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Spectral phase retrieval from interferometric autocorrelation by a combination of graduated optimization and genetic algorithms Wenlong Yang*, Matthew Springer, James Strohaber, Alexandre Kolomenski, Hans Schuessler, George Kattawar and Alexei Sokolov Institute for Quantum Studies and Department of Physics & Astronomy, Texas A&M University, College Station, TX 77843-4242 USA. *[email protected]

Abstract: We describe a method for retrieving spectral phase information from second harmonic interferometric autocorrelation measurements supplemented by the use of the observed spectral intensity. By applying a combination of graduated optimization and genetic algorithms, accurate phase retrieval of laser pulses as short as a few optical cycles was obtained from the measured autocorrelation and spectral intensity. The effectiveness of the combined algorithms is demonstrated on a set of significantly different femtosecond pulse shapes. © 2010 Optical Society of America OCIS codes: (320.7100) Ultrafast measurement; (320.5550) Pulses; (100.5070) Phase retrieval.

References and Links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18.

J.-C. Diels, and W. Rodulph, Ultrashort Laser Pulse Phenomena, (Elsevier, 2006), chap. 9. T. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, (Kluwer Academic Publishers, 2000). C. Iaconis, and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23(10), 792–794 (1998). M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). K. Naganuma, K. Mogi, and H. Yamada, “General Method for Ultrashort Light Pulse Chirp Measurement,” IEEE J. of Quan. Elec. 25, 6 (1989). M. Mitchell, An Introduction To Genetic Algorithms, (MIT, 1998). R. S. Judson, and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992). 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(1-2), 151–158 (1997). T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997). A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, “Adaptive control of pulse phase in a chirped-pulse amplifier,” Opt. Lett. 23(24), 1915–1917 (1998). E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24(7), 493–495 (1999). T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and twophoton transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000). B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001). F. G. Omenetto, A. J. Taylor, M. D. Moores, and D. H. Reitze, “Adaptive control of femtosecond pulse propagation in optical fibers,” Opt. Lett. 26(12), 938–940 (2001). J. W. Nicholson, F. G. Omenetto, D. J. Funk, and A. J. Taylor, “Evolving FROGS: phase retrieval from frequency-resolved optical gating measurements by use of genetic algorithms,” Opt. Lett. 24(7), 490–492 (1999). R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003). E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene RingOpening,” Ultra-fast Phenomena 88, 249–251 (2007). C.-W. Chen, J. Y. Huang, and C.-L. Pan, “Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm,” Opt. Express 14, 22 (2006).

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Received 17 May 2010; revised 14 Jun 2010; accepted 17 Jun 2010; published 29 Jun 2010

5 July 2010 / Vol. 18, No. 14 / OPTICS EXPRESS 15028

19. K.-H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271(1), 169–177 (2007). 20. J.-H. Chung, and A. M. Weiner, “Ambiguity of Ultrashort Pulse Shapes Retrieved From the Intensity Autocorrelation and the Power Spectrum,” IEEE J. Sel. Top. Quantum Electron. 7(4), 656–666 (2001). 21. D. A. Bender, J. W. Nicholson, and M. Sheik-Bahae, “Ultrashort laser pulse characterization using modified spectrum auto-interferometric correlation (MOSAIC),” Opt. Express 16(16), 11782–11794 (2008). 22. D. A. Bender, and M. Sheik-Bahae, “Modified spectrum autointerferometric correlation (MOSAIC) for singleshot pulse characterization,” Opt. Lett. 32(19), 2822–2824 (2007). 23. A. Rosenfeld, Multiresolution Image Processing and Analysis (Springer, 1984).

1. Introduction Measurement of ultrashort laser pulse shapes is an important non-trivial task. There are several methods allowing for the full characterization of the electric field [1–3]; the most common of these measuring methods include second harmonic interferometric autocorrelation (SHIAC) [1], frequency resolved optical gating (FROG) [2] and spectral interferometry for direct electric-field reconstruction (SPIDER) [3]. FROG uses a generalized projection method to retrieve both the spectral phase and amplitude [1]. SPIDER uses noniterative calculations [4] to get the spectral phase and amplitude. Naganuma et. al. [5] have shown that the knowledge of both SHIAC trace and spectral intensity are sufficient for pulse reconstruction. To this end, they provided a Gerchberg–Saxton like algorithm to retrieve spectral phases from the autocorrelation and spectral intensity measurements. Genetic algorithms (GA) [6] are a class of universal evolutionary searching algorithms for solving complex problems. They have been successfully used in femtosecond optics research [7–17]. Of particular interest to the subject presented here, GAs have also found application in the retrieval of spectral phase information from SHIAC [18, 19]. In [18], the spectral phase of individual frequencies are used as genes, as was originally done in [7]. This method allows the adjustment of the phase for local perturbations, but it will take many steps to compensate a big global phase change, such as a big linear chirp and higher-order chirps. In [19], the spectral phase is expanded in a Taylor series around the center frequency, as was originally done in [8, 9]. This method can retrieve the linear chirp and higher-order chirps very effectively, however, it lacks the ability to find local sharp perturbations of the spectral phase. As is shown in [20], although SHIAC and spectral intensity are sufficient to retrieve the spectral phase, a small difference in SHIAC may correspond to very different pulse shapes. Therefore, in order to avoid spurious pulse shapes, a high accuracy phase retrieval algorithm is highly desirable. The GA used in our work combines the two methods discussed above [18,19] and is able to retrieve spectral phases with both large chirps and rapid local perturbations. GAs are noted for their parallel-search ability in solving multi-parameter optimization problems and their ability to avoid local optima; although, in some pathologic cases, they too can be trapped in a local extremum. Also, for a big parameter space, GA is very slow to converge. Bender et al. have shown that a good starting point is helpful for an iterative algorithm for the phase retrieval of modified spectrum auto-interferometric correlation (MOSAIC) [21]; a sequential searching algorithm [22] has been used to seed the iterative algorithm in their work. In our work, the graduated optimization algorithm (GOA) [23] is employed to provide a starting point for GA. GOA as a global searching algorithm, reduces a complex problem into a much simpler one. Then it makes a more detailed optimization in the vicinity of the extrema of the previously simplified problem and repeats this process until the original problem is optimized to a specified precision. In this way, GOA can search a big parameter space and provide GA with a good starting point that is around the global optimum. In this paper, a method is presented that combines GOA with GA so as to avoid local extrema and reach an accurate optimization solution efficiently. We find this algorithm to be well-suited for retrieving phase information from SHIAC and spectral intensity measurements.

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This paper is organized as follows. In Section 2, the method of the algorithms is presented. In Section 3, the results of the algorithm implementation are shown by successfully retrieving the phases of two experimentally available femtosecond (fs) pulses and four noisefree simulated pulses from their SHIACs and spectral intensities. In Section 4, conclusions and discussions are given. 2. Method To obtain the spectral phase of a short pulse, the GOA is used to make a rough estimate of the spectral phase and the GA is then employed to fine-tune the spectral phase to reach the desired accuracy. The spectral phase of a short pulse can roughly be expressed in a polynomial form:

ϕ p (ω ) = a2 (ω − ω0 ) 2 + a3 (ω − ω0 )3 + a4 (ω − ω0 ) 4 ,

(1)

where ω0 is the central angular frequency and ai (i = 2, 3, 4) are chirp parameters. Commonly, these parameters contribute most to the non-trivial part of the spectral phase. The constant and linear part of the spectral phase a0 + a1 (ω − ω0 ) are omitted in Eq. (1) because they are the absolute phase and delay, which are not determined by autocorrelation measurements. Values of ai (i = 2, 3, 4) of the spectral phase are first coarsely searched by the GOA over ai ’s whole parameter space. In fact, there are higher-order expansion terms, or in other words, more local details in the spectral phase that cannot be described by these three parameters. That is why we use the GA to find the details of the spectral phase and make fine tunings of the ai . Before giving a detailed description of the algorithm, the definition of the problem and the data preparation are described in the following subsection. The algorithms are described in later subsections. 2.1 Description of problem and data preparation

In reference [5], it was shown that the measured SHIAC I ac (τ ) together with the spectral 2 intensity Eɶ (ω ) can uniquely determine the spectral phase of an ultrafast pulse. This is the

basis of our algorithm. The first step of our algorithm is to prepare the data. In order to compensate for the linear distortion of the measured SHIAC, I ac (τ ) is modified to I ac _ inuse (τ ) I ac _ inuse (τ ) = I ac (bτ ),

(2)

where b can be determined initially by visually matching the Fourier transform of I ac _ inuse (τ ) and the measured spectral intensity, as was explained in reference [5]. The GA will be used to refine this parameter later. A given ϕ (ω ) and the measured spectral intensity can be employed to calculate the pulse shape. The pulse shape is subsequently used to calculate the autocorrelation I ac _ cal (τ ) . The difference between I ac _ cal (τ ) and I ac _ inuse (τ ) is described by the root mean square (rms) error ∆ :

∆=

Nτ

∑  I i =1

2

ac _ cal

(τ i ) − I ac _ inuse (τ i )  / Nτ ,

(3)

where Nτ is the number of sampled points of τ . This error can be understood as the average difference between the retrieved SHIAC and the measured or simulated SHIAC. The units of the rms error ∆ are the same as the units of SHIAC, whose peak is chosen to be 1. The GOA + GA combined algorithm was designed to minimize ∆ . #128510 - $15.00 USD

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2.2 Graduated optimization algorithm

In our program, ai were optimized one by one. The time unit was fs and the angular frequency unit was rad/fs. Thus a2 , a3 , and a4 have units of (fs/rad)2, (fs/rad)3 and (fs/rad)4 respectively. When optimizing a2 , we set a3 and a4 to zero. For example, we searched a2 from 0 to 100 with a step size of 10. In this way, we were able to find a minimum within a region of 20. Then we reduced the step size to 1 and search this smaller region. The one giving minimum error is called a2 _ opt . Next, we set a2 to a2 _ opt , a4 to zero and gradually optimized a3 . After we obtained a3 _ opt , we set a2 to a2 _ opt , a3 to a3 _ opt and obtained a4 _ opt by the same procedure. These optimized ai _ opt were used as starting values in the GA. The range of the searching process may differ from one experiment to another. The general rule is that these numbers must be set large enough to guarantee the optimization of ai to give a global minimum of ∆ and they must not be so large that the calculated pulse is too long to be characterized by SHIAC. This searching sequence was also used in phase retrieval from MOSAIC in ref [22]. 2.3 Genetic algorithm

The GA belongs to the class of evolutionary algorithms. It was inspired by natural evolutionary biology and is commonly used in multi-parameter optimization problems. Generally, an optimization problem is to extremize a fitness function by tuning its variables. In GA, the variables form a genome where each variable is a gene. Each gene can vary in a certain range. A generation is formed by individuals whose genes have different values. Fitness values are calculated for individuals from the fitness function. Only those with the extreme fitness values will survive, and they are called parents. Parents will produce the next generation using the following three methods: 1) inheritance, a process in which parents transfer their genes to their children (individuals of the next generation) without any change in the genes’ value; 2) mutation, a process in which parents pass somewhat modified genes to their children; 3) crossover, which means that parents’ genes are combined to form the genome of their children. As the next generation is produced, they will go through the process of selection and produce another next generation. In this way, individuals in later generations on average have better fitness values. The problem is thus optimized by the evolutionary process, which proceeds toward “survival of the fittest”. Most pulse spectra can be considered as distributed in a finite frequency region; namely ωstart < ω < ωend . We took Nω equally spaced points ωi from ωstart to ωend , and ϕ r (ω ) was the interpolation function of ϕ r (ωi ) . Our GA’s goal is to optimize the phase ϕ r (ω ) and polynomial phase ϕ p (ω ) so that the total phase ϕ (ω ) = ϕ r (ω ) + ϕ p (ω ) will minimize the rms error ∆ in Eq. (3). Since we have already found roughly a spectral phase close to a global minimum of ∆ from the GOA, with this starting point of GA, we are more confident that the GA would not stagnate at a local minimum. However, in order to optimize the result globally, ai are also used as optimization parameters. This makes the spectral phase of ωi not only change independently, but also collectively. This also fits the physical situation well, because when pulses are transmitted through optics such as windows or lenses, the change of phase can be well represented by ai . Thus, our GA’s parameters are the combination of the parameters of GAs used by Chen et. al. [18] and by K.-H. Hong, et al. [19]. Furthermore, in order to 2 optimize b in Eq. (2), a shift parameter r of the spectrum Eɶ (ω ) is introduced in the GA. For generation g and individual i, the phase is calculated by

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ϕ( g ,i ) (ω ) = ϕ p ,( g ,i ) (ω , a2,( g ,i ) , a3,( g ,i ) , a4,( g ,i ) ) + ϕr ,( g ,i ) (ω ),

(4)

where the ϕ r ,( g ,i ) (ω ) was the interpolation function of ϕ r ,( g ,i ) (ω j ) . Then the electric field was obtained by iϕ (ω ) E( g ,i ) (t ) = ∫ Eɶ (r( g ,i ) + ω ) e ( g ,i ) eiωt d ω.

(5)

The SHIAC was calculated according to ∞

I ac _ cal ( g ,i ) (τ ) =

∫

−∞

 E( g ,i ) (t − τ ) + E( g ,i ) (t ) 

2 2

dt.

(6)

(τ i ) − I ac _ inuse (τ i )  / Nτ ,

(7)

Thus, the rms error can be obtained ∆ ( g ,i ) =

Nτ

∑  I i =1

2

ac _ cal ( g , i )

where Nτ is the number of total sample points in the error calculation. Equation (7) is the same as Eq. (3), except that (g,i) is added as subscript to indicate the numbers of generation and individual. ∆ ( g ,i ) was used as a criterion for judging which individual could survive in GA. In summary, ϕ r ,( g ,i ) (ω j ) , a2,( g ,i ) , a3,( g ,i ) , a4,( g ,i ) and r( g ,i ) are the genes in our GA. We call them xk ,( g ,i ) . Our goal of GA is to adjust xk ,( g ,i ) to minimize the fitness function ∆ ( g ,i ) . Figure 1 shows the scheme of our GA.

Fig. 1. The scheme of the GA. Each parent will reproduce itself as one of the individuals in the next generation without any change to genes, which is called inheritance. Each of them also produces a certain number of individuals of the next generation by mutation and crossover.

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3. Results

3.1. Results for experimentally available pulses As the first example of GOA + GA, we retrieved the spectral phase of a sub-10 fs pulse generated by a mode-locked Ti:Sapphire oscillator (FemtoLasers, Rainbow). The SHIAC was measured by a commercial interferometric autocorrelator (FemtoLasers, 5 – 150 fs). The spectral intensity was measured by an Ocean Optics spectrometer (550 – 1050 nm). In the computation, each pulse is characterized by 2048 points; the time step we used was 0.25 fs, and the time window was 512 fs. In the GOA, a2 was scanned from 0 to 500, a3 from −1000 to 1000 and a4 from −1000 to 1000. The numbers we chose above are more or less arbitrary. This process takes about 6.6 seconds on a computer with Intel® Core 2 Duo E8400 CPU. After the graduated optimization, the retrieved SHIAC and the measured SHIAC (after data preparation) are shown in Fig. 2 (a). There are still visible discrepancies between them. For GA, we used the following parameters: the number of individuals 50, the number of parents 5, children produced from inheritance 1, mutation 7, and crossover 2, the number of generations 200, a mutation rate of 10% and a crossover rate of 25%. The mutation process is like this: with 10% mutation rate, 10% of a parent’s genes will change their values and form an individual of the next generation. The process of crossover is like this: with crossover rate of 25%, a parent takes 75% of its genes and combined with 25% genes from other parents to form an individual of the next generation. The spectral phase was sampled by Nω = 80 points. The choice of Nω is discussed in sSection 4. The GA was programmed with parallel computing function and took about 413 seconds on the same computer (described above). The result of the GOA + GA is shown in Fig. 2. Black dotted curve in (a) and (b) are measured SHIAC after data processing. The red solid curve in (a) is the calculated SHIAC with the spectral phase obtained from GOA coarse global search. The red solid curve in (b) is the calculated SHIAC from the spectral phase obtained by GA. Measured spectral intensity and retrieved spectral phase from GOA and the GA after GOA are shown in (c). The error of each individual was calculated by Eq. (7) and the minimum of errors of individuals in each generation was plotted in Fig. 2 (d). As a second example, we sent the pulse, described above, through a thin solution of an infrared absorber contained between two cover slips. The infrared absorber is Exciton NP800 and the solvent is 1,1,2,2-tetrachloroethane. The infrared absorber has a very strong absorption peak around 800 nm. After the pulse transmitted through the absorber and cover slips, its spectral intensity around 800 nm is greatly reduced due to the absorption of the absorber, and the spectral phase is also modified due to the dispersion from the absorber and the cover slips. The pulse’s spectral phase and intensity are more complicated than those in the first example which provides a challenging test of our combined algorithms. In the GOA, a2 was scanned from 0 to 500. a3 was scanned from −1000 to 1000. We scanned a4 from −5000 to 5000. These numbers are also more or less arbitrary. Parameters used in the GA were the same as those used in example 1. The computation time for GOA was about 6.9 seconds and the GA took about 407.8 seconds to finish. The result is shown in Fig. 3. It can be seen from Fig. 3 (b) that the retrieved SHIAC matched the experimental SHIAC quite well.

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Fig. 2. First example of phase retrieval by the combined GOA and GA. (a) The red solid curve is the calculated SHIAC with optimized phase from GOA. (b) The red solid curve is the calculated SHIAC with retrieved phase from GA with GOA’s result as starting point. In (a) and (b), the black dotted curve is the experimental SHIAC after adjustment. The retrieved autocorrelation is symmetric, therefore only half of the curve was plotted. (c) The measured spectral intensity (black), the retrieved phase from GOA (blue) and the retrieved spectral phase from GA (red). (d) The evolution of minimum error for each generation. The unit of the error is the same as the unit in (a) and (b).

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Fig. 3. Second example of phase retrieval by the combined GOA and GA. (a) The red solid curve is the calculated SHIAC with optimized phase from GOA. (b) The red solid curve is the calculated SHIAC with retrieved phase from GA with GOA’s results as starting point. In both (a) and (b), the black dotted curve is the experimental SHIAC after adjustment. The retrieved autocorrelation is symmetric, therefore only half of the curve was plotted. (c) The measured spectral intensity (black), the retrieved phase from GOA (blue) and the retrieved spectral phase from GA (red). (d) The evolution of minimum error for each generation. The unit of the error is the same as the unit in (a) and (b).

Using the inverse Fourier transform, we can calculate the retrieved electric field (Fig. 4) from the measured spectral intensity and the retrieved spectral phase. As is expected, the center peak of the electric field E (t ) is sharp and strong. However, the tails, which are expected to be weak, are stronger than expected. This will be explained in the discussion part in Section 4.

Fig. 4. (a) Retrieved electric field from combined GOA + GA in example 1. (b) Retrieved electric field from combined GOA - GA in example 2.

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GOA + GA’s results are listed in Table 1. ∆ GOA is the minimum rms error achieved by GOA and ai _ opt are chirp parameters for the optimized pulse shape of GOA. b is defined by Eq. (2). ∆ GOA − GA is the minimum rms errors achieved by GOA + GA. However, GA’s results ai are not very meaningful, because the spectral phase is a combination of low order chirps and phases for individual frequencies. For example, the optimized spectral phase in Fig. 2 (c) looks noisy and cannot be well described by ai . We list the ai here just to show that our GA can fine-tune ai and r . Table 1. Results of GOA and GOA + GA for example 1 and 2 GOA’s results

a2 _ opt

a3 _ opt

22

−182

a2 _ opt

a3 _ opt

33

653

a4 _ opt

Results for example 1 in section 3.1 GOA + GA’s results

b*

∆ GOA

a2

a3

a4

r

−2

1.04 2.2 × 10 22 194 0 −90.28 Results for example 2 in Section 3.1 GOA’s results GOA + GA’s results 194

a4 _ opt −1877

b* 1.04

∆ GOA 4.5 × 10

−2

∆ GOA − GA 9 × 10−3

a2

a3

a4

r

∆ GOA − GA

33

653.48

−1878.4

−0.0055

2.3 × 10−2

* b is not obtained from GOA.

The necessity of GOA can be shown by comparing the results of GA with and without GOA on both examples in this section. We applied the GA with and without the help of the GOA six times for each example, and the minimum errors for each generation in the six runs are shown in Fig. 5. For the GA without GOA, the starting individuals have random phases. For example 1 (Fig. 5(a)), the average error at 50th generation of GOA + GA is the same as the average error of 56th generation of GA without GOA. (Here, the average means the average of the minimum errors of the six runs.) The computation time for one generation is about 2.1 seconds and the GOA takes 6.6 seconds. So GOA + GA leads GA without GOA by 5.8 seconds. In Fig. 5(a), the chirp of the pulse is small, thus the GA without GOA catches up with the GA with GOA after a couple of generations, which also shows the strong searching ability of our GA. For example 2 (Fig. 5(b)), GA without GOA’s average error at 200th generation is the same as the average error of GOA + GA at 106th generation, which means GA without GOA lags GOA + GA by 188 seconds. The error of GOA with GA at 100th generation is close to the error of GOA + GA at 70th generation, which means GA without GOA lags GOA + GA by 60 seconds. In this example, because the pulse has a bigger chirp, GOA + GA is much faster than GA without GOA.

Fig. 5. (a) Comparison of GA with and without GOA with the data in example 1. (b) Comparison of GA with and without GOA with the data in example 2. The unit of the errors is the same as the unit of (a) and (b) in Fig. 2 and Fig. 3. The scales of (a) and (b) are made different due to the different ranges of the data.

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3.2. Results for noise-free simulated pulses The strength of GOA + GA can be better shown by using GOA + GA to retrieve the spectral phase of noise-free simulated pulses. An 800nm 40 fs transform limited pulse was used as the seed pulse. Arbitrary chirps ( a2 , a3 and a4 ) were added to the seed pulse and GOA + GA was then used to retrieve the spectral phases of them. For a pulse with only linear chirp a2 = 483 the GOA gave exact value of the chirp with zero error, this is as expected because GOA starts from searching a2 , so if the pulse has only linear chirp, GOA can find the chirp with zero error. The results of another three simulated pulses are listed in Table 2. From the data in Table 2, the GOA + GA can make a retrieve with error between 10 −3 to 10 −4 on these noise-free simulated pulses. The ai retrieved by GA is not listed here because the total phase is the sum of the polynomial part and the phase for individual frequencies. Table 2. GOA + GA’s results for noise-free simulated pulses Pulse 1’s parameters

a2

a3

a4

23 120 1532 Pulse 2’s parameters

a2

a3

a4

135 −1563 −2831 Pulse 3′s parameters

a2 135

a3 563

a4 −11953

GOA’s results

a2 _ opt 48

a2 _ opt 132

a2 _ opt 68

a3 _ opt

a4 _ opt

17 64 GOA’s results

a3 _ opt

a4 _ opt

−1420 −1908 GOA’s results

a3 _ opt −292

a4 _ opt −116

GOA + GA’s results

∆ GOA − GA

∆ GOA −4

3.1 × 10−4 GOA + GA’s results

∆ GOA

∆ GOA − GA

1.2 × 10−3

1.6 × 10−4 GOA + GA’s results

∆ GOA

∆ GOA − GA

9.3 × 10

2.5 × 10

−3

1.04 × 10−3

3.3. About the usefulness of crossover Crossover is considered as one of the most important features of GA. It is worthy to study how helpful crossover process is in phase retrieval problems. We ran the GA without crossover six times and the minimum errors of each generation of the six runs are plotted in Fig. 6 together with those of the six GA runs with crossover. We found that there is no big difference whether we use crossover or not in GA in our phase retrieval problem.

Fig. 6. GA with and without crossover. The red dashed lines are the minimum errors of each generation in the six runs of GA without using crossover and the black solid line are that of GA with crossover. There is no obvious difference between the results of GA with or without crossover. There is an unlucky case in the GA with crossover whose minimum errors are higher than those of any other runs.

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4. Conclusions and discussion

The GOA + GA combined algorithm can effectively avoid local extrema and retrieve the spectral phase from the SHIAC and spectral intensity successfully. Chirp parameters ai were introduced into our GA, which strengthened the ability of our GA to search for global extrema. For the noise-free simulated strongly chirped pulses, the rms errors between retrieved SHIAC and the simulated SHIAC can reach 10 −3 to 10 −4 . For the experimentally available pulses, errors around 10 −2 are achieved. For pulses with small chirp parameters, the GA itself works as well as GOA + GA. For pulses with big chirps the GOA + GA converges faster than GA without GOA. The crossover is not found to play an important role in our GA. There are several issues that need to be noticed. (1). Due to time reversal invariance, it is impossible to know the sign of the total phase; thus, we defined a2 to be a positive number and a3 and a4 were scanned from negative to positive numbers. (2). Due to the Nyquist– Shannon sampling theorem, Nω determines the pulse’s temporal length that can be retrieved accurately. However, increasing Nω will also decrease the step size used in the discrete Fourier transform, which will increase the runtime of the algorithm. In our experiment, we are mainly interested in the central part of the pulse and the measured SHIAC is between −100fs to 100fs, so 80 points is sufficient for our application. However, this will cause the retrieved electric fields (see Fig. 4.) to have some spurious tails. (3). GA is highly parallelizable. The parallelized program is about 25% faster than serial program on the dual core computer described in the second paragraph of Section 3.1. With multi-core computers, the parallelized program should be faster. However, the communication between CPUs also takes time; thus, the efficiency of this program on multi-core computer needs further investigation. (4). The correction of the SHIAC data is crucial to the successful retrieval of the spectral phase. The spectrum was not only a set of data that was used in the algorithm, but was also used as a criterion to prepare the SHIAC data. We corrected only the linear shift of the SHIAC. (5). The GOA can also be done more than once by using optimized ai instead of zeros as starting values. (6). As is easily notice in Fig. 5 and Fig. 6, one set of err for GOA + GA converges slower than all others. It can be regarded as an unlucky case, which shows that luck can also affect the result of GA. (7). Given the spectral intensity, our GOA + GA combined algorithm provides a general method to retrieve spectral phase from measured data, and could be used to retrieve spectral phase from FROG or MOSAIC. In order to retrieve the phase from another kind of measurement, one only needs to change the error function to be the difference between the calculated result and measured results. Acknowledgments

We thank Benjamin D. Strycker, Chao Wang, Yongrui Wang for helpful discussions. We thank the support from the Office of Naval Research (ONR) under contract N00014-08-10037 and the Defense University Research Initiative Program (DURIP) under contract N00014-08-1-0804, the National Science Foundation (NSF) (PHY 354897, 0555568 and 722800), the Texas Advanced Research Program (010366-0001-2007), the Army Research Office (W911NF-07-1-0475), the Air Force Office of Scientific Research (FA9550-07-10069) and the Robert A. Welch Foundation (A1546 and A1547).

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