1. Introduction

Chirped pulse amplification (CPA) is an established technique to generate high energy ultra-short laser pulses by reducing the pulse peak-power during amplification to avoid distortions due to nonlinear effects and/or damage of optical elements [1]. The influence of self-phase modulation (SPM), which is a consequence of the intensity dependence of the refractive index (Kerr-effect) is particularly detrimental [2]. Despite stretching of the pulse, the onset of nonlinearity during amplification can cause a degradation of the peak-power at the output of the amplification system [3]. CPA-systems in which the accumulated phase-shift due to SPM (i.e., the B-integral [3]) exceeds π rad are referred to as nonlinear CPA-systems.

Compared to bulk laser amplifiers, fiber amplifiers are particularly sensitive to nonlinear effects due to small mode-areas and long interaction lengths. To lower the influence of nonlinearity, substantial efforts have been made on refining fiber designs, e.g. by increasing the mode-area and decreasing its absorption length. However, state-of-the-art large-mode area fibers, such as the recently developed microstructured fibers, are limited in core-diameters to less than 100µm [4]. In addition, the temporal stretching of the sub-picosecond pulses is limited to a few nanoseconds, and stretching beyond this range becomes increasingly impractical. However, the paramount potential of fiber-based CPA-systems has been demonstrated by generating femtosecond pulses with energies in the mJ-range at repetition rates of tens of kHz [5].

In principle, applying techniques that control the influence of nonlinearity during amplification would allow the extraction of even higher energies from ultra-fast fiber laser systems. Approaches, such as spectral amplitude, e.g. Cubicon amplification [6] and parabolic pulse amplification [7], and spectral phase shaping [8] have been reported. All these methods deal with the impact of SPM on the pulse envelope and its compensation [9]. However, nonlinear fiber-based CPA-systems operated at B-integrals of tens of radians have also been reported [10, 11, 12], typically showing an enhancement of weak spectral modulations that are superimposed on the envelope of the launched spectrum. It is noticed that the modulations at the input and output of the nonlinear amplifier have the same spectral modulation period and that the ripples are enhanced the higher the output power.

To identify the reason for this observation we intentionally produced a defined spectral amplitude modulation by a post-pulse. It was generated by a double reflection in a glass plate that was placed in between the oscillator and the stretcher of a fiber CPA-system. In Fig. 1(a) the spectrum, centered at 1030 nm (red curve), and the numerical reproduction (blue curve) are shown. The delay of the post-pulse is about 10ps and the ratio of post to main pulse is about 2.2%. The pulse is stretched to about 170ps in an optical fiber (without any impact of SPM), and is then amplified in an Ytterbium-doped fiber. During amplification the pulse acquires SPM. The B-integral is estimated to be about 3 rad. The resulting spectrum is distorted as shown in Fig. 1(b). In the numerical simulation of the nonlinear amplification process we assume exponential growth and include only the Kerr-effect. From the good agreement between experiment and theory it can be concluded that the Kerr-effect is the main cause for the enhancement of spectral modulations. We observed similar behavior of the nonlinear amplifier when starting without the glass plate, i.e. operating the system with the master oscillator’s inherent pulse contrast and at higher B-integrals.

 

Fig. 1. (a) Experimental and simulated spectrum of a main pulse and with a post pulse, (b) corresponding spectra after nonlinear amplification. The B-integral is about 3 rad.

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In this contribution we will show that the deformation of the spectrum, which is easy to observe experimentally, is only one aspect of a rather extensive distortion mechanism. The action of SPM with weak modulations superimposed on the pulse envelope will also affect the pulsecontrast. Recently, the influence of weak phase-modulations on the stretched state [13] and the resulting decrease of pulse-contrast in nonlinear chirped-pulse amplification systems [14] have been studied. Furthermore, the impact of weak amplitude modulations has been experimentally analyzed in regenerative and multipass amplifiers [15].

We present a complete analysis of distortions due to weak initial amplitude modulations. Both time and frequency domain are studied. The pulse-degradation consists of the generation of pre- and post pulses. We derive simple formulas describing the build-up of side pulses. Since the expressions are also valid for high B-integrals and low initial pulse-contrasts, they permits an accurate quantitative description of systems that compensate for the impact of the Kerr-effect with the pulse envelope at high B-integrals [7]. The relative magnitudes and temporal delays of the side pulses can be determined. The results hold for any type of chirped-pulse amplifier (e.g., regenerative, multipass, and fiber-based configurations) in which the impact of the Kerr-nonlinearity arises.

2. Analysis

Before presenting a detailed analysis of the distortion mechanism at work, we would like to provide a brief description of its basic principle. As commented before, the presence of a weak post pulse causes a modulated spectrum. This spectrum is mapped by the stretcher of the nonlinear CPA-system into time-domain. In a nonlinear CPA-system, the resulting modulated stretched pulse acquires self-phase modulation due to the intensity-dependency of the refractive index (Kerr-effect). Because of the sinusoidal intensity modulation, the nonlinear temporal phase includes a sinusoidal temporal phase modulation, too. In other words, the weak post pulse induces a temporal phase modulation on the stretched main pulse. This leads to pulse splitting. A similar phenomenon has been observed in the field of optical fiber communication [16]; and the spatial analogue is diffraction of a beam by a phase-grating [17]. In our case, the pulse-energy is transferred from the main-pulse to side-pulses. As we will show, the efficiency of this process is determined by the modulation-depth of the sinusoidal temporal phase modulation. The lower the initial pulse-contrast and the higher the B-integral of the nonlinear amplifier, the higher the energy transfer to the side pulses (both pre- and post-pulses).

In the following we will identify and relate the physical quantities that are responsible for this pulse-distortion in nonlinear CPA-systems. We focus on obtaining analytical results, as they are quite valuable in the design of practical systems. To validate the assumptions and approximations made in achieving the analytical results, numerical simulations are performed.

Given the temporal coherence of two optical pulses that are separated by a temporal delay Δt, the initial state is a superposition of the temporal amplitudes of both pulses and is given by A_in=A ₀(T)+√rA ₀(T-Δt). The temporal intensity-ratio of the post- to the main-pulse is r. The corresponding spectrum before nonlinear amplification shows a spectral interference with spectral modulation frequency Δt, and it is given by

Where s(Ω) is the normalized shape of the spectrum of the main pulse, s(Ω)=|∫dT exp(iΩT)A ₀(T)|²/F ². And F is the peak of the spectral amplitude. Fig. 2 shows the joint time-frequency representation of the stretched state of the main pulse and its weak post pulse. In Fig. 2(c) the spectral interference can be seen. The spectrum is unchanged during stretching. It is worth to emphasize that in the initial state a weak side-pulse can cause a strong modulation in the spectrum. For instance, the peak-to-valley modulation of the spectrum shown in Fig. 2(c) is about 4%, and it corresponds to a 40dB contrast of the main- to the post-pulse. The stretched state of the main pulse and its weak post-pulse is given by [2]

Where the second derivative of the stretching phase is denoted as ϕ ⁽²⁾, Ω is the difference between the angular frequency and the central angular frequency of the spectrum (Ω=ω-ω ₀). The time T is in a reference frame moving at the group velocity (evaluated at ω=ω ₀). The third-order dispersion during the stretching process has little impact and is, therefore, neglected. This formula allows obtaining simple analytical results that reveal the key mechanisms at work. The integrals occurring in Eq. (2) can be solved using the method of stationary phase [18, 19, 20]. This is based on the assumption that there are no significant contributions to the Fourier-integral from fast varying phase-terms with frequency, but only from the stationary points of the phase. With this assumption Eq. (2) can be written as

 

Fig. 2. Stretched state prior to nonlinear amplification: (a) the temporal pulse-profile is determined by the spectrum. The temporal FWHM of the stretched pulse is 500ps. (b) the linear stretching chirp maps the spectrum into time-domain, (c) the spectrum remains unchanged during the stretching. The spectral modulations have a frequency of Δt=10ps. (d) parabolic spectral phase, and (e) parabolic spectral phase corresponding to the linear chirp. The corresponding state after nonlinear amplification is shown in Fig. 3.

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The intensity of the double pulse in the stretched state, which is important to evaluate the effect of SPM, can be obtained from Eq. (3) and it is given by

This analytical description agrees with the numerical result shown in Fig. 2(a). In the case of a single pulse (i.e., r=0) the initial spectrum directly determines the temporal intensity profile of the stretched pulse. For an arbitrary initial double pulse, in general, the shape of its spectrum, given by Eq. (1), can not be used for the intensity profile of the stretched pulse, but the result of Eq. (4) must be used instead. Given large temporal broadening compared to the temporal delay of the post pulse, ϕ ⁽²⁾ΔΩ>Δt where ΔΩ is the bandwidth of the spectrum, in Eq. (4) the term [s(T/ϕ ⁽²⁾)s((T-Δt)/ϕ ⁽²⁾)]^-1/2 can be replaced by s(T/ϕ ⁽²⁾). Where it is also assumed that the spectrum is slowly varying. Thus, for a modulated initial spectrum Eq. (1) the corresponding intensity distribution of the stretched double pulse according to Eq. (4) is modulated and it mimics the shape of the spectrum. This is illustrated in Fig. 2. The chirp is the temporal evolution of the instantaneous frequency (offset the carrier) along the pulse. The linear relation between frequency and time can be also seen from the linear chirp that is shown in Fig. 2(b). The corresponding parabolic phases in both time and frequency-domain are plotted in Fig.2(d) and (e), respectively.

As already pointed out, because of the Kerr-effect during the nonlinear pulse-amplification, the pulse will acquire a temporal phase-profile that is proportional to the shape of the temporal intensity-profile (i.e., self-phase modulation) [2]. Thus for any arbitrary pulse:

The growth of signal intensity along the gain-medium is approximated by an exponential functionality. The gain-coefficient, g, is assumed to be spectrally uniform. The physical length of the amplifier is L and the nonlinear parameter is denoted as γ [2]. The effective length of the amplifier is given by L_eff=(exp(gL)-1)/g.

It should be stressed that in theWith these parameters the B-integral of the nonlinear amplifier can be defined as B=γL_effmax[|A(T)|²]=γL_effF ²/(2πϕ ⁽²⁾). In the last term, we assumed that the peak-power of the main pulse is much higher than that of the post-pulse.

It should be stressed that in the derivation of Eq. (5) it has been assumed that dispersion plays only a minor role during nonlinear amplification. This assumption is also accurate for practical nonlinear fiber CPA-systems. Eq. (5) states that the temporal intensity profile after nonlinear amplification is unchanged. The stretched state of the double pulse, Fig. 2(a), has modulations. According to Eq. (5), these intensity modulations superimposed on the envelope of the pulse are transferred to the temporal phase. The nonlinear phase described by Eq. (5) is shown in Fig. 3(e). The total temporal phase consists of the stretching phase, -T ²/(2ϕ ⁽²⁾) according to Eq. (3), and the nonlinear phase contribution due to SPM. In Fig. 3(e) we only plot the nonlinear phase term, since the magnitude of the stretching phase is substantially higher than the magnitude of the nonlinear phase (20 rad for the example shown).

To discriminate between the effects induced by the envelope and the modulations superimposed on it, a parabolic envelope has been used in Fig. 2 and Fig. 3. Under these circumstances the contribution due to the nonlinear action of the envelope is linear chirp. As a consequence, the superposition with the linear stretching chirp causes a linear overall chirp of varying slope, Fig. 3(b). This chirp can be efficiently removed by the compressor of the CPA-system. On the other hand, the nonlinear rippel of the chirp are solely due to the modulation which is superimposed onto the envelope. The non-parabolic part of the temporal nonlinear phase is the magenta curve shown in Fig. 3 (e). Using Eqns. 5 and 4 it can be shown that in the center of the stretched pulse, this oscillating phase contribution has a peak-to-valley modulation depth of B4√r. From Eq. (4) can be seen that the amplitude of the modulation follows the shape of the spectrum. This is shown by the magenta curve in Fig. 3(e). This sinusoidal modulation can be interpreted as a temporal phase-grating.

As briefly mentioned before this temporal phase-grating causes a energy transfer from the main pulse. To evaluate this energy transfer we make a few assumptions. In general, the sinusoidal phase modulation acts on both the main and the post-pulse. However, the energy is primarily contained in the main pulse. Thus, we neglect the initial weak post pulse in the calculation although its impact is included via the temporal phase modulation. In addition to that, the nonlinear temporal phase is reduced to the sinusoidal contribution (i.e., only the last term in Eq. (4) is kept). Besides, we neglect the slowly varying shape of the spectral envelope that determines the amplitude of the modulation, i.e. we set s(T/ϕ ⁽²⁾)≈1 in the expression that describes the phase, Eq. (4). The kept phase is the grey curve that is plotted underneath the magenta cuve in Fig. 3(e). This way, the approximation would only be exact at the center of the pulse and it would result in higher intensities of the side pulses. To account for the profile of the pulse we introduce a profile factor p, which is about 0.7 for standard spectral profiles. This value can be regarded as a weighting factor [22]. With these approximations made to Eqs. (3) and (4), we can write Eq. (5):

 

Fig. 3. State of the pulse after nonlinear amplification when starting with the state shown in Fig. 2: (a) temporal pulse. The temporal FWHM is 500 ps. In the numerical calculation of the nonlinear amplification, the dispersion is neglected. The SPM does not affect the intensity distribution. (b) chirp. Modulations are superimposed on the linear stretching chirp because of SPM, (c) spectrum of the pulse, which is modified due to the influence of SPM. The spectrum was obtained by numerical calculation. (d) residual non-parabolic spectral phase at the output of the CPA-system due to the nonlinear action. (e) temporal self-phase modulation (blue curve), this phase adds onto the existing stretching phase shown in Fig. 2(e), and its non-parabolic part (magenta curve).

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Where pB2√r is substituted by the parameter a. Using the definition of the generating function for the Bessel-functions [21],

Equation (6) can be transformed into

where the temporal phase for each pulse is given by

From this equation it can already be seen that the main stretched pulse is split into several subpulses (including both pre- and post-pulses). Even if the number of terms in the expansion 7 is infinity, for most experimental configurations only a few terms in the series of Eq. (7) are relevant (e.g., m=-2,-1,0,1,2). In Fig. 5(a) the relative intensities of the subpulses within the compound are shown.

A Fourier-transform of Eq. (8) has to be performed for the description of the compression of the stretched state of this multi-pulse. To evaluate the Fourier-integrals, the method of stationary phase can be applied again [20]. For each pulse, the stationary temporal points must fulfil the condition, $\mathrm{\Omega }+{\frac{\mathrm{d}{\mathit{\phi }}_m\left(T\right)}{\mathrm{d}T}\mid }_{T_s}=0$ , so they are given by

Thus, the spectral amplitude is given by

where

In Fig. 3(c), we plotted the square modulus of this analytic solution for the spectral amplitude (grey curve) together with the numerical result (red curve).

By removing the parabolic spectral phase in Eq. (11), i.e. multiplication with the term exp (-iϕ ⁽²⁾Ω²/2), and Fourier-transforming the resulting expression, a description for the compressed multi-pulse at the output of the nonlinear CPA-system can be obtained. Taking into account that the integral

is a convolution with a δ-function in time-domain, the final expression for the recompressed multi-pulse is given by

where

In Eq. (14) the parameter a is pB2√r, and it describes the pulse-degradation. Fig. 4 shows the numerical calculation (based on the Fast-fourier transform) of the pulse-contrast at the output of the nonlinear CPA-system as well as the analytical result (using Eq. (14).

 

Fig. 4. Comparision between the numerical calculation of the pulse after the nonlinear CPA-system (B=20 rad) and the analytical result (B=20 rad) according to Eq. (14). The input pulse is also shown (B=0 rad).

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Fig. 5. (a) Relative intensities of the pulses contained within the multi-pulse as a function of the parameter a=0.7B2√r, (b) total intensity that is in the side pulses relative to the intensity in the main pulse as a function of the value of the B-integral and for different initial pulse-contrasts r [dB].

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In Fig. 5(a) the distribution of the relative intensities of the main pulse and the side pulses is plotted. In particular, the transfer of the energy into several pulses is solely determined by the amount of accumulated nonlinear phase-shift (i.e., the B-integral) and the initial pulse-contrast, r. The energy transfer can be described using Bessel-functions. Thus, applications relying on a high pulse contrast demand a low B-integral of the amplification system and a master-oscillator that exhibits an excellent pulse-contrast. At an initial pulse-contrast of r=10^-4, 10^-3 and 10^-2, no energy will be left in the zero order (the principal pulse) at B-integrals of 120 rad, 38 rad and 12 rad, respectively. In Fig. 5(b) the total intensity in the side pulses relative to the intensity of the main pulse, 1-J ² ₀(0.7B2√r), as a function of the B-integral and for different initial pulse contrasts is shown. It is important to note that the pulse degradation of the main pulse, shown in Fig. 5(b), is solely due to the high frequency modulations. The nonlinear action with the envelope of the pulse causes an additional decrease in peak-power [3, 7].

 

Fig. 6. (a) Spectral modulation superimposed on the envelope of the spectrum at the output of the nonlinear amplification stage (B=20 rad) the intial contrast relative to the post-pulse is r=40dB, the horizontal line corresponds to the state shown in Fig. 3(c), (b) corresponding residual spectral phase at the output of the nonlinear CPA-system, the horizontal line corresponds to the state shown in Fig. 3(d), (c) pulse-contrast at the output of the nonlinear CPA-system

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In Fig. 6 we illustrate and verify by numerical calculation that the pulse-contrast at the output of the nonlinear CPA-system is not a function of the delay of the post-pulse Δt (i.e., the spectral modulation frequency). For the example, the B-integral, the temporal duration of the stretched pulse and initial pulse-contrast r are fixed, they have the values 20 rad, 500 ps and 40 dB, respectively. It can be seen in Fig. 6(a) that the spectrum at the output of the amplifier changes significantly. Correspondingly, the residual non-parabolic spectral phase at the output of the system changes. A low deformation of the phase corresponds to a high amplitude distortion, and vice versa. This is shown in Fig. 6(b). In particular, the joint action of both the amplitude and phase deformation determines the pulse-contrast. The resulting pulse-contrast remains constant, Fig. 6(c). This is in agreement with our analytical model, for which the relative intensities of the main and the side pulses do not depend on the delay of the post-pulse.

Thus, from the deformation of the spectrum alone it is not possible to estimate the amount of pulse-degradation at the output. It is worth to note that the deformation of the spectrum and the phase are related to the functionality of the chirp (Fig. 3(b)). The chirp-modulations superimposed on the linear stretching chirp can be calculated using Eq. (4). It can be shown that the parameter, 2ϕ ⁽²⁾/((Δt)² B4√r), allows an estimation of the deformation. Thus, besides the B-integral and the initial pulse-contrast, the observable distortions depend also on the post-pulse delay and the stretching.

3. Conclusion

The degradation of pulse contrast in a nonlinear CPA-system by initial high frequency modulations in the spectrum is discussed. We could show that during nonlinear amplification, the stretched pulse acquires a temporal phase modulation due to the Kerr-effect. This causes splitting of the pulse into multiple pulses. Even though our description is valid for an arbitrary form of accumulated temporal phase modulation due to nonlinearity, we considered analytically the case of sinusoidal phase modulation. An advantage of the analytical results is that they enable the extraction of the governing parameters. It is found that the intensity distribution of the pulses within the multi-pulse is determined by the accumulated nonlinear phase-shift (i.e., the B-integral) and the initial pulse-contrast. The relative intensities can be described using Bessel-functions. The pulse degradation does not depend on the stretched pulse duration and the modulation frequency. The analytical predictions are verified by numerical simulations. The analytical results should prove useful in the estimation of pulse contrast degradation in nonlinear CPA-systems. In particular, this will be of significance in laser applications, such as high field physics where pre-pulses can cause an unwanted ionization of matter.

Compared to bulk laser amplifiers, fiber amplifiers are particularly sensitive to nonlinear effects due to the small effective mode-area and the long length. Thus, the generation of pulses with high energies and high pulse-contrasts is more challenging. The design of nonlinear CPA-systems starting with a fiber-based master-oscillator should be revised in this context [12, 23].