω*,

*F*(

*ω*)=(2

*π*)

^{-1/2}∫

^{∞}

_{-∞}

*f*(τ)

*e*dτ and

^{-iωτ}*a*=Δ

*ω*is the time-bandwidth product.

_{p}t_{p}For short incident pulse and large stretch factor *t _{s}*/

*t*≫1, when the pulsewidth of stretched field is

_{p}*t*≅Δ

_{s}*ω*=

_{p}/α*a/t*, the stretcher(compressor) converts frequency to time:

_{p}/αwhere
$S\left(\omega \right)={\left(2\pi \right)}^{\frac{-1}{2}}\underset{-\infty}{\overset{\infty}{\int}}E\left(\tau \right){e}^{-i\omega \tau}d\tau $
is the Fourier transform of *E*(*t*).

If an input field consists of several short pulses,
$E\left(t\right)={E}_{0}\sum _{j=0}^{N}{r}_{j}{f}_{j}\left(t-{t}_{j}\right){e}^{i{\omega}_{j}t},$
, an output of the ideal stretcher with *φ _{s}*(

*t*)=(

*ω*+

_{s}*tα*/2)

*t*will be

Here
${r}_{j}^{\prime}={r}_{j}\mathrm{exp}\left(-i\left({\omega}_{s}-\frac{\alpha}{2}{t}_{j}\right){t}_{j}\right)$
and *t*
_{0}=0.

Terms exp(-*iαt _{j}t*) modulate stretched pulse amplitude due to the interference of the chirped fields in time domain after the stretcher. This intensity modulation will cause phase modulation of the fields due to the refractive index nonlinearity. The phase modulation of the stretched field is equivalent to the phase modulation in spectral domain. As a consequence, it results in satellites in temporal domain after compressor since the latter performs the inverse Fourier transform.

For the pulse and its weak replica (*r*
_{0}=1, *r*≡*r*
_{1}≪1) delayed to *T*=*t*
_{1}-*t*
_{0}, the nonlinear phase (*B*-integral *B*(*t*, *z*)=2*πλ*
^{-1}∫^{z}
_{0}
*n*
_{2}(*z*′)|*E*(*t*, *z*′)|^{2}
*dz*′, where *z* is the position in beam direction) accumulated in a medium with nonlinear refractive index *n*
_{2} and thickness *L* is

where *B̃*(*t*)=*B _{m}* cos(

*αT*),

*B*=2

_{m}*r*(2

*πn*

_{2}

*L/λ*)|

*E*′(

*t*)|

^{2},

*S*

_{0}(

*ω*)=

*S*(

*ω*)/max(|

*S*(

*ω*)|) and hence the modulated field will be

Assuming that the amplitude of the oscillating term is small and that *B _{m}*∝

*r*≪1, expanding the exponent in the right-hand term of (6) and keeping terms

*B*up to

^{p}_{m}r^{q}*p*+

*q*=2 one obtains relative amplitude of the stretched field of the 1

^{st}prepulse:

Finally, to calculate the satellite parameters after compressor one should take into account that the main stretched pulse produces shorter stretched satellite of smaller radius since

If the pump beam intensity profile inside nonlinear medium is a Gaussian *I*
_{0}(*t*, *ρ*)=exp(-(*t*/τ_{0})^{2}-(*ρ*/*ρ*
_{0})^{2}), then both radius and pulse width of the 1^{st} prepulse will be smaller in 3^{1/2} times. This is valid if overall nonlinear phase (*B*-integral) is less than ~1 rad. In this case phase distortion is small and both pulsewidths after compressor and beam radii are determined only by intensity profile like Eq. (8).

One should note that the intensity and fluence contrast of the focused beam may differ from that of the unfocused one since the larger divergence of the narrower prepulse beam. If the nonlinear phase is accumulated in the medium placed near the waist of the beam, like in most regenerative amplifiers and moderate-energy multipass amplifiers, the contrast will be the same. However, if the *B*-integral is accumulated in the near-field (like in a parallel beam of a high-energy amplifier), and the beam intensity profile is not uniform (say, a bell-shaped), then the contrast will be higher (Table 1).

For comparison with experiment, the plane-wave numerical calculations of the nonlinear satellites due to nonlinearity of the refractive index will be done. A short pulse field is convolved with stretcher response Eq. (1,2), then a phase modulation due to *n*
_{2} is added to the resulting stretched pulse, and, finally, the result is convolved with compressor response.

To obtain the power contrast, one should take into account the transverse profile of the beam. The plane-wave results should be corrected for different radii of the prepulses and the main peak. Consider an incident pulse with Gaussian temporal and radial intensity profile and a pulsewidth τ_{0}. The ratio of the prepulse beam area to that of the main pulse is *A _{s}*/

*A*

_{0}=(τ

_{0}/τ

*)*

_{n}^{2}, where τ

*is the prepulse pulsewidth. Hence, contrast of the*

_{n}*n*

_{th}prepulse calculated by the plane-wave simulations and multiplied by (τ

_{0}/τ

*)*

_{n}^{2}is the power contrast of the

*n*

_{th}prepulse. This simple method is exact for power nonlinearity and can be applied here since the nonlinearity in our case is smooth and very close to the power one.

Figure 2 shows that the analytical results agree well with numerical ones for moderate values of *B*-integral (≤1 rad) and contrast of the incident postpulse (≤10^{-3}). These plots can be used for an arbitrary pulse width *t*
_{s} and the postpulse delay *T* provided *T*≪*t*
_{s}. At large delays *T*≫*t*
_{s} the satellites’ magnitudes tend to zero due to weak interference of virtually non-overlapped main pulse and postpulse.

Note that if total *B*-integral in a CPA system is large, the nonlinearity of refractive index can generate 1^{st} prepulse whose energy and power are even higher than that of the incident postpulse.

## 2.2 Satellites produced by the gain nonlinearity

Consider an intensity modulated field *E*(*t*)=*I*
^{1/2}
_{0}
*g*(*t*)(1+*m*cos(Ω*t*))^{1/2} exp(^{iωt}) with pulsewidth *t*
_{s}=∫^{∞}
_{-∞}|*g*(*t*)|^{2}
*dt*, max|*g*(*t*)|=1. Saturated amplification in an active medium with peak cross-section *σ*, non-depleted inversion *N*
_{1}-*N*
_{2} and the Lorentz gain formfactor centered at *ω*
_{0} results [11] in an additional intensity modulation *H* and a phase shift *φ* as functions of modulation depth *m*:

$$\equiv {I}_{0}{\mid g\left(t\right)\mid}^{2}\left(1+m\mathrm{cos}\left(\mathrm{\Omega}t\right)\right)\left(1-Q\left(t,z,0\right)\right)H\left(t,z,m\right)\phantom{\rule{.2em}{0ex}},$$

If the amplified pulse is much longer than the period of modulation *t _{m}*, Ω

*t*≫1, then

_{s}*s*(

*t*,

*m*)≅

*s*(

*t*, 0)+

*s*

_{0}|

*g*(

*t*′)|

^{2}

*m*Ω

^{-1}sin(Ω

*t*). One can obtain corresponding small oscillating terms in nonlinear modulation provided the depth of incident intensity modulation is also small,

*m*≪1:

The intensity of the 1^{st} prepulse generated under saturated amplification of the intensity modulated field is

In a CPA system, a short femtosecond pulse with pulse width *t _{p}* is stretched to the duration

*t*=

_{s}*a/t*, where

_{p}/α*a*=Δ

*ω*is the time-bandwidth product of the short pulse. In this case the intensity modulation results from the interference of the stretched pulse and its weak replica delayed to the time

_{p}t_{p}*T*. Then the modulation period is

*t*=2

_{m}*π/α/T*.

Finally, the magnitude of the 1^{st} prepulse generated under saturated amplification of the intensity modulated field is

For the conditions of our experiment (*m*=6.8·10^{-2}, *t _{p}*=30

*fs*,

*T*=5

*ps*, Δ<1) the intensity of the 1

^{st}prepulse is

*I*

_{-1}/

*I*

_{0}≤3·10

^{-9}which is negligibly small compared to both calculated and measured levels of the satellite generated due to

*n*

_{2}(≈10

^{-4}).

## 3. Experimental arrangement

The CPA laser system used in our experiment consisted of a femtosecond oscillator, a stretcher, a regenerative (or multipass) amplifier and a compressor. Kerr-lens mode-locked oscillator produced pulses with duration of 25 fs, energy of 3 nJ, repetition rate 80 MHz. Laser pulses from the oscillator were stretched by an all-reflective stretcher to about 120 ps (50 ps for multipass amplifier) in which a grating with 1200 grooves/mm (600 grooves/mm) was used before seeding the regenerative (multipass) amplifier. A Faraday isolator was used to block the backward ASE from the amplifier.

Before amplification in the MPA a single pulse was selected from 80 MHz pulse train using a Pockels cell. The MPA consisted of two confocal mirrors with 90- and 100-cm radius of curvature. The MPA was pumped by 10 mJ, TEM_{00}, 20 Hz lamp-pumped Q-switched frequency-doubled Nd:YAG laser. The seed beam was focused into the crystal by the first (R=100 cm) mirror, recollimated by the second (R=90 cm) mirror and then sent back to the first mirror. After 8 passes through the amplifier crystal the energy of selected pulse was boosted up to 1.8 mJ. The amplified pulses were double-passed through two parallel gratings with 600 grooves/mm. For the elimination of optical damage to the gratings, the beam diameter was expanded to 5 mm by using a convex/concave mirror pair. After compression, optical pulses with energy of 1.1 mJ, pulse duration of 30 fs, wavelength of 810 nm and repetition rate 20 Hz were obtained.

The RA was designed in ring-cavity configuration. It consisted of two concave mirrors with 50 cm radius of curvature, two flat mirrors, two thin film polarizers and a Pockels cell. Both the polarizers and the electrooptical crystal of the Pockels cell were wedged at 0.2 degrees. A distance between concave mirrors is 565 mm and full cavity length is 2560 mm. A 10 mm long Ti:sapphire crystal is placed at distance of 265 mm from one of the concave mirrors. The RA was pumped by 1 mJ, TEM_{00}, 2 kHz diode-pumped Q-switched frequency-doubled Nd:YAG laser. One Pockels cell was used both for single pulse injection into the RA and for amplified pulse ejection from the RA. After 24 passes through the amplifier crystal the energy of the selected pulse was boosted up to 150 µJ. The amplified pulses were double-passed through two parallel gratings that had 1200 grooves/mm. For the elimination of optical damage to the gratings, the beam diameter was expanded to 5 mm by using a convex/concave mirror pair. After compression, optical pulses with energy of 100 µJ, pulse duration of 35 fs, wavelength of 810 nm and repetition rate 2 kHz were obtained.

The scan of contrast ratio was made with the TOAC (Model COMET800, Avesta Project Ltd.) based on sum-frequency generation technique (SFG, *ω*
_{3}=*ω*
_{1}+2*ω*
_{1}). The fundamental *ω*
_{1} generates a second-harmonic (SH) 2*ω*
_{1} in a BBO crystal. Then the second harmonic and the residual part of the fundamental are focused onto another thin BBO crystal (thickness d=180 µm) for SFG. Sum frequency signal at variable optical delay between the interacting fields at *ω*
_{1} and 2*ω*
_{1} yields the autocorrelation function. High gain and low noise of electronic detector unit and calibrated attenuation of the incident energy at fundamental frequency provide high dynamic range in SFG signal measurement. Two TOAC units of slightly different design and thickness of the crystals were used, one in the RA experiment and the other in the MPA and fused silica ones.

## 4. Experimental results and discussion

#### 4.1 Nonlinear prepulses in regenerative amplifier

The RA described above was seeded with a pulse accompanied by a postpulse. For each number of passes in the RA the compressor was adjusted to compensate for cavity dispersion and to achieve the shortest output pulse. The pulsewidth was measured with a single-shot 2^{nd}-order autocorrelator (Model ASF-20, Avesta Project Ltd.).

To create a weak postpulse in the seed beam, a 570µm thick uncoated plane-parallel plate (PP) made of fused silica was inserted into the beam perpendicular to its axis.

In the absence of nonlinear effects, when the PP was placed right in front of the TOAC for calibration, the contrast of the 1^{st} and 2^{nd} postpulses was close to the levels due to Fresnel reflection, 1.16×10^{-3} and 1.35×10^{-6}, respectively (Fig. 3(a), black line). In this case, the 1st “prepulse” is an artifact of mixing of the 2^{nd} harmonic of the 1^{st} postpulse and the fundamental of the main peak. The contrast of this artifact should be equal to 1.35×10^{-6}, the square of the 1^{st} postpulse contrast. The second TOAC artifact of magnitude 1.8×10^{-12} is below the detection level.

However, when the PP was placed between the oscillator and the RA, magnitude of both 1^{st} and 2^{nd} prepulses rose monotonically with the number of passes in the RA (Fig. 3(b), Fig. 4(a)). Note that level of the 1^{st} prepulse is up to three orders of magnitude higher than that of the artifact (Fig. 3(b), black line). Also, the nonlinear addition results in significant growth of the magnitude of the 1^{st} postpulse.

The observed rise of prepulse magnitude is explained by accumulation of the nonlinear phase in subsequent passes of the intense pulse through the TiS crystal in the amplifier (Fig. 4(e)). Fig. 4(a) shows measured and calculated contrast of the satellites for various numbers of passes in the RA. Calculated diameter of the RA cavity mode on the crystal is 0.146 mm FW1/eM intensity. The stretched pulsewidth is 120 ps, beam path in the crystal is 10 mm and the nonlinear refractive index of TiS is *n*
_{2}=3.4×10^{-16} cm^{2}/W [12]. These parameters result in the nonlinear phase *B*
_{0}=1.32 rad accumulated at pass 24.

For this calculation, the beam diameter in the TiS crystal and the intensity inside it were assumed to be constant. Surprisingly enough, even this simplest model of the amplifier results in a good agreement for the 1^{st} prepulse (Fig. 4(a)).

However, measured power of the 2^{nd} prepulse is initially 2 times lower than the simulation and after pass 25 the behaviour of the 2^{nd} prepulse changes abruptly.

The explanation is that the power of the 2^{nd} prepulse is more sensitive to the nonlinear phase. But real intensity profile along the beam inside the TiS crystal is not uniform due to amplification, and is obviously lower than the output one which is used for the simulation.

Moreover, while the central portion of the beam practically does not change in intensity due to gain saturation, the beam wings still continue to grow. This increases the beam diameter and causes significant departure of the 2^{nd} prepulse magnitude from the calculated curve. The reason is that the intensity *I- _{n}* of

*n*

^{th}prepulse depends strongly on the diameter

*d*

_{0}and intensity

*I*

_{0}of the main beam (see Sec.2.1 and Fig.2):

*I*~

_{-n}*B*

_{0}

^{2|n|+1}~

*I*

_{0}

^{2|n|+1}~

*d*

_{0}

^{-(4|n|+2)}. As an example, just 33% of diameter growth results in tenfold reduction of the prepulse magnitude.

Another important point is the phase distortion (both temporal and spatial) of the beam due to *n*
_{2}. At pass 26 the calculated phase is as large as *B*
_{0}=1.8 rad (see Fig.4(e)). This may mean significant space aberration and the temporal broadening of all compressed pulses.

Figure 4(d) shows the pulsewidths of the satellites, including the TOAC response time. The pulsewidth of the 1^{st} prepulse is larger than that of the main peak but less than that of the 2^{nd} prepulse, as it follows from their nonlinear origin. The point for 2^{nd} prepulse at pass 22 may be distorted by the noise because of the small prepulse magnitude.

An interesting feature is the delays of the satellites (Fig. 4(b)). The absolute value of the delay of the 2^{nd} prepulse should be larger than that of the 1^{st} one (see Sec. 4.4). However, their sign difference is completely unexpected. In addition, the behaviour of the curves changes at the same pass 25. All this may be attributed to the combined effect of deep gain saturation and large nonlinear phase after pass 25.

## 4.2 Nonlinear prepulses in multipass amplifier

Figure 5 shows the nonlinear prepulses in the MPA described above. To create a weak postpulse in the seed beam, a 570 µm thick uncoated plane-parallel plate (PP) made of fused silica was inserted into the beam perpendicular to its axis. Contrast of the 1^{st} seed postpulse is 1.16×10^{-3}. When the plate was inserted in the beam before the MPA a powerful prepulse was generated (Fig. 5).

## 4.3 Prepulse experiment with a fused silica plate as a nonlinear element

To identify whether the nonlinearity of refractive index alone is responsible for the observed effect, an intense stretched pulse accompanied by a postpulse was directed to a 5 mm thick plane-parallel sample made of UV-grade fused silica (KU-1).

The MPA described in the previous section was used. To achieve the highest intensity on the sample, the stretched output of the amplifier was focused as close as possible onto the front surface of the silica plate, slightly below the observed damage threshold. After passing the sample, the beam recollimated by a mirror was directed to the compressor and further to the TOAC.

The energy and pulsewidth of the stretched pulse incident to the sample were 1.8 mJ and 50 ps FWHM intensity respectively. The beam diameter on the focusing mirror was 2.7 mm FW1/eM intensity and distance to the waist was 375 mm. The sample was placed after the waist and the distance between its front surface and the waist was 23 mm.

An intense prepulse appears (Fig. 6) when a 0.57-mm thick uncoated plane-parallel plate made of fused silica is inserted between the MPA and the focusing mirror. Moreover, there is another much weaker prepulse at an approximately doubled time interval.

The measured contrast of the 1^{st} and 2^{nd} prepulses is 1.4×10^{-4} and 1.0×10^{-7} respectively.

An interesting feature is that the exact positions of the prepulses do not coincide with the multiples of the 1^{st} postpulse delay, 5.57 ps. Namely, both 1^{st} and 2^{nd} prepulses are delayed to 100 fs and 430 fs respectively, which is out of the error of the translation stage used. As it will be explained below, this is due to the temporal and spectral asymmetry of the stretched pulse. Note that position of the TOAC artifact of the plane-parallel plate (see Sec.4.1) is very close to −5.57 ps.

For the 1.8 mJ pulse and an ideal Gaussian beam the calculated diameter on the sample (0.18 mm FW1/eM intensity) results in fluence as high as 7.06 J/cm^{2} which equals to the laser-induced damage threshold (LIDT) of silica for the 50 ps pulse [13]. Since the real beam was very close to the diffraction-limited one but not exactly Gaussian, we set the fluence approximately 20% below the threshold for the calculations of satellite intensity.

The numerical calculations were made with the following parameters: wavelength 810 nm, *n*
_{2}=3×10^{-16} cm^{2}/W [14], silica thickness 5 mm, incident fluence 5.7 J/cm^{2} (0.82 to LIDT), short and stretched pulsewidths (FWHM intensity) 30 fs and 50 ps, postpulse delay 5.6 ps, relative intensity of the incident postpulse 1.16×10^{-4}. The calculated power contrast of the 1^{st} and 2^{nd} prepulses is 1.5×10^{-4} and 0.71×10^{-7} respectively (at resulting nonlinear phase (*B*-integral) *B*
_{0}=1.21 rad).

Ratio of calculated to measured contrast for the 1^{st} and 2^{nd} prepulses is 1.1 and 0.71, respectively. Taking into account large sample-to-sample variations of the nonlinear index of refraction in fused silica, *n*
_{2}=(2.5-3.5)×10^{-16} cm^{2}/W [14], we conclude that the agreement between theory and experimental data is very good.

## 4.4 Discussion

Our experiments confirm that the nonlinearity of the refractive index is responsible for the generation of the pre- and postpulses, and magnitudes of the satellites agree well with the simulation.

However, even in the fused silica experiment the satellites are not equidistant. The satellites are delayed with respect to the positions calculated from the measured delay of the 1^{st} (linear) postpulse. Also, these delays are less for the 1^{st}-order satellites than that of the 2^{nd} ones. Numerical modeling with use of a symmetric (Gaussian) pulses could not predict this feature, but this effect can be understood qualitatively.

Due to its nonlinear nature, the satellite is generated in the area of the highest intensity, i.e. near the top of the stretched pulse. The mean wavelength of a pulse is determined by the temporal position of its “center of gravity”. If the amplified stretched pulse is asymmetric, then the mean wavelength of the topmost part differs from that of the pulse as a whole. And this difference in wavelength will be converted to the time delay because of wavelength-dependent path length in the compressor.

This explains also why higher-order satellites have larger delays. The higher is the power of a nonlinear process, the more intense part of the stretched pulse is converted. Consider, for instance, the step-like (threshold) nonlinearity and “asymmetric triangle”. In this case, the nonlinearity converts a small triangle above the threshold, and the time shift of the center of gravity of this triangle is linearly proportional to the threshold.

We note that the change of the sign of chirp in a CPA design can reverse the signs of the relative delays of the satellites, provided the shape of the amplified spectrum will be the same in both systems. This is due to the fact that the path length difference between, say, the “reddest” and the “bluest” parts of the spectrum will be of opposite signs in the two compressors. Therefore, the signs of the relative delays will be reversed as well.

In the saturated regenerative amplifier the satellites’ delays behave in a more complicated manner and can not be explained by *n*
_{2} mechanism alone, since the gain saturation distorts the pulse shape and thus changes both the mean wavelength and position of the pulse maximum. In particular, the direction and magnitude of the shift of the maximum depends strongly on the steepness of the pulse front. Moreover, the gain saturation adds a monotonic phase sweep to the stretched pulse. This sweep in the frequency domain (before compressor) results in the pulse shift in time (after compressor). If the sweep slope is not symmetric relative to the pulse maximum, then the delays may be different even if the stretched pulse intensity profile is symmetric in time.

Whereas this shift of the satellites’ delays is interesting and not completely understood, it seems to be of little value to the CPA design, as compared to the effect of prepulse generation itself. It is always important to avoid strong self-focusing and concomitant damage to the optics while keeping the high efficiency of the amplifier. But if the design objective is a moderate-sized, moderate-priced CPA system, then the overall dimensions and the cost of large gratings may put severe limitations on the maximum duration of the stretched pulse. As a result, a relatively large *B*-integral value close to and above unity is often considered to be acceptable.

Such *B*-integral value leads to an inevitable consequence: the *n*
_{2} nonlinearity will “mirror” all time structures, lying within the stretched pulse duration after the peak, to the corresponding structures of comparable magnitudes before the peak. This, in turn, will result in significant degradation of contrast, if an intense pedestal and/or postpulses exist after the main pulse.

## 5. Summary

We have demonstrated that the refractive index nonlinearity of an active medium can significantly deteriorate the contrast of a chirped-pulse amplifier seeded with a pulse and a single weak postpulse.

Our calculations show that the gain saturation, another source of the nonlinearity, can also produce the nonlinear satellites. However, level of these satellites is negligibly small.

Several non-equidistant satellites at levels up to and above that of the seed postpulse can be generated. The power contrast of the most powerful 1^{st} prepulse equals to *B*
^{2}
_{0}
*I _{p}*/3 for the hat-top and

*B*

^{2}

_{0}

*I*/9 for Gaussian intensity profiles, where

_{p}*B*

_{0}is the nonlinear phase due to the nonlinearity of the refractive index (

*B*-integral) and

*I*is the intensity contrast of the incident postpulse.

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