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The generation of high-fidelity femtosecond pulses is experimentally demonstrated in a fiber based chirped-pulse amplification (CPA) system through an adaptive amplitude and phase pre-shaping technique. A pulse shaper, based on a dual-layer liquid crystal spatial light modulator (LC-SLM), was implemented in the fiber CPA system for amplitude and phase shaping prior to amplification. The LC-SLM was controlled using a differential evolution algorithm, to maximize a two-photon absorption detector signal from the compressed fiber CPA output pulses. It is shown that this approach compensates for both accumulated phase from material dispersion and nonlinear phase modulation. A train of pulses was produced with an average power of 12.6W at a 50MHz repetition rate from our fiber CPA system, which were compressible to high fidelity pulses with a duration of 170 fs.
©2008 Optical Society of America
1. Introduction
Rare-earth doped fiber laser systems are a promising alternative to bulk ultrafast laser systems [1], whose power scaling is not straightforward due to their generally low single-pass gain, coupled with thermo-optical issues. Ytterbium-doped fibers are particularly interesting due to their broad emission spectrum, allowing the generation and amplification of ultrashort optical pulses, although the gain medium does not support the generation of pulses as short as can be obtained with Ti:sapphire. The fiber geometry offers good thermo-optical properties, a high single-pass gain, excellent output beam quality, and coupled with continuous-wave diode pumping, has allowed the realization of compact ultrafast laser systems with more than 100W average power at various repetition rates [2, 3]. Energy levels reaching the millijoule regime have also recently been demonstrated, thanks to the utilization of novel fiber designs, such as photonic crystal-fiber (PCF) [4].
The chirped-pulse amplification (CPA) technique is the preferred way to achieve high peak-power ultrashort pulses in ultrafast laser systems [5]. However, a multitude of factors can degrade the pulse quality, such as uncompensated material dispersion, nonlinearity, and a non-uniform spectral gain profile with finite width. The pulse quality degradation is most notably manifested in the presence of a pedestal, which limits the maximum peak intensity of the pulses that can be achieved [6]. Two factors leading to pulse degradation are uncompensated higher-order spectral phase between stretcher, compressor, and amplifier materials, and self-phase-modulation (SPM), which leads to a nonlinear spectral phase [7].
In ultrafast bulk solid-state laser systems, a great deal of effort has been spent to design appropriate stretcher and compressor pairs to minimize the uncompensated higher-order spectral phase [8]. However, this approach requires careful characterization and design of the system. Furthermore, it is non-adjustable and thus prevents easy reconfiguration of the system. Ultimately, programmable femtosecond pulse shaping [9] offers the possibility of almost arbitrary modifications of the phase and amplitude of ultrashort optical pulses, and thus eliminate the need for meticulous characterization and design of the entire system. In particular, in combination with adaptive learning loop utilizing optimization algorithms, i.e. adaptive pulse shaping, it has allowed for the generation of high-fidelity pulses [10, 11, 12, 13, 14]. This approach also allows for the generation of arbitrary pulse shapes, using either phase-only modulation [15], or phase and amplitude modulation [16]. Nevertheless, femtosecond pulse shaping typically suffers from low throughput, associated with various losses in the setup, and usually a low damage threshold from the programmable modulator, e.g. a pixelated liquid crystal array. In order to circumvent this problem, the pulse shaper has to be incorporated into the system before any high power amplification stages. In Ti:sapphire laser systems, a phase-only pulse shaping setup has been successfully incorporated as part of the stretcher with [17, 18, 19], or without [20], global optimization algorithms to minimize the pulse duration at the output. The generation of arbitrary shaped pulses has also been demonstrated using amplitude and phase shaping and a global optimization algorithm [21].
In ultrafast fiber laser systems, material dispersion, nonlinearity, and a non-uniform spectral gain profile with finite width, are critical considerations, because of the optical confinement and long interaction length in the fiber geometry. As temporal stretching is physically limited by the finite size of the grating compressor, while the scaling of large-mode-area (LMA) fibers will eventually undermine the advantages of fiber geometry, most of the work has concentrated at managing the SPM, by compensation of the nonlinear phase induced by SPM via the third-order material dispersion [22, 23, 24, 25], or utilization of the interplay between the SPM-induced spectral broadening and gain shaping [26, 27]. These approaches, however, cannot fully compensate the nonlinear phase-modulation due to the SPM, and require careful design of the laser system. Recently, another method was proposed to actively compensate for the SPM using phase modulation of the stretched pulses [28, 29], but the phase modulator can only impose a limited amount of phase shift and is limited in terms of complexity of phase profile that can be applied. Finally, the effects of the non-uniform spectral gain profile with finite width also becomes more prominent, further degrading the pulse quality. In fiber CPA systems, multiple amplifier stages are usually necessary to achieve the desired power level. Hence, fiber CPA systems present more technical challenges in producing high-fidelity pulses.
In view of the work in ultrafast bulk solid-state laser systems, adaptive pulse shaping prior to amplification has been adopted in fiber CPA laser systems with some success. Recently, amplitude-only shaping has been demonstrated to control the nonlinear-phase modulation induced by SPM at low energy [30], but it cannot compensate for higher-order spectral phase due to the material dispersion. Our group recently demonstrated a phase-only shaping in a high energy fiber laser system [31].
In this paper, we demonstrate, for the first time to our knowledge, the implementation of adaptive amplitude and phase pre-shaping in a fiber-based CPA system to generate high-fidelity compressed femtosecond pulses. A pulse shaper based on a dual-layer liquid crystal spatial light modulator (LC-SLM) was implemented in our fiber CPA system for amplitude and phase shaping prior to amplification. The LC-SLM is compatible with pulses at any repetition rate, making it the preferred choice for our experiments, and contrasts with a recent post-amplification shaping work utilizing dazzler with limited repetition rate and output efficiency [32]. The LC-SLM was controlled using a differential evolution (DE) algorithm, to maximize a two-photon absorption detector signal produced from the compressed fiber CPA output pulses. We show that our approach compensates for both accumulated phase from higher-order material dispersion and nonlinear phase modulation. A train of pulses with an average power of 12.6W at a 50MHz repetition rate was produced from the fiber CPA system, compressible to high fidelity pulses with a 170 fs temporal full-width at half-maximum (FWHM).
This paper is organized as follows. The fiber CPA experimental setup and the implementation of the adaptive loop pulse shaping are described in Section 2. The experimental results are presented and discussed in Section 3. Finally, we conclude the paper in Section 4.
2. Experimental Setup
2.1. Ultrafast fiber laser system
The experimental setup is schematically illustrated in Fig. 1. The seed source was a passively mode-locked Yb-doped fiber oscillator [33], operated in the self-similar regime [34]. The laser produced a train of chirped pulses with a duration of 2 ps at a 50MHz repetition rate, and a 16nm spectral FWHM at a 1042nm central wavelength. The spectrum of the pulse train had a sharp truncation at the edges, a signature of the self-similar regime, with a 20 dB spectral width of 23 nm. The oscillator produced an average power of 30mW, corresponding to a pulse energy of 0.6 nJ. Using a fiber coupler, part of the output power was routed for this experiment, such that 2mW of average power was launched into a pre-amplifier, while the rest was used for another experiment. A 1.7m long single-mode (SM) core-pumped Yb-doped fiber was used in the first pre-amplifier to boost the average power of the seed pulses to 80mW. The pulses were then sent into a pulse shaper [9], which will be explained in detail later, via a free-space optical circulator. The pulse shaper had a throughput efficiency of 40%, resulting in an average power of 30mW launched into the next pre-amplifier. The second pre-amplifier, comprised of a 1.5m long SM core-pumped Yb-doped fiber, further amplifying the pulses to an average power of 150mW. Both of the pre-amplifiers were pumped by SM fiber-coupled diodes at 976nm in a co-propagating scheme, using wavelength division multiplexers to combine both the seed and pump into the amplifier fibers. The fiber ends were cleaved at an angle to avoid parasitic lasing, and optical isolators were placed at appropriate places to prevent any back-propagation through the system. The pulses were then routed by dichroic mirrors before being launched into the final amplifier. A 1.7m long double-clad LMA polarization-maintaining (PM) Yb-doped photonic-crystal fiber (Crystal-Fibre DC-200/40-PZ-Yb-01), with an active core diameter of 40µm (NA = 0.03) and an inner cladding diameter of 200µm (NA = 0.55), was used for this final amplifier. The Yb-doped PCF had 10 dB/m absorption at 976nm for light launched into the cladding, and was pumped by a multi-mode fiber-coupled diode generating up to 30W power at 976nm in a counter-propagating scheme. The input facet of the PCF was hermetically sealed, while the output facet was spliced to a very short length of coreless fiber in order to reduce the intensity at the facet. Both ends were polished at 5° angle to avoid parasitic lasing. At the output of the power amplifier, the beam was passed through a telescope arrangement to collimate the diverging beam and passed through an optical isolator. The train of pulses, at this point, had a maximum average power of 12.6W, corresponding to a pulse energy of 252 nJ. Finally, a fraction of the output, taken from the Fresnel reflection of a wedge, was compressed using a pair of gold-coated 900 lines/mm holographic gratings. A home-built second-harmonic generation (SHG) FROG, utilizing a 300µm thick β-barium borate crystal and a spectrometer, was employed to characterize the pulses. In addition, the pulse spectra at various points in the fibre CPA system were measured using an optical spectrum analyzer with a resolution of 0.1nm.
Fig. 1. Schematic illustration of the ultrafast fiber laser system. HWP: Half-wave plate, QWP: Quarter-wave plate, OI: Optical isolator, WDM: Wavelength division multiplexer, SM: Single-mode, PBS: Polarizing beam splitter, FR: Faraday rotator, APP: Anamorphic prism pair, G: Gratings, CM: Cylindrical mirror, SLM: Spatial light modulator, PCF: Photonic crystal fiber, T: Telescope arrangement, TPA: Two-photon absorption detector, SHG FROG: Second-harmonic generation frequency-resolved optical gating, DE: Differential evolution algorithm on a computer.
The pulse shaper consisted of a 4 f setup [9] arranged in a reflective configuration. The collimated output from the first pre-amplifier had a 0.9mm 1/e 2 intensity half-width, which was expanded three-times by an anamorphic prism pair, before being sent into the pulse shaper setup. In the pulse shaper, the beam had its spectral components spatially dispersed by a 1200 gr/mm gold-coated plane-ruled grating, which were then focussed onto the Fourier plane by a 25.8cm focal length cylindrical mirror. The grating was operated at a quasi-Littrow condition with a vertical tilt such that the beam was steered to another horizontal plane, where the cylindrical mirror lies [35]. A dual-layer LC-SLM (CRi SLM-128) was placed at the Fourier plane, allowing for the modulation of both the phase and amplitude of the spectral components of the pulses. However, since the two LC layers in the SLM have slightly different thicknesses, they do not yield the same phase retardance for the same voltage applied, making it difficult to impart phase-only or amplitude-only shaping. Both layers of the LC-SLM had N s = 128 pixels, spanning 13.1mm. The 23nm 20 dB-spectral-width of the pulse occupied 97 pixels on the SLM, and the spectrum at the Fourier plane varied linearly with wavelength at a rate of 0.23 nm/pixel. The calculated time window for the pulse shaper optical setup (excluding the SLM) was 23.9 ps with a complexity of 346 [9]. No attempt was made to operate the pulse shaper at precisely zero-dispersion, because any small offset should be accounted for by the adaptive shaping method. An adaptive loop to control the phase and amplitude modulation applied by the LC-SLM to the pulse train was implemented to maximize the signal from a GaAsP TPA detector, which is essentially ∝∫I 2(t)dt, where I(t) is the pulse intensity, using a global optimization algorithm, which will be explained in more detail in the next section.
2.2. Adaptive loop
Fig. 2. Illustration of the applied voltage on the two layers of the LC-SLM, M0 and M1, as interpolated from the pixels controlled by the adaptive algorithm, indicated by the open circles.
Each pixel of the dual-layer LC-SLM was driven by up to 10V electrical voltage with a 212 level of digitization, but with a usable level between 600 and 4095, due to the nonlinear mapping between voltage and phase. Direct pixel-by-pixel optimization using global optimization algorithms would be too large a space to be efficiently searched. Many previous works, therefore, parameterized the search space using a truncated Taylor series [17, 18, 19, 31]. In this experiment, we instead optimized N c pixels of the SLM, where N c≤N s, which were then interpolated onto N s pixels using the piecewise cubic interpolation method, as illustrated in Fig. 2. During the optimization, the N c controlled pixels can be gradually increased [14]. Since each layer was controlled independently in the optimization procedure, there is no distinction between the phase and amplitude shaping.
We argue that this approach has advantages over parametrization with a truncated Taylor series. Firstly, in terms of optimization, changing the value of a parameter in a Taylor series would completely change the entire profile, while in the case of interpolation, the change would be local. Secondly, in practice, the continuous profile inferred from the Taylor series does not necessarily correspond to the one applied to the SLM, due to the SLM spatial pixellization and driving voltage discretization. Finally, while it is easy to individually calculate the boundaries that must be placed on each Taylor series coefficient due to the physical limitations imparted by the pixellization of the SLM, it is not easy to calculate such limits when many coefficients need to be applied simultaneously.
The problem was formulated as max{f(X)|X}, where X is an integer vector of 1 × 2N c parameters, bounded between 600 and 4096, i.e. each of its component X j∈ [600,4096], where j = 1,…,2N c, and f(X) is the evaluated TPA detector signal from the applied voltage. Note that X is a concatenation of the N c pixel voltages from the two layers of the LC-SLM. The DE algorithm [36] was implemented as our global optimization algorithm in Matlab, which was also used to control the SLM, and to read the TPA detector signal. The DE algorithm was chosen, because it has been shown to consistently outperform simulated annealing or genetic algorithms in most cases [36, 37], which we have confirmed both in many simulations and experiments.
The DE algorithm that was applied to solve this maximization problem can be summarized as follows. At each generation g, a population of N p candidate solutions are maintained, denoted as X i,g, i = 1,…,N p. The initial population consists of random integer vectors, which would then undergo mutation, crossover, and selection to form the next generation. The mutation operator that we implemented used a combination of dither and trigonometric operators [38]. During mutation, three randomly selected individuals from the population, , where r 1,r 2,r 3∈ [1,N p], are used to generate the i th mutated individual V i,g+1, with a condition r 1≠r 2≠r 3≠i, according to the following rule:
where
and
In the above equations, u g,u′ g∈ [0,1] are uniformly distributed real random numbers, drawn every generation, and F is called the scaling factor, which is a control parameter of the DE algorithm.
The mutated individual V i,g+1 may not be a vector of integers, in which case its components will be converted to integers, before being subjected to a crossover operation with the current population member X i,g to form candidates for the next generation population. The crossover operator works on a component by component basis, as follows:
where w∈ [1,N p] is a random uniformly distributed integer, and C r∈ [0,1] is called the crossover rate, another control parameter of the DE algorithm. If W j,i,g+1 goes beyond the boundary, then a random integer will be drawn between the X j,i,g and the boundary. Finally, the members of the next generation population are selected from the current generation and the candidate individuals of the next generation in order to maximize f(X), according to the following rule:
In all of our experiments, we chose F = 0.75, C r = 0.5, and N p = 30. In our experiments, the optimization took an average of 0.145 minutes per generation, which was mostly spent updating the SLM, integrating the detector signal, and on electronic communication with the instruments. Therefore, we did not implement a stopping condition based on convergence criteria, but, instead, we ran the DE algorithm for a specific number of generations. In each optimization, we monitored the diversity among the individuals in the population to check for the convergence.
3. Results and discussions
Fig. 3. Normalized measured spectra at the output of the oscillator, after the first preamplifier and the pulse shaper, and after the second pre-amplifier in logarithmic (a) and linear (b) scale.
Figure 3 shows typical measured spectra at the output of the oscillator, after the first preamplifier and the pulse shaper, and after the second pre-amplifier. It is worth noting, at this point, that the fringes that appear on the oscillator spectrum, are possibly due to the etalon effect. The fringes grow in subsequent amplifier stages, owing to the SPM [39]. These fringes could not be eliminated experimentally. The effect of the non-uniform spectral gain profile with finite width is evident, as the spectral FWHM of the pulse was reduced from 16nm at the output of oscillator, to 12 nm after the first pre-amplifier, and to 11nm after the second-preamplifier. Nevertheless, the 20dB spectral width of 23nm was maintained.
Firstly, the pulses from the compressor were characterized without intentional shaping by the pulse shaper (by not applying any voltage to the SLM). The final amplifier was pumped to produce a train of pulses with an average power of 2.3W prior to the compressor, and then the separation of the grating pair in the compressor was adjusted to maximize the intensity of the TPA detector, and the second harmonic signal from the SHG FROG setup when the pulses from both arms were temporally overlapped. This resulted in a grating separation of 10.2 cm. The output pulses were then characterized at two power levels using the SHG FROG;
at 2.3W and at 12.6W average power, without changing the grating separation. We shall refer to these as the low and high average power, respectively, throughout the remainder of this paper. Figure 4(a) and (b) show the square-root of the measured SHG FROG traces, after interpolation onto a 128 × 128 Fourier grid, for both cases. Plotting the square root of the FROG trace was aimed at emphasizing the detail at low intensity. The spectral and the temporal intensity were then retrieved, as well as the group delay [dϕ (ω)/dω] and the instantaneous frequency [-1/(2π)×dϕ(t)/dt], from these traces, as shown in Fig. 4(c) to (f). The root-mean-square (rms) retrieval errors of these traces were less than 8 × 10-3. The excellent agreement between the retrieved and measured spectra, in Fig. 4(c) and (d), demonstrates the quality of the home-built SHG FROG.
Fig. 4. (a,b) Contour plot of square-root of measured SHG FROG traces, after interpolation onto a 128 × 128 Fourier grid, of the pulses after the compressor without intentional shaping. The contour lines represent levels [0.02,0.06,0.1,0.2,…,1]. (c,d) Retrieved spectral intensity (blue curves), spectral group delay (green curves), and measured spectra (red curves). (e,f) Retrieved temporal intensity (blue curves) and instantaneous frequency (green curves). The figures correspond to measured average powers of 2.3W (a,c,e) and 12.6W (b,d,f) prior to the compressor.
The mainly parabolic profile of group delays shown in Fig. 4(c) and (d), and the side-lobes in the temporal intensity profiles shown in Fig. 4(e) and (f), are a strong indication that the dominant effect on the output pulses of the CPA system was the accumulated third-order dispersion (TOD). These results are expected, since there was no compensation element for the TOD placed in our system. In addition, the departure of the group delay profiles from parabolic at both low and high average power levels [Fig. 4(c,d)] suggests the presence of nonlinear phase-modulation, whose amount increases with power level, as signified by the spectral broadening that accompanied the increase in the average power. This increase in accumulated nonlinear phase modulation causes the side-lobes of the temporal profile in the high power case to not decrease monotonically, as seen in Fig. 4(f).
The accumulated nonlinear-phase modulation evident in the above characterization is expected, since there was no specific stretching of the pulses prior to amplification in our fiber CPA system. The estimated upper limit of the accumulated nonlinear phase was ϕNL=π rad at low power, and ϕNL = 1.6π rad at high power, of which the contribution prior to the pulse shaper was 0.7π rad. In this calculation, an exponential amplification with a constant gain per unit length with flat spectral gain profile in the fiber amplifiers was assumed. The accumulated nonlinear phase, coupled with non-uniform spectral gain profile, caused the large change in spectral intensity after the different amplifier stages. It is worth noting that we observed nonlinear polarization rotation (NPR) in both of our pre-amplifiers. Due to the NPR, environmental changes, particularly of temperature, induce small fluctuations in the output of our system, explaining the discrepancy between the measured spectra after the pulse shaper and the second pre-amplifier in Fig. 7 and 10. However, this fluctuation is a very slow process that happens on a day-to-day timescale, and thus did not affect our experiments.
Having characterized the pulses without intentional shaping, the optimization was then performed using the DE algorithm, as described in the previous section, initially at low average power. For this experiment, every 8th pixels of the SLM was controlled from pixel number 16 to 112, corresponding to where the spectrum of the pulses was located on the SLM. In addition, pixels number 12 and 116 were also controlled to avoid overshoots at the edges of the pulse spectrum after interpolation. Thus, a total of N c = 15 pixels were controlled on each layer of the SLM. The optimization algorithm was run for 350 generations, which took 51 minutes to complete.
Figure 5 shows the evolution of the TPA detector signal, normalized to the case without intentional shaping, evaluated from the applied SLM voltages of the best individual in the population at each generation. Note that since the algorithm was run from a random initial condition, the best individuals at the early stages of the optimization had lower TPA detector signal than the case without intentional shaping. After 350 generations, there was not much diversity among individuals in the population, indicating that the algorithm had converged to a solution, and the TPA signal was improved by a factor of 4.2. The mask corresponding to the best individual in the population was then applied to the SLM, and the pulses were then characterized using the SHG FROG. Figure 6(a) shows square-root of the measured FROG trace, after interpolation onto a 128 × 128 Fourier grid. The retrieved spectral intensity and group delay are shown in Fig. 6(b), while the temporal intensity and instantaneous frequency are shown in Fig. 6(c). The rms retrieval error was less than 2 × 10-3, and the temporal FWHM of the retrieved intensity is 195 fs. The calculated Fourier transform-limited profile is also shown in Fig. 6(c). The excellent agreement between the retrieved and calculated transform-limited profiles down to the -20 dB level, highlights the success of the optimization algorithm. The measured spectra at various points in our system are shown in Fig. 7, before and after optimization, as well as the calculated transmission and group delay applied by the SLM. The optimization yielded a spectral broadening toward shorter wavelengths after the second pre-amplifier, as well as in the final spectrum.
Fig. 5. Two-photon absorption detector signal, normalized to the case of without intentional shaping, and evaluated from the best individual in the population as a function of generation, for average power of 2.3W (blue dots) and 12.6W (red dots). For the low average power case, a constant number of controlled pixels, N c = 15, was used throughout the optimization. For the high average power case, the number of controlled pixels N c were increased from 15 to 51 during the optimization, at positions indicated by the arrows (see text).
Fig. 6. (a) Contour plot of square-root of measured SHG FROG trace, after interpolation onto a 128 × 128 Fourier grid, of the compressed low average power pulses, after the maximization of the TPA signal by controlling every 8th pixels of the SLM (see text). The contour lines represent levels [0.02,0.06,0.1,0.2,…,1]. (b) Retrieved spectral intensity (blue curve), spectral group delay (green curve), and measured spectrum (red curve). (c) Retrieved temporal intensity (blue curve) and instantaneous frequency (green curve), as well as the calculated transform-limited intensity profile (red curve).
Fig. 7. (a) Calculated applied group delay (blue curve) for the low average power pulses overlayed with the negative of the retrieved group delay for the case without intentional shaping [Fig. 4(c)]. The measured shaped spectrum after the shaper (shaded grey) is shown for reference. (b) Calculated transmission (black curve) of the SLM after maximizing the TPA detector signal. Normalized measured spectra before (shaded) and after (curves) optimization after the pulse shaper, after the second pre-amplifier, and after the compressor are shown, as indicated by the labels.
The system was then operated at its highest power, generating 12.6W average power prior to the compressor, without changing the grating pair separation. The optimization algorithm was run with the same parameters as in the previous case for 350 generations, and the pulses were then characterized. Figure 8 shows the measured SHG FROG trace and the retrieved temporal and spectral data. The rms retrieval error was less than 4 × 10-3. The temporal FWHM of the retrieved profile is 200 fs. It can be easily seen from Fig. 8(c) that the pulses exhibit some pedestal on their trailing edge. Furthermore, the pulse has a lower intensity and a longer duration compared to the calculated Fourier transform-limited profile. In an attempt to improve the temporal intensity profile and to reach the Fourier transform-limit, the optimization algorithm was extended to 450 generations and the control parameters of the optimization algorithm (F and C r) were varied, but this was not successful. Therefore, this inability to fully compress the pulse did not stem from shortcomings of the optimization algorithm.
Fig. 8. Same as Fig. 6, but for high average power, after the maximization of the TPA signal by controlling every 8th pixels of the SLM (see text). Note that the temporal intensity in (c) is shown in linear scale.
Finer control over the pulse shaping was then attempted by increasing the number of controlled pixels, from every 8th pixel, to every 4th pixel at generation 51, and finally to every 2nd pixel at generation 101, between pixels 16 to 112 of the SLM. In addition, pixels number 12 and 116 were still controlled, resulting in a final total of N c = 51 controlled pixels on each layer of the SLM. The DE algorithm was run with this condition for 450 generations, taking 65 minutes to complete. The evolution of the TPA signal evaluated from the best individual in the population, normalized to the case without intentional shaping, at each generation is shown in Fig. 5 (red dots).
After the optimization, there was little diversity among the members of the population, and the TPA signal was improved by a factor of 4.3. As in previous experiments, the pulses resulting from the optimization were characterized by the SHG FROG as shown in Fig. 9. The rms retrieval error was less than 1.5 × 10-2, which was mainly due to the large area the trace occupies on the Fourier grid. The temporal FWHM of the retrieved profile is 170 fs. Although the square-root of the measured SHG FROG trace exhibits wings that extend up to 2 ps delay, they have very low intensity, below 0.5%, and most of the trace mass is concentrated at the center, unlike the case of Fig. 8(a), for which case only every 8th pixel was controlled by the DE algorithm. Therefore, the temporal profile shows a high quality pulse with the main pulse having an excellent agreement with the calculated Fourier transform-limited profile. The pedestal of the retrieved temporal intensity is less than -20dB everywhere, except for the small satellite pulse at t ≃-1.5 ps. The measured spectra at various points in our system are shown in Fig. 10, before and after the optimization, as well as the calculated transmission and group delay applied by the SLM. As in the lower average power case, the optimization also yielded a spectral broadening toward shorter wavelength after the second pre-amplifier and in the final spectrum. In contrast to the low average power case, the use of more control points causes the the pulse spectra to exhibit more oscillations after the nonlinear propagation through the second pre-amplifier and the final amplifier.
Fig. 9. Same as Fig. 6, but for high average power, after the maximization of the TPA signal with increasing numbers controlled pixels of the SLM (see text).
Although the DE algorithm started with random initial candidate solutions, the optimization results had a high reproducibility. The resulting applied phase and transmission profiles showed little dependence on the initial condition, yielding similar compressed pulse profiles in each case, which implies that our results have to be in the vicinity of the global optimum. In fact, there is a consistency in the optimized spectra after the shaper in both low and high power cases, as shown in Fig. 7 and 10, i.e. suppression of the part of the pulse spectra between 1035 and 1040 nm. In order to obtain the true global optimum, more generations would be required, but with diminishing returns that may not justify the effort. It is important to note that it is not necessary to start from random initial candidate solutions. It is possible to feed a previously optimized data as one of the initial candidate solutions in order to reduce the optimization time.
In interpreting the results of our experiments, it is worth remembering that the pulses were amplified in three stages, using two different fibers having their own gain profile, nonlinearity, and dispersion. We believe that this is the main reason that the optimization did not yield group delays that strictly compensated for the group delay of the unshaped pulses in both low and high average power case (see Fig. 7 and 10), apart from the fact that the pulse shaper optical setup (without the SLM) may impart a non-zero dispersion. Furthermore, the algorithm found the necessity to shape the spectrum of the pulses prior to amplifications because of the interplay between gain shaping and spectral broadening due to nonlinear-phase modulation in the amplifiers. Therefore, phase-only modulation only, such as in our previous experiment [31], might not be sufficient to achieve the pulse fidelity that we have presented in this work.
Fig. 10. Same as Fig. 7, but for the case of high average power, after the maximization of the TPA signal with increasing controlled pixels of the SLM [Fig. 9].
Since the pulse propagation was nonlinear, it follows directly that the search landscape was highly nonlinear as well. However, it is evident that this did not pose a problem to our optimization algorithm. In fact, it highlights its superior performance. In addition, since the absolute phase does not matter, and thus one can apply similar phase profiles with a constant offset, one may argue that the search landscape was highly multi-modal, i.e. having multiple optima. However, this is not strictly true experimentally, since the pulse shaper exhibits a nonlinear mapping between voltage and phase, with a limited range, as well as having discrete phase values. All of these factors reduce the modality of the problem. This has the benefit of reducing the time required to search for the global optimum.
In order to achieve high fidelity pulses, maximization of a TPA detector signal or, similarly, a SHG signal, is the simplest objective function to be implemented. Note that the maximization of TPA signal does not guarantee the generation of a specific spectral shape at the output. The approach in Ref. [30] to produce a parabolic shaped spectrum after nonlinear amplification via an amplitude-only pre-shaping in order to obtain high fidelity pulses after compression can work to a certain extent. However, as the authors themselves have pointed out, this method requires more effort to compensate for accumulated higher-order phase from the material dispersion. Furthermore, when a more complex setup is involved, comprising multiple amplifier stages such as in our case, this approach is less likely to be successful.
4. Conclusion
In conclusion, we have successfully demonstrated an adaptive amplitude and phase pre-shaping technique for producing high-fidelity femtosecond pulses in a fiber CPA system. We have demonstrated that this technique is very robust, effective, and efficient, in compensating for both accumulated phase from higher-order material dispersion and nonlinear phase-modulation. We did not have to perform painstaking characterization and design to optimize our fiber CPA system to produce the high-fidelity femtosecond pulses presented here. We found that, with increasing nonlinearity, a finer control over the pulse shaping was necessary to achieve high fidelity pulses upon compression. This technique should enable power-scaling to higher energy and/or average power at various repetition rates. In addition to producing high-fidelity compressed pulses, this technique has the potential to produce arbitrary shaped pulses necessary in various applications, including coherent control [40].
Acknowledgment
This work was supported by EPSRC Instrument Grant EP/C009479/1. J.H.V. Price is supported by a Royal Academy of Engineering/EPSRC research fellowship.
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