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We both theoretically and experimentally investigate the optimization of femtosecond Yb-doped fiber amplifiers (YDFAs) to achieve high-quality, high-power, compressed pulses. Ultrashort pulses amplified inside YDFAs are modeled by the generalized nonlinear Schrödinger equation coupled to the steady-state propagation-rate equations. We use this model to study the dependence of compressed-pulse quality on the YDFA parameters, such as the gain fiber’s doping concentration and length, and input pulse pre-chirp, duration, and power. The modeling results confirmed by experiments show that an optimum negative pre-chirp for the input pulse exists to achieve the best compression.
©2012 Optical Society of America
1. Introduction
Yb-doped fiber amplifiers (YDFAs) feature superior power scalability, high electrical-to-optical conversion efficiency, large single-pass gain (~30 dB), excellent beam quality, as well as compactness and robustness. To avoid detrimental effects from fiber nonlinearities (e.g., self-phase modulation (SPM), stimulated Raman scattering etc.) when amplifying ultrashort pulses, YDFAs normally operate in a low-nonlinearity regime using chirped pulse amplification (CPA) [1, 2], in which the spectral bandwidth of the amplified pulse only changes slightly during the amplification.
For some applications, strong nonlinear effects are preferred in YDFAs such that the amplified pulse acquires substantial extra bandwidth, and therefore can be compressed much shorter than the pulse prior to the amplification. This operation regime is of particular importance for implementing high repetition-rate (>1 GHz) master-oscillator-power-amplification (MOPA) systems with applications including optical arbitrary waveform generation, high-resolution spectroscopy, high-speed analog-to-digital conversion, and precision calibration of astronomical spectrographs to search for Earth-like extra-solar planets [3], to name a few. The ideal front-end of such a MOPA laser system is a fundamentally mode-locked fiber oscillator with multi-GHz repetition rate. Primarily limited by the available pump power (~1W) from single-mode laser diodes and low intra-cavity pulse energy, these multi-GHz fiber oscillators normally produce sub-picosecond/picosecond pulses with <100 mW average power. The poor performance on pulse duration and average power prevents multi-GHz fiber oscillators from most nonlinear optical applications that usually demand femtosecond (~100 fs) pulses of several nano-joules for pulse energy. Such a limitation can be overcome by the nonlinear amplification in YDFAs which broaden the input pulse spectrum. The spectrally-broadened, power amplified pulses are then de-chirped by a subsequent optical compressor to much shorter duration. However, relatively narrow gain bandwidth (~40 nm) of Yb-doped fibers and the gain narrowing effect during the power amplification usually generate compressed pulses >100-fs with considerable pedestal. In this paper we both theoretically and experimentally study the dependence of the compressed pulse quality on the parameters of the fiber amplifier (e.g., doping concentration and length of the gain fiber) and the input pulse (e.g., pre-chirp, duration, and power) from the fiber oscillator. The results from detailed numerical modeling are presented in section 2 and 3. We find that the input pulse pre-chirp—a quantity that can be easily tuned in experiments—is critical for optimizing the system to achieve high-quality compressed pulses. Section 4 presents the experimental results confirming the existence of an optimum negative pre-chirp that leads to the best-quality compressed pulses, as predicted by numerical modeling. Finally section 5 concludes the paper.
2. Modeling nonlinear amplification of femtosecond pulses in YDFAs
Amplification of femtosecond pulses in an YDFA involves nonlinear interaction among the pump, signal (i.e., the input pulses to be amplified), and Yb-fiber. The process can be accurately modeled by coupling two sets of eqs [4–6]: (1) the steady-state propagation rate eqs. that treat the Yb-fiber as a two-level system and (2) the generalized nonlinear Schrödinger equation (GNLSE) that describes the evolution of the amplified pulses. Such a model, with all the modeling parameters experimentally determined, enables us to study and therefore optimize a femtosecond MOPA system.
YDFA can be pumped in the co-propagating scheme (i.e., pump and signal propagate in the same direction.), counter-propagating scheme (i.e., pump and seed propagate in the opposite direction.), or both. In this paper, we focus on co-propagating pumping scheme which is normally adopted in monolithic femtosecond nonlinear fiber amplifiers. Under this scenario and with amplified spontaneous emission (ASE) neglected, the model includes the following steady-state propagation-rate eqs [7–9]. and the GNLSE:
In Eq. (1), the amplifier is treated as a homogeneously broadened two-level system. ρ accounts for the ion density per unit volume. N2 and N1, satisfying N2(t,z) + N1(t,z) = 1, denote the ion fraction for the upper state and the ground state. Ps(λ,z)and Pp (λ,z) represent the signal and pump power at position z for wavelength λ. Γp (Γs) corresponds to the modal overlap factor between the pump (signal) mode and the ion distribution [10]. σe and σa denote the emission and absorption cross-sections. The spontaneous decay rate from level 2 to 1 is 1/τ21; τ21 is the characteristic fluorescence lifetime. The stimulated absorption/emission rate and pumping rate for a fiber core-area A are Wij = Γs(λ)σe,aPs/hυsA and Rij = Γp(λ)σe,aPp/hυpA, respectively. For the steady-state solution, by setting , we get
When writing Eq. (3), we assume that the inverse of pulse’s repetition-rate is much shorter than the relaxation time of the upper level (~1 ms), which holds in the paper.The nonlinear propagation of the signal pulse is governed by the standard GNLSE (i.e., Eq. (2)), taking into account the gain, dispersion, self-phase modulation, self-steepening, and stimulated Raman scattering. A(z,t), βn, γ, and ω0 are the amplitude of the slowly varying envelope of the pulse, n-th order fiber dispersion, fiber nonlinearity, and central frequency of the pulse, respectively. R(t) denotes both the instantaneous electronic and delayed molecular responses of silica molecules, and is defined as R(t) = (1-fR)δ(t) + fR(τ12 + τ22)/(τ1τ22)exp(-t/τ2)sin(t/τ1) where fR, τ1, and τ2 are 0.18, 12.2 fs, and 32 fs, respectively [11]. The GNLSE is solved by the split-step Fourier method implemented using the fourth-order Runge-Kutta method in the interaction picture [12] with adaptive step-size control [13].
Flow chart in Fig. 1 illustrates the iteration in numerically solving Eq. (1) and (2). In each step, the signal power spectrum Ps(ω, z0) is first calculated from Ã(ω, z0) and then substituted into the rate equation Eq. (1) to obtain the signal power Ps(ω, z0 + Δz) at the next position. The gain spectrum g(ω, z0) is derived from Ps(ω, z0 + Δz) and Ps(ω, z0), and fed into Eq. (2) to derive Ã(ω, z0 + Δz) from Ã(ω, z0). Repeating this iteration generates the longitudinal distribution of pump power and population inversion as well as the nonlinear evolution of the seed-pulse along the gain fiber.
Table 1 lists the key simulation parameters. For a single-mode step-index fiber with its fundamental mode approximated by a Gaussian, the overlap of the optical mode and ion distribution becomes Γ = 1-exp(−2a2/w2). a is the ion dopant radius or core radius and w the mode field radius at 1/e2 power intensity approximated by the Whitely model: w = a(0.616 + 1.66/V1.5 + 0.987/V6) [10], where V is the normalized frequency. The wavelength dependent emission cross-section σe and absorption cross-sections σa are adapted from Fig. 2 in Ref. 4.
Table 1. Amplifier Parameters Used in the Simulation
Fig. 2 (a) Pump and signal power as a function of Yb-fiber length. (b) RMS duration of the optimum compressed- pulse and the transform-limited pulse as a function of Yb-fiber length.
As an example, we study an YDFA seeded with an 80-MHz pulse train centered at 1.03 µm with 20-mW average power corresponding to 0.25-nJ pulse energy. The amplifier is constructed from a single-mode Yb-fiber with the ion density of 1025 m−3 pumped by a 976-nm diode with 600-mW average power. The amplified pulse is then compressed using a pair of diffraction-gratings (600 grooves/mm for line density) operating at 45° incident angle which introduces a negative group-delay dispersion (GDD) of −1408 fs2/mm and a third order dispersion of 2180 fs3/mm. For the simulation results in Fig. 2, the input signal is a transform-limited Gaussian pulse of 170-fs root-mean-square (RMS) (or 200-fs full width at half maximum (FWHM)) duration. Figure 2(a) plots the absorption of pump (blue, solid line) and amplification of signal (red, dashed line) as a function of Yb-fiber length. For 2-m Yb-fiber, the pump is almost completely absorbed and the signal power reaches the maximum of 560 mW. Note that in this paper, the input pulse is characterized by FWHM duration which is normally used for a Gaussian-shaped pulse. For a compressed pulse, we use RMS duration which provides better information on both the compression quality and the pulse pedestal. The input RMS duration can be calculated from the input FWHM duration by τFWHM = 1.178τRMS. At each Yb-fiber length, the grating-pair separation is adjusted to compress the amplified pulse to the optimum (i.e., shortest) RMS duration, plotted in Fig. 2(b). As a comparison, the corresponding transform-limited RMS duration is plotted in the same figure as well. While the transform-limited duration monototically decreases due to the continuous increase of the pulse’s bandwidth, the optimum compressed duration reaches a minimum value of 63-fs at the Yb-fiber length of 100-cm. At this length, the compressed duration coincides with the transform-limited duration indicating a close-to-perfect compression. Nonetheless, the signal power is only amplified to 140 mW. Although a longer Yb-fiber length leads to higher amplified power, the compressed-pulse duration deviates farther away from the transform-limited duration, implying a worse compressed pulse quality. Therefore, achieving both high amplified power and good compressed-pulse quality requires optimizing more parameters (e.g., input pulse chirp, bandwidth, and power, Yb-fiber length and doping concentration, and grating pair separation, etc), which is discussed in the next section.
3. Optimization of different amplifier parameters
3.1 Optimization of the pre-chirp
We first use our model to investigate the effect of the input pulse chirp on the pulse compression with the Yb-fiber length fixed at 2-m. As shown in Fig. 3 , the signal pulse is pre-chirped, then amplified by an YDFA, and finally compressed to its minimal RMS duration by a diffraction-grating pair. The positive pre-chirp is introduced by propagating the pulse through fused silica with β2 = 18.9 fs2/mm and β3 = 16.5 fs3/mm; the negative pre-chirp is generated from a diffraction-grating pair the same as the one in the previous section, with β2 = −1408 fs2/mm and β3 = 2180 fs3/mm. In the simulation, other parameters remain unchanged from section 2.
Figure 4 summarizes the simulation results with the total GDD varying from −7 × 104 to 4 × 104 fs2 for pre-chirping the input signal pulse prior to amplification; hereafter, we refer to such GDD as pre-chirping GDD. While the transform-limited RMS duration (green, dashed line in Fig. 4(a)) slightly varies, the optimum RMS duration for the compressed pulse strongly depends on the pre-chirping GDD. Insets of Fig. 4(a) show the compressed pulse (blue, solid line) and transform-limited pulse (green, dashed line) for three different GDD: (I) −5 × 104 fs2, (II) 0, and (III) 1.9 × 104 fs2. Figure 4(b) illustrates the evolution of RMS bandwidth along the Yb-fiber for these three pre-chirping cases. For a negatively pre-chirped input-pulse (e.g., case I), its spectrum (blue line) experiences an initial spectral compression [5,6] and then subsequent spectral broadening. While the case II (i.e., zero pre-chirp) generates the broadest spectrum in terms of RMS bandwidth, both its transform-limited pulse and the optimum compressed-pulse are longer than the other two cases due to the existence of strong pedestals. This simulations suggests that varying the input pulse pre-chirp leads to different spectra, some of which exhibit considerable temporal pedestals that severely limit the quality of the compressed pulse. This can be clearly seen in Fig. 4(c), in which the output spectrum corresponding to negative pre-chirp (shown as blue curve) exhibits less steep wings than the other two cases; the resulting transform-limited pulse has minimal pedestals in general. In this case, there exists an optimum negative pre-chirping GDD that leads to a compressed pulse with the shortest duration close to its transform-limited duration (see case I inside Fig. 4(a)).
Fig. 4 (a) Optimum RMS duration of the compressed-pulse and the corresponding transform-limited RMS duration as a function of pre-chirping GDD for the input pulse. Insets: compressed pulses and transform-limited pulses for three different pre-chirp. (b) bandwith evolution inside the Yb-fiber amplifier. (c) output sepctra for three different pre-chirp.
In fact, this phenomenon appears in mode-locked Yb-fiber oscillators as well. Depending on the amount of net cavity GDD, an Yb-fiber oscillator may operate in different mode-locking regimes (i.e., stretched-pulse, similariton, and dissipative soliton) [14]. For a stretched-pulse Yb-fiber oscillator that features a net cavity GDD close to zero, the intra-cavity pulse prior to entering the gain fiber acquires negative chirp from dispersion over-compensation by optical elements with negative GDD (e.g., diffraction-grating pair, chirped fiber Bragg grating, and hollow-core photonic-crystal fiber). On the contrary, the intra-cavity pulse in similariton or dissipative soliton Yb-fiber oscillators exhibits positive chirp before entering the gain fiber. Our simulation results presented in Fig. 4, despite modeling a single-pass Yb-fiber amplifier, explains why stretched-pulse mode-locking produces better pulse quality than the other two regimes of mode-locking.
3.2 Optimization of the input power and optical bandwidth
In this subsection, we study the compressed pulse quality as we vary the FWHM duration from 200 fs to 600 fs (i.e., different input optical bandwidth) for an input Gaussian pulse before being pre-chirped. The input RMS duration can be calculated from the input FWHM duration by τFWHM = 1.178τRMS. Other parameters such as the fiber length are the same as those in the previous section. For every input power and input transform-limited pulse duration, we scan the pre-chirp to find the optimum compressed-pulse duration.
Figure 5 illustrates the optimum compressed duration versus input power for an input pulse with different transform-limited duration. The optimum compressed RMS duration occurs at ~20-mW input power, corresponding to a pulse energy of ~0.25 nJ. For an input power less than 20mW, the optimum compressed duration starts to increase significantly because the accumulated nonlinearity is too small to broaden the spectrum enough for achieving substantial pulse compression. Even for an input pulse duration of 500 fs, it can be compressed to 63 fs, a factor of 6.7 in RMS duration shortening. That is, as long as we apply a suitable pre-chirp, the RMS duration of the compressed pulse varies slightly as the transform-limited input pulse duration changes from 200 fs to 500 fs. These results indicate that, while the amplified output power is nearly the same (slight difference due to wavelength dependent emission/absorption cross-section) for all cases (~560 mW), the best compressed pulse is achieved by optimizing both the pre-chirp and the input power.
Fig. 5 Calculated RMS duration for optimum compressed-pulse as a function of input signal power for five different spectral bandwidth corresponding to transform-limited pulse FWHM duration of 200 fs, 300 fs, 400 fs, 500 fs, and 600fs.
3.3 Optimization of the Yb-ion doping concentration
Furthermore, at 20-mW input seed power and 600-mW pump power, we study the effect of three different Yb-ion doping concentrations: 1025/3 m−3, 1025 m−3, and 3 × 1025 m−3. Varying the doping concentration allows us to better understand the interplay between nonlinearity and the amplifier gain. To get the same amplified power for these three cases, the fiber lengths are chosen to be 6 m, 2 m, 2/3 m, respectively since the amplified power is determined by the product of the Yb-fiber length and its doping concentration. Figure 6 shows that, for different transform-limited input-pulse duration, the optimized compressed pulse duration depends on Yb-fiber doping concentration. For a higher doping concentration with shorter fiber-length, the accumulated nonlinearity is reduced. This is a consequence of the reduced effective length of the amplifier. In general, Yb-fiber with higher doping concentration is preferred for the input signal pulse with shorter transform-limited duration.
Fig. 6 Calculated RMS duration of the optimum compressed pulse as a function of the FWHM pulse duration of the transform-limited Gaussian input pulse for three doping levels: blue-triangle curve for high doping at 1025/3 m−3; red-circle curve for medium doping at 1025 m−3; and black-square curve for low doping at 3 × 1025 m−3. The purple-diamond curve shows the compressed-pulse RMS duration obtained with a low-doping Yb-fiber amplifer seeded with transform-limited pulses.
For a long Yb-fiber (here 6 meters) with low doping concentration, the resulting nonlinear fiber amplifier works in a well-studied regime known as parabolic similariton amplification; that is, any input pulse to an amplifier constructed from sufficiently long Yb-fiber will asymptotically evolve into a positively chirped pulse with a parabolic temporal profile and propagates inside the fiber amplifier self-similarly [15, 16]. Most existing investigations employed a transform-limited pulse at the input, corresponding to the purple-diamond curve in Fig. 6. The curve shows that achieving shorter compressed pulses prefers longer transform-limited pulses at the fiber input for low-doped Yb-fibers. The compressed pulse duration can be further reduced if the input pulses are optimally pre-chirped; see the black-square curve in Fig. 6. Note that parabolic similariton amplification suffers from limited gain bandwidth and stimulated Raman scattering [17]. These limitations have been numerically studied assuming a finite Lorentzian gain bandwidth [18]. With the gain derived from the rate eq. in our model, we can perform a more accurate optimization of parabolic similariton amplification.
4. Experimental results on pre-chirp management for optimizing compressed pulse quality
In this section, we experimentally investigate the effect of input pulse pre-chirp on the compressed pulse quality after amplification. Figure 7 schematically illustrates the experimental setup consisting of an Yb-fiber oscillator serving as a seed source, a diffraction-grating pair to adjust the pre-chirp of the input pulse launched into the Yb-doped fiber amplifier, and finally another diffraction-grating pair to compress the amplified pulse to the shortest pulse duration. The oscillator of 280-MHz repetition rate is passively mode-locked with a saturable Bragg reflector and cavity dispersion is managed by highly dispersive mirrors [19]. The output pulse has a duration of 2.1 ps with ~6 nm FWHM bandwidth centered at 1030 nm. Since the pulse is positively chirped, use of a grating pair (600 grooves/mm) providing negative chirp allows to continuously tune the input-pulse pre-chirp from positive to negative by changing the separation of the grating pair. The pre-chirped pulses are coupled into the YDFA with >60% efficiency. The 10% tap from the splitter monitors the seed power into the YDFA including a 2-m Yb-fiber (Nufern, SM-YSF-LO) pumped by two laser diodes combined with a polarization beam combiner. The amplified pulses are compressed by the second grating pair to the shortest pulse duration.
Fig. 7 Experimental set-up, OSC: oscillator, HP: half waveplate. M: mirror, G: Grating, LS: lens, FBS: fiber beam splitter, LD: laser diode, PBC: polarization beam combiner, WDM: wavelength division multiplexer, YDF: Yb doped fiber, QP: quarter waveplate, OSA: optical spectrum analyzer, AC: autocorrelator, P.M.: power meter.
With the fixed 20-mW input power, we achieved amplified pulses of 600-mW average power at 1-W pump power. We varied the pre-chirp by changing the separation of the first grating pair and then adjusted the second grating pair to compress the amplified pulses to its shortest FWHM duration measured by an autocorrelator. We recorded the compressed-pulse spectra as we varied pre-chirp and then calculated from these spectra the RMS duration of the transform-limited pulses. Such a RMS duration was plotted in Fig. 8(a) as a function of pre-chirping GDD. Figure 8(b) plots the input pulse spectrum (black dashed curve) and three compressed-pulse spectra corresponding to different pre-chirping GDD: −6.3 × 104 fs2 (black curve), −1.8 × 104 fs2 (blue curve), and 1.0 × 104 fs2 (red curve).
Fig. 8 (a) Calculated RMS pulse duration of optimum compressed pulses from the measured spectrum (black scattered) and simulated curve (green line). The shortest autocorrelation (AC) trace is achieved at the lowest RMS pulse duration. τAC (FWHM) = 134 fs for a pre-chirping GDD of −6.3 × 104 fs2, τAC (FWHM) = 149 fs for −1.8 × 104 fs2, and τAC (FWHM) = 169 fs for 1 × 104 fs2 respectively. (b) The spectra corresponding to three AC traces in (a). The pulse has the largest spectral bandwidth and smoothest edge for the pre-chirping GDD of −6.3 × 104 fs2.
For all three cases, the input spectrum is broadened from ~6 nm to >20 nm. However, the shape of the amplified pulse spectra varies substantially with different pre-chirp, which in turn supports different transform-limited pulse duration. The black scattered curve shows that the minimum duration is achieved at a negative pre-chirping GDD of −6.3 × 104 fs2. Also plotted in Fig. 8(a) are the autocorrelation traces for the three compressed pulses corresponding to these three pre-chirping GDDs. The FWHM of these autocorrelation traces are 134 fs, 149 fs, and169 fs, respectively. Apparently, the best compression quality occurs at the pre-chirping GDD of −6.3 × 104 fs2 with a measured autocorrelation trace of 134-fs, suggesting a de-convolved pulse of ~100-fs. Deviation from this optimum pre-chirp degrades the compressed-pulse quality featuring an increased temporal pedestal. The spectra in Fig. 8(b), similar to the results shown in Fig. 4(c), show that, as we vary the pre-chirping GDD from the optimum value of −6.3 × 104 fs2 to −1.8 × 104 fs2 and then to 104 fs2, the corresponding spectra start to develop sharper edges, which leads to larger pedestal for the compressed pulses. Note that there are two sections of HI1060 single mode fiber before (2.45m) and after (0.33 m) the YDFA. These two fibers would introduce excessive nonlinearity and hence further optimization of other parameters other than pre-chirp is needed to achieve even shorter pulse duration.
The green curve in Fig. 8(a) shows the simulation results by numerical modeling the experimental setup. It can be seen that the modeling and experimental measurements agree well for negative pre-chirp. The difference between experiment and simulation can be mainly attributed to the fact that the emission/absorption cross-section, the doping concentration, and the modal overlap factors of the pump/signal mode, Γp and Γs, are just assumed rather than experimentally derived. In the simulation, we neglected the higher-order (>2) spectral phase of the initial 2.1-ps pulse. Nevertheless, our model correctly predicts the overall tendency of the measurements: there exists an optimum pre-chirp that results in the shortest compressed pulse.
5. Conclusion and discussion
In this paper, we use a model —which couples the steady-state propagation-rate eqs. and the GNLSE—to study nonlinear amplification of femtosecond pulses in Yb-fiber amplifiers configured in the co-pumping scheme. We investigated these femtosecond Yb-fiber amplifiers to achieve better compressed pulse quality by optimizing the parameters, such as the input pulse pre-chirp and power, input pulse bandwidth, and Yb-fiber doping concentration. The results show that, in general, a negative pre-chirp exists to achieve the best compression which is verified experimentally.
The presented study has practical applications. For example, we recently demonstrated a fundamentally mode-locked Yb-fiber oscillator emitting ~206 fs pulses with 15-pJ pulse energy and 3-GHz repetition rate [20]. Stabilization of the oscillator’s repetition rate and the carrier-envelope phase offset will result in a femtosecond frequency comb with 3-GHz line spacing. Such a large line spacing is desired in many spectral-domain applications, e.g., frequency combs optimized for precision calibration of astronomical spectrographs [21–26]. Stabilization of the carrier-envelope phase offset using the well-known 1f-2f heterodyne detection technique involves generation of low-noise supercontinuum using ~100-fs (or even shorter) pulses with ~1-nJ pulse energy. Both our theoretical and experimental results enable us to construct an optimized Yb-fiber amplifier followed by a proper compressor to achieve these pulse requirements from the initially long (~206 fs) and weak (15 pJ) oscillator pulses.
Acknowledgment
This work was supported by the National Aeronautics and Space Administration (NASA) through grants NNX10AE68G, NNX09AC92G and by the National Science Foundation (NSF) through grants AST-0905592 and AST-1006507 and the Center for Free-Electron Laser Science.
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