Interest in optical pulse propagation in water has been renewed due to recent experiments demonstrating two orders of magnitude increased transmission while propagating at near-IR wavelengths [1
1. A. E. Fox and U. Österberg, “Observation of non-exponential absorption of ultra-fast pulses in water,” Opt. Express 14, 3688–3693 (2006). [CrossRef] [PubMed]
]. Several groups have subsequently investigated optical pulse propagation in water [2
2. J. Li, D. R. Alexander, H. Zhang, U. Parali, D. W. Doerr, J. C. Bruce III, and H. Wang, “Propagation of ultrashort laser pulses through water,” Opt. Express 15, 1939–1945 (2007). [CrossRef] [PubMed]
5. A. V. Sokolov, L. M. Naveira, M. P. Poudel, J. Strohaber, C. S. Trendafilova, W. C. Buck, J. Wang, B. D. Strycker, C. Wang, H. Schuessler, A. Kolomenskii, and G. W. Kattawar, “Propagation of ultrashort laser pulses in water: linear absorption and onset of nonlinear spectral transformation,” Appl. Opt. 49, 513–519 (2010). [CrossRef] [PubMed]
], but were not able to repeat the transmission increase in the linear propagation regime. Many groups, however, chose laser parameters that were inconsistent with that of other researchers.
In this paper, we use an optical pulse shaper to mold both the amplitude and phase of optical pulses in the linear propagation regime. In this way, many previous propagation experiments can be replicated by appropriate tuning of our laser parameters – their experimental parameters are potential solutions within our search space. Given such control, we use an evolutionary algorithm to search the space of all possible pulses (within our system’s constraints) for those pulse shapes that maximize propagation through our water setup.
The EA requires experimental results for each potential solution that it creates during the optimization process. Thus, we must simply automate the trial-and-error process of the EA by coupling the physical experiment with the algorithm. Once this is accomplished the optimization can be left to run with no human intervention. The EA will generate new potential solutions based on previous solutions, eventually converging to an optimum. We note that EAs have previously been used in combination with optical experiments in molecular electronic population transfer [6
6. C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280, 151–158 (1997). [CrossRef]
], in improving second harmonic generation [7
7. T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000).
], and in subwavelength control of nano-optical fields [8
8. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. Garca De Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007). [CrossRef] [PubMed]
In our experiment we seek to increase the transmission efficiency (ratio of output to input optical power, Pout /Pin) when propagating pulses through a 2.9-m water cell. An optical pulse shaper controls the spectral amplitude and phase in 128 wavelength bins of a spatial light modulator. The 100 possible amplitude values and 120 possible phase values for each wavelength yields a search space of (100 × 120)128 potential solutions! Nevertheless, by experimentally measuring the ratio of input and output optical powers for a given pulse shape, we can use an EA to modify the pulse’s amplitude and phase and efficiently search for those parameters that maximize the optical pulse transmission. Although the search space is large, the EA converges (unattended) upon a solution within a few hours.
The experimental setup consists of a mode-locked Ti:sapphire laser, an optical pulse shaper and an optical water cell, as shown in Fig. 1
. Absorptive neutral density filters are used to adjust the optical power after the pulse shaper, and this is followed by a spatial filter before the optical water cell. The water cell is constructed of a glass cylinder 8”×14” in dimension and fitted with two 11-mm thick BK-7 end caps both with Ag mirrors and protective SiO coatings. The diameter of one mirror is 6” concentric with the glass window to provide a toroidal clear aperture for the beam’s entrance and exit from the water cell. This water cell design allows the optical path length to be varied as a function of input angle into the cell. In these experiments, a 4.6° input angle provides a 287±0.1-cm optical pathlength involving four passes through the water cell (three and four reflections off the front and back mirrors respectively). The cell is filled with 18+MΩ water, achieved through processing with several filters (including charcoal, porous, and reverse osmosis filtration).
Fig. 1 Experimental water propagation setup using a femtosecond laser followed by an optical pulse shaper, neutral density (ND) filters, spatial filter (SF), 4-pass water cell, and two photodiodes measuring laser reference power (Pref) and transmitted power (Pout) which are fed into an evolutionary algorithm (EA) to control the pulse shaper.
The laser’s bandwidth is 60 nm and by using the pulse shaper is capable of producing pulses as short as 12 fs, as measured by BioPhotonic Solution’s MIIPS system. The maximum width is set by a combination of spectral amplitude and phase, which are controlled by the spatial light modulator with values of ~1–100 percent in amplitude and ±2π phase per spectral element (~1.6-nm bandwidth). The light transmission efficiency through the water is measured by the ratio of two Ophir PD-300 photodetectors: the first monitors the output of the water cell; the second monitors the input power via a reflection from an ND filter after the pulse shaper.
Our experimental goal is to find the optical pulse that experiences the largest transmission efficiency through the 287-cm water path. Toward this end we define our objective function as the transmission efficiency and seek the input pulse shape that maximizes this function. An EA is useful for such an optimization because we have little knowledge of how the solution space is structured and the existence of local optima in such a space could quickly lead a gradient-based optimizer to a sub-optimal solution. EAs, however, are designed to avoid sub-optimal convergence and are more likely (but not guaranteed) to find the global optimum within a given parameter region. In addition, EAs are direct search optimization routines; that is to say, they require only direct calls to the objective function, in this case, the experiment itself. Conversely, gradient-based methods require derivative information, which is infeasible in this context because numerically calculating the gradient at each step of the algorithm would require, at a minimum, two calls to the objective function for each of the 256 parameters.
Briefly, evolutionary algorithms are a class of optimization procedure that employ a population of randomly generated solutions and principles inspired by Darwinian evolution to efficiently search a parameter space for optimal solutions. The fitness (if maximizing) or cost (if minimizing) of each solution in the population is measured against an objective function and those solutions that best meet the objective are selected to “breed” and pass their characteristics on to the next generation of solutions. Over many generations the population will converge to an optimum solution. The stochastic nature of the search and the presence of many solutions in the population helps the algorithm efficiently search the space while avoiding convergence to locally optimal solutions.
In contrast to genetic algorithms which require binary encoding of solutions, we use differential evolution (DE), a type of evolutionary algorithm designed to work with real-valued vector solutions [9
9. R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optimization 11, 341–359 (1997). [CrossRef]
]. The algorithm is easily implemented and can efficiently search non-smooth parameter spaces with many local optima. Each solution is encoded as a vector with 256 phase and amplitude values which control the pulse shaper. A population of 30 solution vectors is randomly initialized and the optical transmission ratio produced by each solution is measured. Once the population is characterized, the next generation of solution vectors is created using the DE algorithm and the process is repeated until the algorithm fails to find improved solutions over many generations (on the order of 200–300 generations). We leave a description of the algorithm to [9
9. R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optimization 11, 341–359 (1997). [CrossRef]
] and use their scale factor and crossover constant of F
= 0.9 and CR
= 0.5, respectively. The choice of population size is dictated by a trade-off between how long the algorithm will run and whether it will prematurely converge to a local optimum. There are no good theoretical prescriptions for population size but empirical studies suggest populations on the order of 20–50. If faster convergence is desired, populations on the order of 10 will usually lead to acceptable improvement. We find that a population size of 30 is a good compromise between the speed of the search and robustness to premature convergence to local optima. Ultimately, the algorithm is not especially sensitive to perturbations in the crossover constant, scale factor, or population size.
Given these parameters, the experimental system is capable of processing about 100 generations per hour. The laser power is kept low (< 3 mW) to avoid both nonlinear optical processes and thermal lensing. With our 4-mm diameter beam (1/e
2) thermal lensing over this length is noticeable at 10-mW average power and by 50-mW changes the spot size by more than 2 mm after 2.87-m water propagation. The starting spectrum is initiated with random values for the 128 amplitudes and 128 phases and within 60 minutes the evolutionary algorithm achieves a transmitted efficiency 40% greater than with the pulse shaper turned off. Ultimately the solution converges to a transmission efficiency 1.8 times that without pulse shaping.
The experiment was repeated several times on different days, each time producing similar results and converging to a maximum solution within 500–800 generations. The range of maximum fitness values (transmission efficiencies) for our different optimization runs were all within 5% of the best discovered solution. The similarity of the converged spectra from the repeated optimization runs provides evidence that we are near the global optimum, but there is still a possibility that a better solution exists. Absent an analytical proof, however, it is unlikely that any other method short of an exhaustive search could find a better pulse shape.
To help quantify the results, our experiments constrained either (1) amplitude, (2) phase, or (3) neither. In the amplitude-constrained case, the spectral amplitudes of all 128 elements were set to 100%, and the 128 spectral phases were varied by the EA. Similarly, in the phase-constrained case, all 128 spectral phases were set to −6.1 radians and the 128 spectral amplitudes were varied by the EA. In the third case, both amplitude and phase were allowed to vary. Finally, the original and shaped optical pulses were characterized in the time-frequency domain using Mesa Photonic’s FROG Scan.
3. Results and discussion
The shaped laser spectra evolve over the first few hundred generations from random spectral amplitudes and phases toward a windowed spectrum near 800 nm. The fitness of a given pulse is the ratio of optical power exiting the water cell to the optical power reference before the water cell (but after the pulse shaper). A typical evolution of fitness as a function of generation for these experiments is shown in Fig. 2
, with both the maximum and average fitness of the population plotted for each generation. As the EA converges toward a solution the maximum fitness eventually stops increasing and the population’s average fitness approaches the maximum fitness.
Fig. 2 Maximum and average population fitness versus generation for a typical optimization experiment with no constraints on the amplitude or phase.
The spectra corresponding to maximum fitness for a few generations are plotted in Fig. 3
to demonstrate the trend for the amplitude shaping occurring as the EA converges towards a solution. Here the initial random spectra evolve into narrow spectral solutions near 800 nm. As a reference, the final spectrum beyond the last generation shows the laser with the pulse shaper disabled (labeled SLMoff
in Fig. 3
Fig. 3 Waterfall plot of shaped laser spectra as a function of generation in the EA. The spectrum labeled SLMoff is that of the laser with the pulse shaper turned off.
The spectral weights applied to the pulse shaper for the best discovered experimental solution are shown in Fig. 4
. In this experiment, the majority of the laser’s spectral power is in the 730- to 860-nm region. The EA, however, converges to a solution with a narrower spectral window near 800 nm. The remaining coefficients within the 730- to 860-nm spectral band are set to zero. In the long-wavelength region of the laser’s bandwidth (> 860 nm) the spectral weights are random since they are not relevant in maximizing efficiency. However, in the short-wavelength region of the laser’s spectrum (< 730 nm), the EA sets the amplitude to maximum. To better understand the spectral amplitude coefficients we need to consider the spectral absorption of water.
Fig. 4 Spatial light modulator masks (amplitude and phase) for the pulse shaper’s optimal solution superimposed with the shaped laser spectrum.
superimposes the spectral absorption coefficient of water [10
10. D. Segelstein, “The complex refractive index of water,” Master’s thesis, Department of Physics, University of Missouri-Kansas City (1981).
] over the shaped spectrum of the best discovered solution. There seems to be a few nanometer shift in wavelength between the published absorption minimum of water and the optical spectral shape. This small wavelength shift is probably not significant since it is likely due to temperature differences and wavelength calibration errors. Nevertheless, it is clear that the EA optimizes the pulse amplitude to the local minimum of water’s absorption near 800 nm. Observing water’s spectral absorption in Fig. 5
, the reason becomes more obvious for the short wavelength (< 730 nm) shaping of the spectrum in Fig. 4
. At wavelengths < 730 nm water’s spectral absorption is less than the local minima near 800 nm. Thus, even though the laser has imperceptible spectral content in this region, the EA still generates a solution that utilizes this energy.
Water absorption coefficient from Segelstein [10
], superimposed with the shaped spectrum of EA’s optimal solution.
As for the phase contribution to the pulse, the spectral phase weights appear to mimic the amplitude weights. After running amplitude-constrained and phase-constrained tests, the results demonstrate the same behavior in the phase-constrained tests (i.e., fixed phase with the EA optimizing just the spectral amplitude). The amplitude-constrained tests (i.e., amplitude fixed at 100%, with the EA searching just the spectral phases) show very little change in optical transmission ratio, most of which may be attributed to residual phase/amplitude crosstalk in the pulse shaper.
To further quantify the shaped optical pulses, Mesa Photonic’s FROG Scan measures the laser’s temporal and spectral characteristics with and without pulse shaping. The results without spectral shaping show the optical pulse has a significant spectral chirp such that the temporal width is almost 5× its transform limit, with a 190-fs temporal width (full width at half maximum). The spectral chirp of the best discovered pulse is similar, but since the bandwidth is drastically reduced the time-bandwidth product is 0.74 with an associated 115-fs temporal width.
A consistent spectral feature is the notch at the short wavelength spectral region, giving the spectra a mitten-like shape. This solution occurs in many different experiments, demonstrating its viability. This spectral notch is a function of the system, which includes the laser, pulse shaper, filters, water, and detectors. While we can easily separate the response from the filters and the detectors, it is more difficult to separate the response of the pulse shaper from that of the water. Further experimentation is required to determine if the spectral notch is due to the pulse shaper or is a narrow spectral feature in the water’s absorption.
In conclusion, an evolutionary algorithm is used in a feedback design within our experiment to optimize the transmission efficiency of a large bandwidth optical pulse when propagating in a water cell. Both the spectral amplitude and phase of the input optical pulse are modified to produce an improved result. The tailored pulse shape evolves to a dramatically narrower bandwidth that coincides with a local minimum (near 800 nm) in water’s absorption. Even though the laser has very little spectral power in its short wavelength wings (< 730 nm), the EA favors these in its optimal result, since water’s absorption is even lower at these wavelengths than at the local minimum near 800 nm.
In this linear propagation regime the average optical power is < 3 mW, which limits thermal lensing and nonlinear optical phenomena. In this case, the optical phase seems to have little importance in optimizing the transmission efficiency. This is reinforced by tests constraining either the amplitude or phase, and also by analyzing FROG Scan measurements of the optimized optical pulse. There is, however, a small phase dependence which potentially arises from phase/amplitude crosstalk in the pulse shaper. Finally, we observe a spectral notch in our solutions, for which further experimentation is needed to determine if it is specific to our experimental setup or is a narrow spectral feature in the water’s absorption.
Our experimental setup employs an evolutionary algorithm to efficiently search a large parameter space of optical pulses centered near 800nm and with 60-nm bandwidth. Repeated optimizations consistently converge to a similar solution. These results offer additional evidence that there is no advantage to using an optical pulse in the linear propagation regime. The solutions imply the transmission efficiency is dominated by the absorption spectrum of the water.
Finally, this experimental setup also demonstrates the consistency and viability of this method for experimental optimization. Direct search optimization techniques (like evolutionary algorithms) can be used to explore many questions of interest when combined with a carefully defined objective function (in our case the transmission efficiency). The EA is blind to whether objective function values come from an experiment or a model. Obtaining objective function values from an experiment conveniently allows for an exploration of both linear and nonlinear regimes. For example, if increased transmission efficiency is the only objective, then extending the current work to improved transmission in the the nonlinear regime could be accomplished simply by increasing the input pulse’s power. Future work will explore nonlinear propagation regimes.