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We demonstrate the adaptation of an iterative Fourier transform algorithm for the calculation of theoretical spectral phase functions required for pulse shaping applications. The algorithm is used to determine the phase functions necessary for the generation of different temporal intensity profiles. The performance of the algorithm is compared to two exemplary standard approaches. i.e. a Genetic Algorithm and a combination of a Simplex Downhill and a Simulated Annealing algorithm. It is shown that the iterative Fourier transform algorithm converges much faster than both alternative methods.
©2001 Optical Society of America
1 Introduction
The ability to produce temporally shaped ultra short laser pulses has become a major issue in different fields of science. Examples are the generation of pulse trains for optical fiber communication systems including en- and decrypting of information [1] or the generation of complex pulse structures for the coherent control of atomic systems [2], of molecular systems [3, 4], and of collective effects in solids [5]. A somewhat more technological application of pulse shaping is the automated correction of residual phase modulations in short pulse laser systems [6, 7] and the compression of chirped pulses with a large bandwidth [8]. Pulse shaping is usually based on linear filtering of the pulses in the Fourier domain [9], i.e. the modulating phase and/or amplitude patterns are applied to the dispersed spectral components of the input pulse. A practical realization of such a scheme is achieved by placing fixed phase or amplitude masks [10] or variable light modulators into the Fourier plane of a 4f-arrangement, where a spatial dispersion of the spectral components of the pulse is provided. The two most common variable modulator types are acousto-optical modulators (AOM) [11] and liquid crystal spatial light modulators (LC-SLM) [12, 13] which recently have been significantly improved [14]. By using for example a nematic liquid crystal modulator it is possible to perform phase modulations or amplitude modulations if a pair of polarizers is employed. Combining two liquid crystal modulators allows for a simultaneous phase and amplitude modulation [15].
If a phase-only shaper is used there exists no unique way to calculate the phase modulation that is necessary to generate a specific user-defined temporal waveform. In recent publications different optimization algorithms have been used to address this problem [9, 10, 16]. The aim of this article is to present a new approach to this problem based on a simple, robust, and fast algorithm.
The code is similar to those employed to extract the temporal profile of ultrashort laser pulses from autocorrelations measurements [17], the construction of speckle-free computer generated holograms [18], or phase objects for spatial amplitude shaping [19]. All these applications base on an algorithm that was developed to determine phases from image and diffraction plane pictures [20]. In the first part the peculiarities of phase-only shaping are summarized and in the second part the principle of the newly adapted algorithm is presented. Finally, the performance of the new approach is compared to state of the art techniques.
2 Phase Shaping
To shape a given input pulse to an arbitrary temporal waveform both the spectral amplitude and the phase have to be modified. Therefore, an ideal pulse shaping device which is able to change both the spectral amplitude and phase provides the necessary complex spectral transfer function
which transforms a given input pulse represented by its spectral electric field Ein(ω) to a specific tailored output pulse Eout(ω). Clearly, the output pulse shape is only restricted by the existence of sufficient spectral components in the input pulse. In reality, the applicable amplitude and phase modulations are limited by the properties of the pulse shaping device. In the case of common liquid crystal spatial light modulators the limiting factors are the number of stripes, the gap to stripe ratio, the maximum phase shift, the maximum contrast of amplitude modulations, and the smallest increment of the stripe properties (bit depth per stripe). Modulators which only change the phase of the spectral components still provide a large variety of possible output waveforms and have the advantage of conserving the pulse energy. Hence, phase-only shapers are preferred if only the temporal intensity profile is of importance not the phase. According to equation (1) the possible transfer functions of an ideal phase-only shaper are subject to the restriction
Any, possibly frequency dependent, amplitude modulation due to reflection at interfaces or absorption losses of a real phase-only shaper may be incorporated in the spectral amplitude of the incident laser pulse. The spectral phases of the input pulse may be changed by a real phase shift function Δ(ω)=arg[T(ω)] corresponding to
Due to the constraint (2) no exact transfer function can be determined for those transformations which require a change of spectral amplitudes. In these cases no general recipe can be given to derive that real phase shift function Δ(ω) which approximates best the desired output pulse |z(t)|2.
To solve this problem for non-trivial cases usually optimization algorithms such as Genetic Algorithms (GA), Evolutionary Strategies (ES) [21] or combinations of Simulated Annealing and Simplex Downhill algorithms (SASD) [22] are employed. The basic principle of these algorithms is to successively modify phase patterns by trial and error depending on their evaluation. In contrast, the proposed algorithm performs repeatedly deterministic transformations on the initial field until an invariant solution is found.
3 Description of the algorithm
The scheme of the newly adapted algorithm is depicted in fig. 1. The algorithm requires the complex spectral electric field E(ω) of the input pulse characterized by the spectral phase Φ(ω) and the spectral amplitude A(ω) and the temporal amplitude z(t) of the target pulse. The absolute magnitude of z(t) is of no importance because the algorithm rescales the amplitude by the Fourier domain constraint.
To initialize the algorithm the amplitude of the input field together with its original or any other arbitrarily chosen phase pattern is inserted in step 1. At this point only the amplitude A(ω) is of importance and the specific shape of the spectral phase is irrelevant. The spectral field is transformed to the time domain by an inverse Fourier transformation (step 2) and the temporal amplitude b(t) is replaced by the target amplitude z(t) (step 3: time domain constraint). The temporal phase Θ(t) remains unchanged, i.e. the algorithm restrains only the temporal amplitude of the pulse not the phase. Next, the Fourier transformation of the pulse back to the spectral domain is performed (step 4). Because the temporal amplitude has been changed in the previous step a modified spectral phase Ψ(ω) is obtained. Then, the spectral amplitude B(ω) is replaced by the spectral amplitude of the input pulse A(ω). Again, the phase Ψ(ω) remains unchanged and the next iteration reconvenes with step 1. The truncation condition is reached if b(t) is approximately equal to z(t). A more practical way to determine the truncation is to test whether the optimization process stagnates at a certain level. The phase difference Ψ(ω)-Φ(ω) constitutes the desired phase modulation which transforms the input pulse as close as possible to the target pulse. This phase pattern may then be transferred to the spatial light modulator.
Fig. 1. Scheme of the iterative Fourier transform algorithm (FT - Fourier transformation). The initial phase may be set to any random distribution of numbers.
This algorithm yields a solution, which can be regarded as being invariant to transformations between time and frequency space and which is subject to constraints in both spaces (available input spectrum and desired temporal shape). In ref. [20] proof is given that after each iteration the residual error of the intermediate result can only decrease or at least remain constant. Therefore, the algorithm must converge to a result.
As mentioned above, the search for optimal phase modulations for non-trivial tasks is usually solved by means of optimization algorithms. They basically rely on a successive improvement of randomly initialized phase-patterns by applying more or less random variations, which are necessary to overcome local optima. Therefore, they all can be considered as trial-and-error-methods and show a rather slow convergence due to the large number of unsuccessful trials. In contrast to all other known approaches to generate specific phase patterns for phase-only pulse shaping the newly adapted algorithm may be regarded as a deterministic method. Therefore it delivers results much faster as compared to common approaches.
4 Examples and comparison with other methods
Three arbitrary examples are shown in fig. 2. The trivial case (fig 2a) has been chosen because it easily allows to verify that the algorithm was able to find the optimal solution. Similar examples as those shown in fig. 2b) and c) have been investigated and reproduced experimentally in reference [16]. In all cases, the algorithm is able to find the phase pattern necessary to approximate the specified waveform.
In order to evaluate the algorithm a nontrivial task was chosen. The result is compared to those obtained by two standard approaches, i.e. a GA and a SASD algorithm. The task was to find the spectral phase modulation which transforms a Gaussian input pulse as close as possible to a rectangular pulse of a given temporal width. While the GA was able to independently modify the phase of each single pixel, the SASD algorithm had to optimize the coefficients of a Taylor series (to fifth order) describing the phase function.
Figure 3a) shows the temporal intensity profile of the target and the optimized output pulse. The corresponding spectral phase is depicted in fig. 3b). In fig. 4 the deviation (root mean square, RMS) of the current temporal profile from the target profile is shown as a function of the iteration. The RMS values are normalized to the RMS value obtained for a pulse experiencing no phase modulation by the SLM. Noteworthy, the final result was obtained after only 10 to 20 iterations. Although the exact number of iterations depends on the initial phase pattern the algorithm converges to the same solution in all cases. Figure 5a) and fig. 5b) show the temporal profile and the corresponding spectral phase of the solution found by a GA. It is obviously better than the solution by the iterative Fourier-transform algorithm but it requires the numerical evaluation of about 6000 generations of 30 individuals per generation (see fig. 6). To ensure that the result is not limited by the 8 bit resolution of the phase values the resolution was increased to 10 bits but only minor improvements were achieved. The SASD algorithm shows worse results than the two previously described algorithms as may be seen in fig. 7. The reason is that such an algorithm can only handle a low number of parameters compared to a GA and, therefore, a description of the phase function by a set of parameters (five Taylor coefficients) was chosen. Convergence is achieved much faster but otherwise this restriction limits the accessible pulse shapes and may not allow to generate any desired waveform.
5 Conclusion
We have presented a new approach to find the spectral phase function which approximates best a user-defined waveform and is based on an iterative Fourier transform algorithm. The performance of the algorithm was compared to standard approaches and it is found that the results nearly reach the quality of those generated by genetic algorithms. More important, the algorithm converges 10 to 100 times faster and produces in a reliable fashion always the same result. In case it should be necessary to refine the results of the algorithm they are well suited as initial patterns for genetic algorithms and would greatly accelerate its convergence.
The authors would like to thank R. Sauerbrey for continuous support of this work and L.-C. Wittig and D. Zeidler for stimulating and fruitful discussions.
Fig. 2. A bandwidth limited Gaussian pulse of 47 fs FWHM is phase modulated to produce a) a stretched pulse with 400 fs FWHM, b) a double pulse with a temporal separation of 480 fs and a FWHM of 80 fs each, and c) a triple pulse with ascending amplitude.
Fig. 3. a) The iterative Fourier transform algorithm was used to approximate a rectangular pulse with a FWHM of 300 fs. b) Spectral phase function found by the algorithm.
Fig. 4. Progress of the pulse shape optimization versus the number of iterations for the problem depicted in fig. 3. The different curves correspond to different initial phase patterns.
Fig. 5. a) The GA was used to approximate a rectangular pulse with a FWHM of 300 fs. The algorithm was able to change independently the phase of all pixels. b) Spectral phase function found by the algorithm.
Fig. 6. Progress of the GA versus the number of generations for different runs and different bit depths (30 individuals per generation).
Fig. 7. a) The SASD algorithm was used to approximate a rectangular pulse with a FWHM of 300 fs. The phase function was expressed in terms of a Taylor series with five coefficients. b) Spectral phase function found by the algorithm.
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