Self-phase modulation compensated by positive dispersion in chirped-pulse systems

Damian N. Schimpf; Enrico Seise; Jens Limpert; Andreas Tünnermann

doi:10.1364/OE.17.004997

1. Introduction

The generation of ultra-short pulses is typically governed by the interplay of dispersion and nonlinearity. Ultrashort pulse have high peak-powers which cause nonlinearities. In particular, nonlinearity can cause wave-breaking of ultra-short pulse, as described in [1]. This effect imposes limitations on the generation and amplification of femtosecond laser pulses. Chirping of the pulse during its formation reduces the peak-power, and thus, increases the onset of wave-breaking [2, 3]. Consequently, oscillators using stretched-pulses can produce higher pulse-energies compared to soliton-lasers. Basically, the same arguments hold for the concept of chirped-pulse amplification (CPA) in comparison to the direct amplification of ultra-short pulses [4]. Since the chirping of the pulses is limited due to practical reasons, the impact of nonlinearity plays a role again with increasing pulse-energies [5].

Ultrafast fiber-laser systems offer several advantages over bulk solid-state lasers: good thermo-optical properties resulting in high power operation, the compact size and stability due to the all-fiber structure. However, they appear sensitive to Kerr-nonlinearity because of the confinement of light to small areas of the single modes (diameter ~ 100μm) and the propagation in material (n ₂ ~ 2.7 × 10^-20 m ²/W at λ = 1.06μm [6]). However, the potential of fiber-based CPA-systems has been demonstrated by generating pulses with energies at the mJ-level and at repetition-rates of tens of kHz [7]. To further scale the energy of such fiber-based systems, there is need for operation in the nonlinear CPA-regime.

In this contribution, we report on a detailed analysis of nonlinear CPA-systems and we show that the Kerr-nonlinearity can be used for the optimization of the spectral phase at the output of nonlinear chirped-pulse systems. Self-phase modulation acquired by the stretched pulse can be compensated by positive dispersion. This technique is of particular importance for systems employing static stretching and compressing elements, e.g. chirped volume Bragg-gratings [8, 9]. The technique is demonstrated with a fiber-based system employing conventional diffraction gratings as well in a system that is based on novel chirped volume Bragg-gratings. Practical requirements on the quality of the stretched pulse are discussed.

2. Analysis of the phase-compensation mechanism

The main principle of the method is compensation of phase-terms due to positive dispersion with the phase due to self-phase modulation (SPM). It is important to emphasize that the method uses chirped pulses. Compensation of SPM with positive dispersion is impossible for initially transform-limited pulses since both effects result in positive chirps, which can not cancel each other.

The impact of the Kerr-nonlinearity on stretched pulses (with standard shapes such as Gaussian and sech ²) effectively results in a negative tilt of the initial chirp due to stretching. Thus, the negative chirp-contribution due to SPM can be compensated with the positive chirp-contribution due to positive dispersion. It is worth noting that the initial chirp can be positive or negative. Dispersion is an effect in frequency-domain whereas the Kerr-effect acts in time-domain. However, for stretched pulses the spectrum is mapped into time-domain, and for smooth spectra and strong stretching it is possible to obtain an expression for the spectral phase due to the SPM. In the following we will derive this relation. Then, this term will be used for the derivation of the condition for the phase-compensation with positive dispersion.

In the analysis the action of the SPM on the stretched pulse is separated from the action of the positive dispersion: Provided the pulse is strongly stretched, this assumption is also applicable to systems in which both effects occur in a single element. Since the magnitude of the stretching phase is substantially higher than the phase-shifts resulting from positive dispersion, the pulse-shape does not change significantly during the action of the Kerr-effect (which is proportional to the pulse-shape). Thus, the two effects can be regarded separately.

2.1. Spectral phase due to self-phase modulation of stretched pulses

At first, we derive an analytical term for the stretched pulse in time-domain. To analytically integrate the Fourier-integral, we use the method of stationary phase. Then, the result is used for the description of the impact of self-phase modulation on the stretched pulse. To obtain a relation for the spectral phase, a Fourier-transform from time-domain into frequency-domain is necessary. The corresponding Fourier-integral is also analytically integrated using the method of stationary phase.

The stretching of a pulse can be described by [6]

A_{st} (T) = \frac{1}{2 π} \int dΩ \exp (- i Ω T) \exp (i \frac{ϕ_{st}^{(2)}}{2} Ω^{2}) {\tilde{A}}_{0} (Ω) .

Where Ω stands for the difference between the angular frequency and the central angular frequency of the spectrum (Ω = ω - ω ₀). The second derivative of the stretching phase (evaluated at ω = ω ₀) is denoted as ϕ_st ⁽²⁾ . The time T is in a reference frame moving at the group velocity (evaluated at ω = ω ₀).

In Eq. 1 it is assumed that the magnitude of the third-order term of the stretching phase is small compared to the magnitude of the second-order term, therefore, the impact of the third-order term is not dominant and can be neglected. In practice, this assumption is valid for the stretching of pulses having bandwidths of a few THz with standard stretchers. The assumption permits the derivation of simple expressions and reveals the key mechanisms at work. The integral in Eq. 1 can be solved using the method of stationary phase, e.g. [10]. This technique assumes 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: $\frac{d}{dΩ} [\frac{ϕ_{st}^{(2)}}{2} Ω^{2} - Ω T] ∣_{Ω_{s}} = 0$ which is equal to Ω_s = T/ϕ_st . Then, Eq. 1 can be written as

A_{st} (T) \approx \frac{1}{\sqrt{- i 2 π ∣ ϕ_{st}^{(2)} ∣}} \exp (- i \frac{T^{2}}{2 ϕ_{st}^{(2)}}) {\tilde{A}}_{0} (\frac{T}{ϕ_{st}^{(2)}}) .

The intensity of the stretched pulse, which is important for the evaluation of the impact of SPM, can be obtained from Eq. 2 and it is given by ∣A_st (T)∣² = F ² s (T /λ_st ⁽²⁾) / (2π∣ϕ ⁽²⁾∣). The normalized spectrum is denoted as s (Ω), and F is the peak of the spectral amplitudes. The shape of a stretched pulse mimics the shape of the spectrum.

Due to the Kerr-effect, the pulse will acquire self-phase modulation during propagation [6]. This effect can be described by

A_{amp} (T) = A_{st} (T) \exp (\frac{gL}{2}) \exp (i γ L_{eff} {∣ A_{st} (T) ∣}^{2}) .

In this description we include also an amplification process. We assume that the growth of the signal intensity along the gain-medium is exponential. This is quite accurate for high gain fiber-amplifiers below saturation [11]. The gain-coefficient, g, is assumed to be spectrally uniform. The physical length of the amplifier is L and the nonlinear parameter is γ [6]. The effective length of the amplifier (with regard to the input) is given by L_eff = (exp (gL) - 1)/g.

With these parameters the B-integral of the nonlinear amplifier can be defined as [5] B = γL_effmax[∣A_st(T)∣²] = γL_effF ²/(2π∣ϕ_st ⁽²⁾∣).Using the expression for the stretched pulse in form of Eq. 2, Eq. 3 is given by

A_{amp} (T) \approx \frac{\exp (\frac{gL}{2})}{\sqrt{- i 2 π ∣ ϕ_{st}^{(2)} ∣}} {\tilde{A}}_{0} (\frac{T}{ϕ_{st}^{(2)}}) \exp (i φ (T)),

where the temporal phase is given by

φ (T) = \frac{T^{2}}{{2 ϕ}_{st}^{(2)}} + B s (\frac{T}{ϕ_{st}^{(2)}}) .

The total temporal phase consists of the part due to stretching, -T ²/(2ϕ_st ⁽²⁾) and the nonlinear phase term due to SPM, which is determined by the shape of the spectrum.

To obtain an expression in frequency domain, and thus, to derive the spectral phase term, Eq. 4 has to be Fourier-transformed:

{\tilde{A}}_{amp} (Ω) \approx \frac{\exp (\frac{gL}{2})}{\sqrt{- i 2 π ∣ ϕ_{st}^{(2)} ∣}} \int d T \exp (i Ω T) \exp (i φ (T)) {\tilde{A}}_{0} (\frac{T}{ϕ_{st}^{(2)}}) .

To evaluate the Fourier-integral, the method of stationary phase can be applied again. The stationary temporal points must fulfil the condition: $\frac{d}{d T} [Ω T + φ (T)] ∣_{T_{s}} = 0$ which is equal to Ω = T_s/ϕ_st ⁽²⁾ - B ds(T/ϕ_st ⁽²⁾)/dT∣_Ts. In the case of smooth initial spectra and strong stretching, the chirp still corresponds to approximately linear one-to-one mapping between time and frequency, and the term -B ds(T /ϕ_st ⁽²⁾)/dT∣_Ts can be neglected. Then, the stationary points are given by T = ϕ_st ⁽²⁾Ω.

Finally, the expression for the spectral amplitude with the spectral phase due to SPM is

{\tilde{A}}_{amp} (Ω) = \exp (\frac{gL}{2}) {\tilde{A}}_{0} (Ω) \exp (i \frac{ϕ_{st}^{(2)}}{2} Ω^{2}) \exp (i B s (Ω)) .

The spectral phase due to SPM is determined by the shape of the (normalized) spectrum and has the magnitude of the B-integral value. It is important to stress that SPM originally acts in time-domain and Eq. 7 is the description for the resulting impact in frequency-domain.

In the cases of non-smooth spectra [12], initial phase ripples [13] or weak stretching of the (2) pulses [1], Eq. 7 is not applicable. In particular, the term -B ds(T/ϕ_st ⁽²⁾)/dT∣_Ts results in a multiple-point-to-point configuration for the chirp. Optical wave-breaking is a consequence. Eq. 7 has to be replaced by a sum over the multiple stationary points [10, 12, 13].

2.2. Compensation of SPM with positive dispersion

Equation 7 includes the phase due to the initial stretching of the pulse, φ_st = ϕ_st ⁽²⁾Ω²/2, and the spectral phase resulting from SPM, φ_SPM = B s(Ω). As mentioned before, in the case of strong stretching, the impact of the Kerr-nonlinearity can be separated from the effect of dispersion. In the following, the phase-contribution due to dispersion during propagation is denoted as φ_disp. To describe the spectral phase at the output of the system, the phase of the compressor, φ_co, has to be added. Since the phase-terms due to stretching and compression are also a result of dispersion, for simplicity, we reduce all the phase-contributions due to dispersion to a single phase-term φ_D = φ_st + φ_disp + φ_co. The residual spectral phase at the output of the nonlinear system is given by φ_total = φ_D + φ_SPM.

To quantify the quality of the pulse at the output of the chirped-pulse system, we use the ratio of the peak-power of the pulse to the peak-power of the transform-limited pulse, i.e. the Strehl-ratio, e.g. [14]:

S = \frac{max {∣ \int dΩ \exp (- i Ω T) \sqrt{s (Ω)} \exp (i [φ_{D} + B s (Ω)]) ∣}^{2}}{max {∣ \int dΩ \exp (- i Ω T) \sqrt{s (Ω)} ∣}^{2}} .

For the evaluation of the pulse only the spectrum s(Ω), the B-integral of the nonlinear propagation and the residual phase due to the stretcher-compressor-mismatch or other dispersion must be known. Compared to numerical simulations based on the fast Fourier-transform, Eq. 8 offers a simple approach to determine the optimal design of a nonlinear chirped-pulse systems. In particular, the stretching and the compression must not be explicitly described. Furthermore, it reveals the important physical parameters.

In the experiment we will match the stretcher and the compressor of the chirped pulse system, i.e. φ_st = -φ_co so as to demonstrate compensation of SPM with positive dispersion. Consequently, φ_D = φ_disp. In the following, the focus is placed on the compensation of the nonlinear phase with the dominating phase-term due to second-order dispersion, i.e. φ_disp = ϕ_disp ⁽²⁾Ω²/2. Soliton mode-locked oscillators emit pulses with sech ²-shape, and hence, the spectrum is also of sech ²-shape. For such shapes, the parabolic part of the nonlinear phase is partially compensated. In particular, the central section of the shape is parabolic. Furthermore, the sign of the second-derivative of the spectrum, s ⁽²⁾, is negative for standard spectral shapes, such as the sech ²-shape. It is worth noting that experimental techniques that use the third-order dispersion of the compressor can be also described with our model [15, 16].

In the experiment, we use the positive dispersion of an optical fiber to compensate for the Kerr-nonlinearity inside the fiber. Furthermore, we focus on the compensation of the nonlinear phase with the dominant phase-term φ_disp = β ⁽²⁾ LΩ²/2. Where the second derivative of the mode-propagation constant and the length of the fiber are denoted as β ⁽²⁾ and L, respectively. For a fixed center wavelength, the best Strehl-ratio is only a function of the bandwidth of the spectrum, the length of the dispersive medium and the B-integral. Numerical evaluation of Eq. 8 with regard to maximum Strehl-ratio allows determining the optimum fiber-length L as a function of both the spectral bandwidth and the B-integral. The corresponding curves are shown in Fig. 1(a).

To illustrate the effect of peak-power enhancement by nonlinearity, in Fig. 1(b) the autocorrelation traces at one fixed fiber-length but different B-integral values are given. We show the configuration corresponding to the experiment with the Öffner-stretcher: a sech ²-spectrum with bandwidth of 2.8 nm (FWHM), 50m-length of fiber, and a B-integral of about 10 rad at the optimum. The autocorrelation trace shows tails. This can be explained with the increase of residual (i.e. uncompensated) phase with the value of the B-integral at the points of best Strehl-ratio, Fig. 2(a). The behavior of the phase corresponds to temporal shifts of the energy away from the center. This results in temporal spreading of the pulse, Fig. 2(b). The strongest phase-shifts are located at the tails of the spectrum. Figure 2(a) also shows that over-compensation with parabolic phase-terms results in higher peak-powers (i.e. the best Strehl-ratio) as compared to perfect compensation of the parabolic terms of the spectral shape. This can be explained with the fact that the residual phase at the output has to be weighted with the spectral amplitudes. Figure 2 was obtained using Eq. 8 and adjusting the parameter ϕ_disp(2) for the best Strehl-ratio for the different B-integral values. The finding is in accord with the experimental results for the spectral phase was needed for improved operation of a nonlinear fiber CPA-system [18].

Fig. 1. (a) The condition for phase-compensation: length of the amplifier as a function of the FWHM bandwidth of the sech ² spectrum for B-integral values 5rad, 10rad and 20rad. The center-wavelength is 1030nm and β ⁽²⁾ is 25ps ²/km. (b) For the case of a FWHM bandwidth of 2.8nm and a 50m length of fiber, the autocorrelation trace at the optimum B-integral of 10rad is plotted (red cross in (a) ). To illustrate the effect of the phase-compensation, autocorrelation traces for B-integrals above and below the optimum are also shown. The traces are normalized to the peak.

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Fig. 2. (a) The residual spectral phase at point of optimum phase-compensation at different B-integral values (top) and the corresponding spectrum of sech ²-shape (bottom), (b) corresponding spread of the temporal intensity with increasing B-integral values.

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2.3. Perfect compensation

Perfect compensation of the nonlinear phase (i.e., negligible residual phase) will be achieved with a parabolic spectrum [17], s(Ω) = 1 - Ω² (√2/∆Ω_fwhm)². Where ∆Ω_fwhm is the full-width-at-half-maximum of the spectrum. In this case, the polynomial expansion of the spectral phase due to SPM is given by

φ_{SPM} = B [s^{(0)} + s^{(1)} Ω + s^{(2)} Ω^{(2)} / 2]

For the phase-compensation the zeroth and first (which is zero anyway) term can be neglected because they imply only an irrelevant constant phase-offset and a temporal delay after compression. It is worth noting that s ⁽²⁾ is negative. With these assumptions, the condition for phase-compensation at the output of the nonlinear fiber CPA-system is given by

B \frac{2}{Δ Ω_{fwhm}^{2}} = β^{(2)} L / 2 .

The relation allows determining the required length of the fiber-amplifier so as to compensate the spectral phase at the output of the CPA-system, which is operated at a certain B-integral. It is worth noting that the phase-compensation is strongly dependent on the bandwidth of the spectrum. According to Eq. 10 a broader bandwidth permits operation at higher values of the B-integral. This is can be also seen from Fig. 3(a). Comparing Fig. 3 to Fig. 1, it can be stated that for high B-integrals, Eq. 10 gives a quite accurate estimate even for non-parabolic spectral shapes.

Fig. 3. (a) The condition for phase-compensation: length of the amplifier as a function of the FWHM bandwidth of the parabolic spectrum for different B-integral values. The center-wavelength is 1030nm and β ⁽²⁾ is 25ps ²/km. (b) For the case of a FWHM-bandwidth of 5nm and a fiber of 15.4 m, the autocorrelation trace at the optimum B-integral of 10 rad is plotted. Autocorrelation traces for B-integrals above and below the optimum are also plotted.

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3. Experiment

In this section results of the experimental demonstration of the compensation of the self-phase modulation with positive dispersion are presented. First, a chirped-pulse system based on a chirped volume Bragg-grating (CVBG) will be considered. In particular, the CVBG is used for both stretching and compression of the pulse. The phase-compensation acts intrinsic during the nonlinear propagation of the stretched pulse in an optical fiber, which is placed between the stretcher and the compressor. The results show that side-effect arise, namely, the degradation of the pulse-contrast due to amplitude modulations. To avoid aberrations, in a second experiment we employ an Öffner-stretcher. With this device, the compensation of SPM with positive dispersion can be clearly demonstrated.

3.1. System based on CVBG

A schematic of the experimental setup is shown in Fig. 4. The setup consists of a passively mode-locked laser (10 MHz, ~ 400 fs, 1030 nm, 1.5 W), a CVBG and a fiber (mode-field diameter is 5μm) in which SPM and dispersion acts. The CVBG has a rectangular 4.2nm-bandpass and the center-wavelength is 1029.6nm. The device has a length of 3cm) and it is used for stretching and compression. The CVBG stretches the pulses to duration (FWHM) of about 190ps.

Fig. 4. Schematic of the experimental setup of the ultra-compact fiber CPA-system with intrinsic phase-compensation. One chirped volume Bragg-grating is used for stretching and compression of the pulses.

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Since the shape of the spectrum is given, the (normalized) peak-power can be calculated using Eq. 8. The resulting 2D map is a function of the fiber-length and the B-integral and it is shown in Fig. 5. The dispersion of the fiber is characterized by the second derivative of the mode-propagation constant β ⁽²⁾ = 25ps ²/km. It can be stated that for the given shape of the spectrum a clear optimum in CPA-performance will be only seen for fiber-lengths L > 20m.

Fig. 5. (a) Spectrum of the pulse before and after passage of the CVBG, and numerically calculated (i.e. ‘simulated’) ideal spectrum (b) Simulated (normalized) Peak-power at the output of the experimental configurations as a function of the B-integral and the fiber-length.

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The fiber-length is varied between the limits of 10m and 50m in steps of 10m by cutting from the 50m-length of fiber. At every length, the input power into the fiber is varied to obtain different B-integrals values. The spectrum and autocorrelation are recorded for each configuration. Figure 6(a) shows the widths of the measured autocorrelations over the B-integral values. At the minima of the curves, almost transform-limited pulses are produced since the dispersion of the fiber and nonlinearity compensate. For the fiber with a length of 40m, Fig. 6(b) shows the autocorrelation traces for the different B-integral values. The optimum corresponds to the B-integral of about 6rad.

Fig. 6. (a) Measured temporal widths (FWHM) of the autocorrelation traces as function of the B-integral of the nonlinear pulse propagation for different fiber-lengths, (b) Autocorrelation traces for characteristic B-integral values for 40m length of fiber. (c) Spectrum at the output at low and high B-integral value.

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However, the effect of phase-compensation is overlapped by the detrimental effect of enhancement of initially weak spectral modulations by the Kerr-nonlinearity. This phenomenon is explained in detail in [12, 13]. The CVBG imposes spectral amplitude modulation with a peak-to-peak value of ~ 10%. For such spectra, B-integral values of few radians leads to a fully modulated spectrum at the output of the fiber, i.e. energy of the main pulse is shifted to side pulses. This effect can also be seen by build-up of modulations in the spectrum with increasing B-integral: Fig. 6(c) shows spectra at B ~ 0rad and B = 6rad.

Figure 7 shows the same simulation as in Fig. 5 but starting with the experimental spectrum after the CVBG, Fig. 5(a). In particular, with increasing fiber-length and B-integral the relative peak-power enhancement is washed out. This agrees with the experimental observation.

Fig. 7. Simulation of performance with the real experimental spectrum: Peak-power (normalized to the energy) at the output of the CPA-system as a function of the B-integral and the fiber-length.

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3.2. System based on the grating-based Öffner-stretcher

To demonstrate the effect for higher values of the B-integral without the generation of temporal side-pulses, a cleaner pulse-shape at the input of the fiber is required [12, 13]. For this purpose, we designed and constructed a grating-based Öffner-stretcher being nearly free of aberrations. The stretching is about 170ps. For the experiment the same oscillator is used. The stretcher and compressor are perfectly matched to each other. A 30cm length of fiber was used during the matching of the stretcher and compressor. Then, the 50m length of fiber is spliced to the input of the small piece of fiber. In this way the alignment of the compressor is also kept with the 50 m length of fiber. Figure 8(a) shows the temporal widths of the autocorrelations as function of the B-integral values for the 50m length (mode-field diameter is 6μm) of the fiber. The corresponding autocorrelation traces are shown in Fig. 8(b). They are cleaner compared to the experiment with the CVBG. Almost no energy is shifted out of the time-window of ±10ps. This is in agreement with the recorded spectra: only minor distortions can be seen for B-integrals > 10rad. The spectra can be seen in Fig. 8(c). Using the formulas in [12], the temporal contrast of the recompressed pulse is estimated to be better than 30dB.

Fig. 8. (a) Measured temporal widths (FWHM) of the autocorrelation traces as function of the B-integral of the nonlinear pulse propagation for the 50m fiber-length. (b) Autocorrelation traces at different B-integral values for 50m length of fiber. (c) Spectrum at the output at low and high B-integral value.

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4. Conclusion

We showed that self-phase modulation can be compensated by the positive dispersion in chirped-pulse systems. The basic principles were described. Simple analytical expressions, which permit characterization of nonlinear chirped-pulse systems, were derived. Using these expressions design-guidelines for the application are obtained. Of particular importance is a relation between the B-integral, the spectral bandwidth, and the second derivative of the phase due to dispersion, Eq. 10. In particular, a larger bandwidth permits nonlinear operation at higher B-integral values. In addition, we demonstrated the method in fiber-based chirped-pulse systems. Intrinsic phase-compensation could be realized in an ultra-compact nonlinear chirped-pulse system using chirped volume Bragg-gratings. The experimental results showed that the method relies on clean stretched pulses. In agreement with previous findings it could be shown that modulations imposed on the spectrum limit the applicability of the method. Generating clean pulses with an Öffner-stretcher, the compensation technique could be demonstrated without any side-effects. In principle, the method could be applied in the different layouts of chirped-pulse systems.

5. Acknowledgment

This work has been partly supported by the German Federal Ministry of Education and Research (BMBF) with project 03ZIK455 ’onCOOPtics’. The authors also acknowledge support from the Gottfried Wilhelm Leibniz-Programm of the Deutsche Forschungsgemeinschaft.

References and links

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Self-phase modulation compensated by positive dispersion in chirped-pulse systems

Abstract

1. Introduction

2. Analysis of the phase-compensation mechanism

2.1. Spectral phase due to self-phase modulation of stretched pulses

2.2. Compensation of SPM with positive dispersion

2.3. Perfect compensation

3. Experiment

3.1. System based on CVBG

3.2. System based on the grating-based Öffner-stretcher

4. Conclusion

5. Acknowledgment

References and links

Cited By

Figures (8)

Equations (10)

Optics Express