1. Introduction

Chirped pulse amplification (CPA) has demonstrated to be a powerful technique to scale the energy of ultra-short optical pulses while avoiding distortions due to nonlinear effects and/or damage of optical elements [1]. The Kerr-nonlinearity manifested in form of self-phase modulation (SPM) is particularly damaging [2]. SPM is proportional to the intensity envelope of the stretched pulse. Despite the lower peak-powers obtained by the temporal broadening (i.e., chirping) of the pulse in CPA-systems, the onset of the SPM causes a non-vanishing residual spectral phase at the output. This is translated into a decrease in peak-power [3, 4]. This problem is particularly severe for amplifiers in which the B-integral, i.e. a quantifier for the amount of accumulated SPM [3], exceeds π.

Fiber lasers are scalable in power; however, they are more susceptible to nonlinear effects than bulk laser amplifiers due to their smaller mode-areas and longer interaction lengths [5]. One way to reduce the impact of nonlinearities consists in modifying the design of active fibers, e.g. by micro-structuring [6]. Current state-of-the-art fiber-based CPA-systems are capable of producing femtosecond pulses with energies in the mJ-range at repetition rates of tens of kHz [7]. In these systems, controlling the influence of nonlinearities during pulse amplification would allow the extraction of even higher energies. Examples of such approaches are the cubicon amplification [8], the parabolic pulse amplification [9], and spectral phase shaping [10]. All these methods deal with the impact of SPM on the pulse envelope and its compensation [4]. In spite of these methods, fiber-based CPA-systems operated at B-integrals of tens of radians (e.g., [11, 12, 13]) typically show modulations on the spectrum at the output. Recently, it has been shown that the build-up of these modulations and the associated pulse-contrast degradation can be due to weak post pulses [14, 15].

In this contribution we show that an intial weak spectral phase modulation can be another source of pulse-contrast degradation at the output of nonlinear CPA-systems. This study is particularly relevant since phase-modulations are a matter of concern in every practical CPA-system. In particular, they can be caused by abberations (e.g., due to poor surface quality) in bulk stretchers [16, 17, 18, 19], or by group-delay ripples in integrated stretching devices, such as chirped fiber or volume Bragg-gratings [20, 21]. The influence of an initial random phase-modulation on the shape of the stretched pulse has been discussed recently [22]. In this article, the impact of an initial spectral phase-modulation on the pulse-contrast at the output of CPA-systems, in which the Kerr-nonlinearity acts, is analysed. In the study analytical formulas for the estimation of the decrease in pulse contrast are obtained. These formulas reveal the main physical parameters. The results hold for any type of chirped-pulse amplifier (e.g., regenerative, multipass configurations, fiber-based) in which phase changes due to Kerr-nonlinearity are accumulated. The pulse-degradation consists of the generation of pre- and post pulses. In particular, the pre-pulses are detrimental for many ultrafast laser applications, such as high field physics [23, 24, 25, 26]. It is also worth noting that the pulse-contrast is important for parametric CPA-systems [27, 28].

2. Analysis

In the following we will identify and relate the physical quantities that are responsible for the degradation of the pulse-contrast when a spectral phase-modulation is present at the input of a nonlinear amplifier. We will focus on obtaining analytical results, as they are quite valuable for the design of practical systems.

There are two kinds of perturbations that affect nonlinear CPA-systems. These can be categorized into spectral amplitude and phase modulations. The first type can be seen in Fig. 1(a–c): if the main pulse is followed by a post pulse, which can be produced by a double internal reflection, a spectral modulation will be superimposed on the intensity spectrum. Since the stretching implies a mapping of the spectrum into time-domain, the stretched pulse will also show modulations. Regarding the second kind of modulation (shown in Fig. 1(d–f)), a sinusoidal spectral phase-modulation and a smooth intensity spectrum result in multiple pulses in the time-domain. Where the number of pulses depends on the modulation-depth. The corresponding stretched pulse, obtained by adding a parabolic spectral phase to the spectral phase-modulation, shows intensity modulations as well [22]. These two types of modulations differ according to the impact of the stretching on the shape of the stretched pulse. For a spectral amplitude modulation, the shape of the stretched pulse (i.e., the structure of the modulations superimposed on the envelope) is independent of the magnitude of stretching. On the other hand, for an initially phase-modulated signal the amount of stretching affects the profile of the stretched pulse. In any case, intensity modulations superimposed on the temporal pulse envelope will cause a pulse-degradation at the output of the CPA-system. The impact of an intitial amplitude modulation is discussed in Ref. [14]. In this contribution, the case of an initial phase-modulation is investigated.

 

Fig. 1. (a) Spectral amplitude-modulation due to a main pulse which is accompanied by a weak post-pulse (constrast of 40dB, delay of 10 ps), e.g. generated by a double internal reflection (b), and the modulated stretched pulse (c) with a full width at half maximum of 500 ps (ϕ ⁽²⁾=33ps ²); (d) Spectral phase-modulation, i.e. a smooth intensity spectrum with a sinusoidal weak spectral phase-modulation (modulation depth is d=0.02, spectral modulation frequency is 10 ps) that corresponds to a transform limited multi-pulse in the time-domain (e), numerical and analytical result of the modulated stretched pulse (f) where the same stretching parameters as in (c) are used.

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In the nonlinear amplification stage of a CPA-system, the pulse acquires self-phase modulation due to the intensity-dependence of the refractive index (Kerr-effect). Because of the modulation superimposed on the intensity envelope, the nonlinear temporal phase will also be modulated. Through this effect the stretched main pulse acquires a temporal phase modulation. This will in turn cause a energy transfer from the main (stretched) pulse to the (stretched) side-pulses. This is analogous to the diffraction of a beam by a spatial phase-grating [29]. With increasing B-integral the modulation depth of this temporal phase-grating rises, and thus, a larger amount of pulse-energy is transferred from the main-pulse to the side-pulses. As we will show, the efficiency of this process is determined by the magnitude of the accumulated nonlinear phase (i.e., the B-integral) of the nonlinear amplifier, the depth and period of the inital spectral phase modulation, and the magnitude of the stretching chirp. To validate the assumptions and approximations made to arrive at the analytical results, numerical simulations have also been carried out.

For the following analysis it does not matter whether the spectral phase ripples are acquired before or in the stretcher of the CPA-system. We assume that the high-frequency spectral phase modulation is sinusoidal, d cos(ΔtΩ-b). This is a simple description of a real modulation but it allows the derivation of useful analytical formulas to reveal the main dependencies. The depth and the spectral frequency of the modulation are denoted as d and Δt, respectively. We also include an arbitrary shift b. The stretched state of the temporal amplitude of the pulse is given by [2]

Where the second derivative of the stretching phase (i.e., the slope of the stretching chirp) is denoted as ϕ ⁽²⁾, and Ω is the difference between the angular frequency and the central angular frequency (Ω=ω-ω ₀). We assume that the sinusoidal spectral phase modulation is a perturbation of the stretching phase, and thus, the modulation depth is small. As an example, in a bulk stretcher the modulation is produced by the surface quality of its elements, which are on the scale of about λ/10 (i.e. 10^-7 m). This amount is small compared to the distance that is needed for the stretching of the pulse to 0.1…1 ns (i.e. 10^-1 m). The magnitude of the stretching phase produced by such stretchers is typically about 10²…10³ rad. As a consequence, the depth of the sinusoidal phase modulation can be assumed to be d<1rad. This assumption is valid for most of the stretchers used in practical CPA-systems. An example is shown in Fig. 1 (d–f). The depth of the phase-modulation is d=0.02, Fig. 1(d). The corresponding transform-limited pulse is plotted in Fig. 1(e) and the modulated stretched pulse can be seen in Fig. 1(f). In Eq. (1), the time T is in a reference frame moving at the group velocity of the pulse (evaluated at ω=ω ₀). The third-order dispersion during the stretching process is of small magnitude compared to the parabolic phase contribution, in addition to that, it is smooth and slowly varying, and therefore, it will be neglected in the following. This allows deriving simple analytical expressions. The initial spectral amplitude, Ã ⁰(Ω), is assumed to be transform-limited (i.e., zero spectral phase). The spectrum, |Ã ₀(Ω)|², is unchanged during stretching as expected for a multiplication with a phase-transfer function.

To obtain a simple but valid expression for the stretched pulse, we expand the sinusoidal spectral phase-modulation Bessel-functions [30]:

With this formulation, the integral in Eq. (1) can be solved using the method of stationary phase [31, 32, 33]. This technique is based on the assumption that the main contributions to the Fourier-integral come from the stationary points of the phase and not from the fast varying phase-terms with frequency. In particular, the stationary points must fulfill the condition $\frac{d}{d\Omega }\left[\frac{-\Omega T+{\varphi }^{\left(2\right)}{\Omega }^2}{2}-m\Delta t\Omega +\mathrm{mb}\right]{\mid }_{{\Omega }_s}=0$ , and thus, they are given by

Consequently, Eq. (1) can be formulated as

i.e. the signal comprises many pulse-amplitudes that overlap in the stretched state. This can be regarded as the primary reason for the modulation seen in Fig. 1(f). The corresponding power-profile, which is needed for the calculation of the intensity-dependent nonlinear phase, is given by

In obtaining Eq. (5), we have assumed once more that the modulation depth is small, i.e. d<1rad, so that in the expansion 2 only the terms with m=-1,0,1 are significant (note that J_-m(d)=(-1)^m J_m(d)). Furthermore, in the derivation of Eq. (5) from Eq. (4) we approximated à ₀(T±mΔt)/ϕ ⁽²⁾) with à ₀(T/ϕ ⁽²⁾). We have also assumed that the temporal durations of the stretched amplitudes are large compared to the spectral modulation frequency Δt, thus, rendering the temporal shift neglible. Moreover, since the modulation depth d is small (and thus J ² ₁(d)<J ₀(d)J ₁(d)), the term with J ² ₁(d) is neglected in Eq. (5). The result agrees with the findings given in Ref. [22].

In Eq. (5) it can be seen that a high-frequency modulation is superimposed on the envelope of the pulse as previously advanced. In particular, this high-frequency modulation shows the same characteristic behaviour as the initial spectral phase-modulation, i.e. $\left(\Delta t\frac{T}{{\varphi }^{\left(2\right)}}-b\right)$ . In addition, the amplitude is strongly dependent on the initial spectral modulation frequency Δt and the slope of the stretching chirp. In Fig. 1(f) both the analyitcal expression (Eq. (5)) and the numerical result (based on the Fast-Fourier-transform) for the stretched pulse are plotted. As mentioned before, because of the Kerr-effect during 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 temporal amplitude, the nonlinear amplification can be expressed as

In this expression, the growth of the signal intensity along the gain-medium is approximated by an exponential function. The gain-coefficient, g, is assumed to be spectrally uniform. The physical length of the amplifier is L and the nonlinear parameter is denoted γ [2]. The effective length of the amplifier is given by L_eff=(exp(gL)-1)/g. It should be stressed that in the derivation of Eq. (6) it has been assumed that dispersion plays only a minor role during the nonlinear amplification process. This assumption is also accurate for practical nonlinear fiber CPA-systems, since the temporal broadening due to dispersion in the amplifier is small compared to the temporal duration of the stretched pulse during amplification. The B-integral of the nonlinear amplifier can be defined as B=γL_effmax[|A_st(T)|²].

According to Eq. (6), the modulation superimposed on the envelope of the stretched pulse (Eq. (5)) is responsible for a high-frequency temporal phase modulation that is created by the Kerr-effect. In the following we will only analyze the impact of the high-frequency modulation. Therefore, we neglect the impact of the envelope that is described by the first term in the square brackets of Eq. (5). Furthermore, we neglect the slowly varing shape of the spectral envelope of the amplitude of the modulation and use a constant amplitude instead, i.e. we set |à ₀(T/ϕ ⁽²⁾)|²≈F ² in Eq. (5). Where F is the maximum of the spectral amplitude à ₀(Ω). 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 [19]. With these approximations, we write the term responsible for the impact of the nonlinear phase in Eq. (6) as

Where B is the value of the B-integral given by γL_effF ²/(2πϕ ⁽²⁾). We have assumed that the modulation depth d is smaller than unity, and thus, the product of J ₀(d)J ₁(d) can be approximated by d/2. As briefly mentioned before, the temporal phase-grating causes an energy transfer from the main pulse to the side-pulses. To evaluate this energy transfer, we make further assumptions. In general, the sinusoidal phase modulation acts on the amplitudes of both the main and the side pulses (e.g., m=±1) in the stretched state, Eq. 4. However, the energy is primarily contained in the main pulse, i.e. J ₀(d)≈1. Thus, we neglect the initial side peaks (m≠0) in the subsequent calculations, although their impact is included via the temporal phase modulation, Eq. 7. With these assumptions we write Eq. 6 as

Where we cast the amplitude of the modulation into a single parameter:

As can be seen, for a given spectral modulation frequency Δt, the impact of the modulation can be strongly influenced by the stretching (ϕ ⁽²⁾).

The term describing the impact of the high-frequency temporal phase modulation in Eq. (8) can be expressed in terms of Bessel-functions according to Eq. (2). Thus, Eq. (8) can be transformed into

where the temporal phase for each pulse is given by

From Eq. (11) it can be seen that the energy is transfered from the main stretched pulse to the side-pulses (both pre- and post-pulses). Even if the number of terms in the expansion is strictly infinity, for most experimental configurations only a few terms in the series of Eq. (2) are relevant (e.g., m=-2,-1,0,1,2).

A Fourier-transform of Eq. (10) has to be performed to describe the compression of this stretched multi-pulse. To evaluate the Fourier-integrals, the method of stationary phase can be applied again [33]. For each pulse, the stationary temporal points must fulfill the condition, $\Omega +\frac{d{\phi }_m\left(T\right)}{dT}{\mid }_{T_s}=0$ . Thus, they are given by

In this way the spectral amplitude is obtained as

where

 

Fig. 2. Spectrum after the amplifcation stage in a nonlinear CPA-system. The modulations have their origin in the weak spectral phase modulation at the input. The corresponding stretched pulse at the input of the nonlinear amplifier is shown in Fig. 1(f).

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Figure 2 shows the analytical result based on Eq. (13) as well as the numerical calculation (based on the Fast-Fourier-transform) of the spectrum at the output of the nonlinear amplifier. The corresponding signal at the input of the nonlinear amplifier is shown in Fig. 1(d–f). The strong spectral modulations shown in Fig. 2 are characteristic of CPA-systems operated at B-integrals of tens of radians [11, 12]. The spectral interference of the stretched sub-pulses is the explanation for this observation.

By removing the parabolic spectral phase in Eq. (13), 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

Equation 16 states that the joint action of the Kerr-nonlinearity and the intial spectral phase-modulation results in the transfer of energy from the main pulse into the side pulses (both pre- and post-pulses). The intensity distribution is determined by the parameter a. If a<1 the intensity of the first pre- and post pulse relative to the main pulse can be approximated by J ² ₁(a)≈a ²/4.

In the derivation of Eq. (16) we have neglected the intensity envelope in the nonlinear temporal phase-contribution (see Eq. (7)). Thus, the pulse-contrast degradation due to the envelope is not included in the analysis. However, Eq. (16) still permits the calculation of the intensities of the enhanced pre- and post-pulses relative to the main pulse for various initial pulse envelopes. In Fig. 3(a) the analytical result is compared with the numerical calculation (based on the Fast-fourier transform) for the case of a parabolic spectrum. The seed signal corresponding to this distorted output of the nonlinear CPA-system is the example shown in Fig. 1(d–f). The initial pulse contrast due to initial sinusoidal phase-modulation (with depth of d=0.02) is 40 dB. At a B-integral of 20 rad of the nonlinear CPA-system the pulse-contrast is significantly reduced: it is only 10dB. Ideally, the parabolic spectrum allows to control the impact of nonlinearity during amplification at high B-integrals because the spectral shape of the nonlinear phase due to the intensity envelope can be efficiently compensated by the compressor [9]. However, high pre-and post-pulses can arise if the seed of the nonlinear amplifier shows a weak phase-modulation. The analytical result for the pulse-contrast of a nonlinear CPA-system, Eq. (16), gives also a valuable estimation for standard-profiles, such as sech ². Although the pulse-degradation due to the Kerr-effect depending on the pulse envelope is not included in the analysis, the intensities of the pre- and post pulses relative to the main pulse are well estimated. The case of a sech ²-pulse is shown in Fig. 3(b). The B-integral is lower since the pulse-degradation due to the envelope is quite strong, and eventually (i.e. at higher B-integrals) prevents a distinction between the decrease in pulse-contrast due to the envelope and the initial high-frequency modulation.

 

Fig. 3. (a) Numerical and analytical result of the pulses at the output of the nonlinear CPA-system for the configuration shown in Fig.1(d–f); (b) intensity-distribution for the case of an inital sech ² profile. The parameters of the phase-modulation and ϕ ⁽²⁾ are the same as in (a).

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The case of an intial phase-modulation differs significantly from the pulse-contrast degradation due to an initial amplitude modulation. As shown above, the intensity distribution of the sub-pulses depends on the stretching. Particularly, the parameter a depends on the term sin ((Δt)²/(2ϕ ⁽²⁾)). The spectral modulation frequency Δt can take values in the range of several picoseconds and ϕ ⁽²⁾ is on the order of about 10ps ²/rad. Thus, in practical systems the ratio (Δt)²/(2ϕ ⁽²⁾) can take any (relevant) value between 0 and 2π. In Fig. 4 the behaviour of the energy transfer into the side-pulses as a function of the product of the B-integral and the modulation-depth d is shown for different characteristic ratios (Δt)²/(2ϕ ⁽²⁾). Pulse-degradation is lower for small Δt and high magnitudes of the stretching chirp ϕ ⁽²⁾, i.e. 2ϕ ⁽²⁾≫(Δt)². Thus, a high stretching chirp can allow a higher contrast for small Δt. However, in practical CPA-systems the magnitude of the stretching chirp is limited to the order of 10ps ².

To describe the relative intensities of the sub-pulses at the output of a nonlinear CPA-system due to an intial amplitude modulation, the argument of the square of the Bessel-function a has to be replaced by 0.7B2√r, where r is the initial pulse contrast [14]. In contrast to initial phase-modulations, there is no dependence on the stretching and on the delay of the post-pulse. The pulse contrast degrades with increasing B-integral. However, in the case of a phase-modulation there are stretching configurations where the pulse-contrast is not degraded with higher B-integrals due to a high-frequency modulation. Particularly, these points are given by (Δt)²/(2ϕ ⁽²⁾)=kπ, where k is an integer. In theory, changing the magnitude of the stretching chirp would allow distinguishing between amplitude- or phase-modulation as the origin of the pulse contrast degradation based on the observed output from a CPA system. However, in practice a different magnitude of the stretching chirp means a different configuration of the stretcher, and thus, a different initial phase-modulation which causes a different behaviour of the nonlinear CPA-system.

 

Fig. 4. (a) Total intensity of the side-pulses relative to the intensity of the main pulse, 1-J ² ₀(a), as a function of the product of B-integral and modulation-depth d times 1.4 and for different ratios (Δt)²/(2ϕ ⁽²⁾) in the sine of the parameter a.

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3. Conclusion

The degradation of the pulse-contrast in a nonlinear CPA-system by intial high frequency spectral phase-modulations has been analyzed. It has been shown that the corresponding stretched pulse is amplitude modulated. During nonlinear amplification, this results in a temporal phase modulation due to the Kerr-effect. In turn, this causes energy transfer from the main pulse into side-pulses. Even though this occurs for any arbitrary initial spectral phase modulation, we considered analytically the case of a sinusoidal phase modulation. In contrast to initial spectral amplitude modulations, the decrease of pulse contrast due to a phase-modulation is found to be strongly dependent on the stretching of the pulses. The formulas derived in this paper allow the calculation of the pulse-contrast at the output of nonlinear CPA-systems. 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 B), the depth and frequency of the inital spectral phase modulation (d and Δt, respectively), and the slope of the stretching chirp, ϕ ⁽²⁾. The relative intensity of the m-th order sub-pulse within the resulting multi-pulse can be described using Bessel-functions, i.e. J ² _m(a), where the governing parameter is given by the expression a=0.7B2d sin((Δt)²/(2ϕ ⁽²⁾). Certain stretching configurations, i.e. (Δt)²/(2ϕ ⁽²⁾)=kπ where k is an integer, show no decrease in pulse-contrast with increasing B-integral. The analytical equations derived in this contribution are of particular importance for the design of lasers for high peak-power applications, such as high field physics, demanding high pulse-contrasts. For expample, pre-pulses prevent the generation of high harmonics from solid targets [26].