It is possible to modify pulse propagation and nonlinear interactions in microstructured fibers using phase-tailored ultrashort laser pulses. We experimentally investigate how pre-shaping of the input laser pulse can be used to alter its evolution and subsequent output characteristics. We also demonstrate how adaptive pulse shaping can be used to control the output properties of the pulse spectrum. Numerical simulations based on the nonlinear Schrodinger equation predict the output spectral profiles of the propagated pulse in good agreement with experimental results, and elucidate the relevant processes producing the optimal output.
©2004 Optical Society of America
The ability to tailor the refractive index and dispersive properties of optical fibers has led to a renaissance in nonlinear optics. By virtue of their small core size, low dispersion in the near infrared spectral region, as well as their unique waveguiding properties over wide bandwidths, photonic crystal fibers allow for the propagation of ultrashort (sub-100 fs) laser pulses to intensities of 1012 W/cm2 over meter lengths using modest pulse energies . This has led to demonstrations of a number of nonlinear optical effects in photonic crystal fibers, including four wave mixing , soliton squeezing , third harmonic generation , and pulse breaking . To date, supercontinuum (SC) generation has been one of the most spectacular manifestations of nonlinear pulse propagation in microstructured fibers . The ability to generate SC directly from femtosecond laser oscillators has led to technologically important applications in the field of precision metrology [7, 8] and optical coherence tomography .
Nonlinear interactions in microstructured fibers, specifically supercontinuum generation, has been investigated both experimentally and theoretically by a number of groups [10, 11, 12, 13]. What has not yet been substantially investigated is the effect the input pulse envelope has on controlling the propagation dynamics or altering the output spectral and temporal characteristics. Control of nonlinear optical processes may at first glance seem difficult, since a nonlinear process (by definition) reacts nonlinearly in response to changes in the input fields. However, when viewed from the standpoint of an initial value problem, it is clear that the input parameters of the pulse as well as the properties of the fiber strongly influence the subsequent evolution of the pulse and nonlinear dynamics. There exist a number of ‘control knobs’ that can be used to influence the nonlinear processes of an optical pulse, such as input power, center wavelength, fiber length, pulse duration, and polarization. The only study existing to our knowledge was reported by Apolonski, et al., where the effects of pulse duration, pulse chirp, and polarization were examined as mechanisms for influencing the shape of the output SC .
Fourier domain pulse shaping  has emerged over the past decade as a powerful tool for exercising control over nonlinear optical processes. When coupled with adaptive or learning algorithms , pulse shaping methods have proven to be very effective in producing a specific ‘target’ nonlinear output state. Specific examples include the enhancement of specific peaks  and combined enhancement and tuning  of the spectrum of high order harmonics, as well as nonlinear broadening  and soliton self-frequency shifting in conventional fibers . In this paper, we investigate for the first time how femtosecond pulse shaping can be used to control the evolution of the temporal and spectral structure of optical pulses propagating in microstructured fibers. In particular, we examine how both intuitive and non-intuitive pulse shapes can influence the shape of the output pulse spectrum. By intuitive pulse shapes, we mean pulses whose temporal envelope can be determined by consideration of the underlying physics. Non-intuitive pulses are those whose time-dependent intensity form can be determined using adaptive pulse shaping methods.
The paper is organized as follows. In Section 2, we describe the experimental details and measurements as well as the numerical modelling methods. In Section 3, we present results on the influence of pre-shaping the temporal envelope on the output spectral continuum. In particular, we examine the what role cubic spectral phase has in modifying the shape of the spectrum and examine more sophisticated temporal envelopes for reducing spectral modulation and asymmetries due to self-steepening. In Section 4, we present our results on adaptive pulse shaping, where we investigate how learning algorithms can be used to increase spectral bandwidth and precisely control the soliton self-frequency shift. We conclude in Section 5.
2. Experimental and theoretical methods
For most of the results presented here, we use a 82 MHz, 30 fs Ti:sapphire laser producing 3 nJ pulses centered at 800 nm. The output pulse train is shaped using an all-reflective 4f pulse shaper consisting of a 128 element programmable liquid crystal spatial light modulator (SLM) which alters the spectral phase of the pulse. The transmission through the shaper is approximately 40% upon leaving the shaper, and the pulse is coupled into a piece of microstructured fiber (MSF) using a 100X objective (Vickers w4017) with a numerical aperture of 1.3. The fiber used in these studies is a honeycomb microstructured fiber consisting of a 1.7 μm diameter silica core surrounded by an array of 1.3 μm diameter air holes in a hexagonal lattice. The zero dispersion point is estimated to be 760 nm and the coupling efficiency is estimated to be approximately 15%. The dispersion introduced by the microscope objective was measured separately and compensated for in our experiments. The optical spectrum is recorded at the fiber output using a 0.25 m spectrometer with a CCD detector. For the more complex pulses used in our intuitive control experiments, a Gerchberg-Sexton algorithm was used to synthesize the phase filter . SHG FROG was used to characterize the shaped pulses before input to the fiber.
For our adaptive control experiments, a genetic algorithm (GA) was embedded in a Lab-view program which simultaneously controlled the pixels of the shaper and read out the spectra from the CCD. The GA in the ‘learning loop’ uses evolutionary paradigms (crossover, mutation, survival of the fittest) to search a variety of spectral phase configurations to produce the desired output spectrum from the fiber. Different parameterizations were used for different control goals. Each spectral phase was represented by an n-element bit (i.e., the spectral filter that determines the pulse phase). Each pulse was then tested by imparting spectral phase on the driving pulse via the SLM and reading the output spectrum from the CCD. The fitness function was chosen to minimize the difference between the point-by-point difference between the actual and the target spectrum. For each optimization, between 128 and 180 individuals were tested the first generation. A mutation rate of 5% was used in all runs. Following application of the genetic operators, a new generation was tested, and this process continued until the algorithm ceased to improve.
To model our experiments, we used an extended nonlinear Schrodinger equation (NLSE) to compute the propagation of shaped pulses in the photonic crystal fiber . The NLSE has been shown to be valid for femtosecond pulses in the limit where the optical bandwidth approaches the center frequency of the pulse [22, 23]:
Here A(z,t) is the slowly varying envelope function of the pulse:
where φ(z,t) is the temporal phase of the pulse, α is the linear absorption coefficient (taken to be 0 since the fiber loss is small over distances used in this experiment), βi are the Taylor series coefficients of the mode propagation constant about the center frequency ω 0, γ is the nonlinearity coefficient, and R(t) is the Raman response function:
where fr = 0.15 represents the fractional contribution of the delayed Raman response and the Raman response function hr (t) for fused silica is approximated analytically as
The time constants τ 1 and τ 2 are chosen to be 12.2 fs and 32 fs respectively.
A standard split-step Fourier algorithm was used to perform all of the simulations. The time resolution is chosen as 0.5 fs, and the sample number is 214 = 16384, which leads to a frequency resolution of 111.1 GHz. The propagation step size is 0.5 mm. The temporal phase of the input pulse φ(0,t) was specified from the Fourier transform of the pulse in the frequency domain. For the fiber dispersion parameters, we used β 2, β 3 and β 4 taken from the literature . The inclusion of higher order fiber dispersion did not significantly change the results of our simulations.
3. Influence of pulse shape on spectral broadening and uniformity
We first consider how intuitive pulse shaping can be used to control the spectral output of a MSF. From Eq. (1), it is evident that the shape of the initial temporal envelope influences the evolution of the pulse as it propagates. The shape is determined by the input phase of the pulse, varphi(z = 0,t). In addition, we investigate how the interplay between the pulse spectral phase and the fiber dispersion influences the propagation.
3.1. Influence of cubic spectral phase on propagation dynamics
First, we examine the influence of cubic spectral phase on the output spectrum of pulses with frequencies near the the zero dispersion wavelength of the MSF. It is known that finite bandwidth pulses centered at the zero dispersion wavelength experience both normal and anamolous dispersion . Near the zero dispersion point of the MSF where L 3D ≈ L 2D, the intrinsic third order dispersion β 3 (TOD) therefore plays an important role in determining the dispersive propagation and the interplay between dispersion and self-phase modulation (SPM). Here, the dispersion length is LND = /|βN | where N is the order of dispersion. In particular, the sign of the input cubic phase, Φ(3) ≡ d 3 ϕ/dω 3 can enhance SPM by compressing the pulse during propagation (if the sign of Φ(3) and β 3 are different) or suppress SPM by acting with the fiber TOD dispersion to broaden the pulse (if Φ(3) and β 3 have the same signs).
Figure 1 plots the spectrum as a function of cubic phase after 5 cm and 70 cm of propagation through the MSF, respectively. A 50 fs nm pulse centered at 770 nm was used in these experiments. For pulse propagation in the 5 cm fiber, we observe an asymmetry in the spectrum as the sign of Φ(3) changes (Fig. 1(a) inset), with negative values of Φ(3) yielding a broader spectrum. This is quantitatively illustrated in Fig. 1(a); the broadest spectrum occurs at Φ(3) = - 500fs 3, and the a broader bandwidth is produced at all - Φ(3). Over longer distances (Fig. 1(b)), this asymmetry disappears, although the 1050 nm soliton achieves a broader bandwidth and slightly larger self frequency shift for nonzero |Φ(3)|.
These results are consistent with a positive TOD for the MSF at this wavelength, and show that the negative cubic phase of the pulse can be used to counteract the intrinsic TOD and produce a larger bandwidth through enhanced self-phase modulation during the initial stages of propagation. Quantitatively, a complete compensation of TOD and thus the broadest spectrum would occur for a |β 3| ≃ |Φ(3)|/Lfiber . However, since we are not operating at the zero dispersion point, the pulse experiences some broadening from GVD. Nevertheless, we find that for Lfiber = 5 cm, this corresponds to a |β 3|= 0.01ps 3/km, in reasonable agreement with literature values . For longer fibers, both proportionally more Φ(3) is needed and the GVD broadening will play a larger role, thus the effects of pulse cubic phase in this case should be minimal, as seen in the experiment. We note that at the true zero dispersion point the pulse cubic phase should play a much greater role.
3.2. Control of self-steepening
As another example of how pulse shape can be used to influence propagation in MSFs, consider self-steepening, whereby the high power peak of an ultrashort pulse experiences a temporal delay relative to the lower intensity trailing edge, leading to a steepening of the of the pulse followed by the formation of an optical shock front. An asymmetric blue shift of the spectral weight from self phase modulation accompanies the steepening. The short nonlinear lengths afforded by MSFs make them an ideal environment for investigating self-steepening.
The parameter governing self-steepening is s ≡ 2/ω 0 Tp , where Tp is the input pulse width. Significant spectral asymmetries can be observed over distances 0.01LNL /s. Here LNL = 1/γP 0 is the nonlinear length. For 30 fs, 3 nJ pulses, this distance is roughly 15 cm. As shown in Fig. 2(a), a transform-limited (T-L) pulse (black curve) experiences severe self-steepening over 70 cm, as evidenced by the large blue-shift of the spectrum. The simulated spectrum (Fig. 2(b) black curve) shows reasonable agreement, with slightly asymmetric broadening toward the blue.
An intuitively simple way to suppress self-steepening is to alter the temporal profile of the input pulse by placing the ‘steepness’ on the leading edge of the pulse. Such a forward ‘ramp’ pulse is shown in the inset of Fig. 2(a), which displays intensity and phase profile of a ramp pulse synthesized using phase-only pulse shaping using a Gerchberg-Sexton algorithm . The output spectrum corresponding from the MSF to this pulse is shown as the red curve in Fig. 2(a). For comparative purposes, P 0 was chosen to be the same as the T-L pulse. In contrast to the T-L pulse, the output spectrum is smoother and shows less broadening toward the blue side. This is in good agreement with the NLSE-simulated spectrum, which shows a comparably broadened spectrum and a modulation depth of the read and blue-shifted components nearly identical to that observed in the experiment. It could be argued that since the ramp pulse is 3–4 times longer in duration with respect to the T-L pulse, the reduction in self-steepening comes from the increased self-steepening length. However, if the polarity of the ramp is reversed (backward ramp) with respect to time, i.e., if the steep edge is placed on the trailing edge of the pulse, the output spectrum exhibits a similar character to that of the TL pulse, with deep spectral modulation and asymmetric broadening toward the blue (shown as the green curve in Fig. 2(a)). This suggests that the actual shape of the pulse rather than the increased duration is responsible for effect. Our simulations of the temporal output of the forward and backward ramp pulses corroborate this conclusion: the temporal structure of the forward ramp pulse shows a smoother envelope and trailing edge when compared with the backward ramp pulse. Our simulations also reveal that the self-steepening can be corrected for different peak powers.
4. Adaptive control of nonlinear interactions in microstructured fibers
We now investigate how adaptive pulse shaping can be applied to manipulate nonlinear interactions in MSFs. Unlike intuitively sculpted pulses, adaptive shaping does not rely on a knowledge of the underlying nonlinear optical process, but proceeds by efficiently searching a large parameter space to design the optimal input pulse shape to achieve the desired outcome.
4.1. Enhancement of spectral broadening via adaptive pulse shaping
As a first test, we examine how adaptive pulse shaping can be used to increase the bandwidth of supercontinuum from MSFs. In this experiment, Taylor series coefficients of the pulse spectral phase (up to the 5th order) were selected for optimization using the GA, with 7-bit (27 = 128) parameterization of each coefficient in the Taylor series expansion. A 2 nJ pulse was coupled into a 70 cm long MSF. The cost function was computed at the integration of the spectral intensity in the wavelength bands 500–570 nm and 950–1020 nm:
The initial generation contained a near T-L pulse, thus insuring that the optimization is non-trivial, i.e., that any change to the input pulse shape producing a broader spectrum is more complicated than a T-L pulse. Fig. 3(a) plots the progression of the cost function during the optimization. During the course of the optimization, the spectral density in the target spectral ranges of the supercontinuum has been increased as evidenced by a doubling of the cost function during the run. After approximately 20 generations, the optimization begins to stagnate. The evolution of the spectrum is shown in Fig. 3(b).
The optimization produces a 10% increase in bandwidth and factor of two increase in integrated spectral intensity in the optimization bands selected in the cost function. It is important to keep in mind that this enhancement is with respect to a T-L pulse, and thus represents a true enhancement of the nonlinear interactions in the MSF caused by the pulse shape. An examination of the evolution of the spectral phase during the optimization (not shown) suggests that the genetic algorithm compensates all orders of higher order dispersion during optimization. The phase evolves visibly during the first two generations, as adaptive learning algorithm compensates residual 2nd order dispersion at the beginning of the run. Further in the optimization process, we observe only slight changes in the spectral phase, evidencing that higher order dispersion is being compensated. As noted above, cubic compensation will work to compensate shorter fiber lengths. In this case, all orders of the dispersion are needed to produce the optimal result.
4.2. Adaptive control of soliton self-frequency shifting
Self-frequency shifting due to Raman scattering plays an important role when soliton durations are less than a few picoseconds. Self-frequency shifts (SFS) following soliton fission also plays a role in the initial stages of the formation of supercontinuum in MSFs . Recent work by Efimov, et al. has shown that it is possible to stabilize soliton propagation in conventional fibers using adaptive pulse shaping . Here, we consider how adaptive pulse shaping can be used to control the magnitude of SFS and duration of the shifted soliton using adaptive pulse shaping. Unlike the work of Efimov, et al., we are not suppressing the SFS but rather interested in generating the soliton SFS with different center frequency and bandwidth in the microstructured fiber.
In these experiments, we parameterized the individual soliton in the spectral domain as a Gaussian function specified by its center wavelength and width. The fitness function was then specified as:
The σ terms in each denominator determined the fitting tolerance to a target peak; the W coefficients provided weights for each of the target criteria. The spectral phase was parameterized such that each of the 128 elements was allowed to vary in the range 0-2π with 8-bit resolution.
Figures 4(a) and (b) show the optimized spectra corresponding to different target soliton wavelengths and widths, respectively. Figure 4(a) demonstrates that the GA is able to synthesize a pulse that correctly selects a specified central wavelength while keeping the spectral width of the soliton fixed at 15 nm. Three different wavelengths are found, each one corresponding to the target wavelength to within 1 nm. Thus the achievement in these experiments was essentially 100 %. In addition, a phase-matched non-solitonic component is seen to the short wavelength side of the spectrum. The data in Fig. 4(b) present the results of controlling the width of the soliton at a fixed wavelength. Again, the learning algorithm is able to synthesize a specific input pulse that achieves the a specific goal, although in the case of widest bandwidth (54 nm corresponding to the green curve), the spectral signature indicates that a higher order soliton is undergoing a fission process into 2 fundamental solitons. To investigate the λtarget ,Δλtarget parameter space in which adaptive pulse shaping is successful, we attempted to produce solitons at a variety of wavelengths and bandwidths and found it possible to produce continuously tunable fundamental solitons in the 900 – 950 nm range with bandwidths of 10 – 40 nm (corresponding to soliton durations of 22 – 90 fs. Outside these ranges, the GA stagnated and was unable to reach the specified target state.
In order to gain insight into why the optimization works, it is instructive to examine the nature of the input pulse. The optimizing pulse producing the 16 nm wide, 925 nm soliton is shown in Fig. 5. The spectral phase that produces the optimized soliton (blue curve, Fig. 5(a)) is quite complex, as evidenced by the lack of smoothness and series of discrete phase jumps from pixel to pixel. The temporal structure of the pulse is shown in Fig. 5 (b) and resembles a ‘noise burst’, characterized by an ultrashort coherent feature embedded in a longer series of lower intensity pulses with random intensity and temporal spacing. Nonetheless, an NLSE simulation of the optimization using the experimentally optimized phase and amplitude from this pulse faithfully reproduces the experimental result, as shown in Fig. 6, although the simulated soliton shifted slightly to the blue. Figure 7 displays a propagation simulation of the temporal and spectral evolution of the soliton in the co-moving frame of the pulse it propagates down the MSF. After approximately 20 cm, the spectrum of the pulse begins to show intensity modulation, and the coherent feature in the temporal domain also shows interference on the leading edge. After 35 cm, the soliton has split off from the spectrum and by 55 cm has almost reached its shifted spectral position. In the time domain, the fission of the soliton is accompanied by a reduction in intensity and temporal broadening of the central feature and coherent beating between the soliton and the residual non-solitonic pulse.
Even though the optimizing spectral phase appears to be random, the magnitude of the SFS and the soliton width depend sensitively on its exact form. Both simulations and experiments reveal the soliton SFS differs by 30 nm from the target value when only 10% of the spectral phase pixels are altered by π. In addition, it is known that changing the input pulse peak power controls the magnitude of the SFS. The nature of the input pulse, the sensitivity to the phase structure, and the peak power dependence of the magnitude of the SFS suggest that adaptive pulse shaping serves two roles in this experiment. First, the phase-only shaper acts as a very sensitive amplitude filter, controlling the absolute SFS by selecting the appropriate peak power of the active ‘noise burst’ component of the input pulse. This happens both through the ‘scattering’ of pulse energy from the main peak into the wings of the pulse (inset of Fig. 5(b)) as well as from the direct amplitude modulation of the pulse due to diffractive scattering off the pixels of the shaper (approximately 25 % of the incident pulse energy). We have performed NLSE simulations which corroborate this aspect of the optimization. By adjusting the peak power of the incident pulse to correspond to the experimental power, we are able to produce an SFS in reasonable agreement with the experiment. However, the algorithm also produces the correct soliton duration. To accomplish this, coherent component of the noise burst with the correct bandwidth and spectral phase such that the soliton fissions from the main pulse to achieve the correct soliton temporal width. This demonstrates that the optimization is nontrivial.
It is interesting to note that by comparing our SFS control experimental results with NLSE simulations, we are able to independently determine the dispersive properties of the MFS. By varying β 3,4 in our simulations, we examine how the simulated SFS and soliton width vary. We find that the best agreement with the experimental results occur for β 2 = -4.579ps 2/km, β 3 = -0.01725ps 3/km and β 4 = 0.0003178ps 4/km. These values are in agreement with values reported for MSFs.
We have performed a series of experiments and simulations investigating how the input pulse shape influences the nonlinear frequency generation process in a microstructured photonic crystal fiber. Experiments pre-shaping the pulse demonstrate that role that cubic spectral phase plays in the initial formation of continuum. Self-steepening can be minimized using pulses which possess asymmetric profiles. Using adaptive pulse shaping, we have shown that the spectral wings of the supercontinuum can be enhanced by optimizing the spectral phase of the pulse. In addition, it is also possible to control the formation of solitons and magnitude of the self-frequency shift using adaptive shaping.
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