The simulated annealing method is used for retrieving the amplitude and phase from cross-phase modulation spectrograms. The method allows us to take into account the birefringence of the measurement fiber and resolution of the optical spectrum analyzer. The influence of the birefringence and analyzer resolution are discussed.
©2004 Optical Society of America
The advance in ultrashort laser pulse technology and development of coherent control has brought with it a need for new diagnostic tools that would allow one to unambiguously and simultaneously measure the amplitude and phase of pulses. A large group of techniques that deals with this problem is usually denoted as Frequency Resolved Optical Gating (FROG) methods [1, 2]. The FROG allows us to fully characterize the pulses in a manner that is general, robust, accurate and rigorous. This measurement is of the autocorrelation type, in which a set of autocorrelator beam spectra, called the spectrogram, is measured and analyzed. Depending on the beam geometry and nonlinearity involved, several types of FROG can be distinguished - the second harmonic generation FROG (SHG FROG), polarization gate FROG (PG FROG), self-diffraction FROG (SD FROG), third harmonic generation FROG (THG FROG), or cross-phase modulation FROG (XPM FROG).
The best established technique is the SHG FROG . It is one of the most sensitive FROG methods and can measure pulses as weak as 1 pJ. The main drawback of the SHG FROG is its ambiguity in the direction of time, which means that both the pulse E(t) and its time-reversed replica E(-t) yield the same SHG FROG trace. There is also ambiguity in the relative phase between two well separated pulses .
A concurrent technique based on cross-phase modulation in optical fiber was proposed by Thomson et al. . This technique has a sensitivity comparable with that of the SHG FROG due to collinear geometry in optical fiber, but without the temporal ambiguity. A standard FROG algorithm of generalized projections (GP) was used for the amplitude and phase retrieval from spectrograms obtained by this method [1, 5].
Retrieval techniques based on usual minimization methods are susceptible to becoming stalled on local minima. A genetic algorithm was applied to avoid this problem . In this paper we describe an application of another global search-type algorithm, the simulated annealing method. This is, like the genetic algorithm, a “naturally” slow algorithm, that is, however, not prone to getting stuck in local minima and allows us to take into account the fiber birefringence and optical spectrum analyzer resolution that are usually ignored in other methods.
We consider a modification of the experimental set-up described in  that is shown in Fig. 1. This is an alignment-free configuration suitable for measurement of picosecond optical pulses. The incident pulse is split by a polarization beam splitter (PBS) into two replicas with orthogonal linear polarizations propagating in two branches. The splitting ratio is controlled by a fiber polarization controller PC at the input of the apparatus. The light incident on Faraday mirrors is reflected backward with orthogonal polarizations and is recombined on the PBS and sent to the measurement fiber. As the measurement fiber a highly nonlinear microstructure optical fiber (MOF) with small dispersion is used, as described elsewhere . Since MOF’s are usually highly birefringent, coherent coupling between the linearly polarized modes is averaged out and only incoherent coupling remains. It is not possible to use the circularly polarized modes with highly birefringent measurement fiber because the circularly polarized states are coupled by linear birefringence of the fiber. So, we do not convert the linearly polarized modes to circularly polarized modes as it was done in [4, 5]. A lower XPM coefficient for linear polarizations can be partially compensated for by intentionally unbalancing the branches in favor of XPM. The probe pulse is selected by a polarizer and the output signal ESIG (t,τ) is sent to the optical spectrum analyzer to generate the spectrogram. The pigtails between the polarizing beamsplitter and Faraday mirrors are assumed to be short enough so that no significant self-phase modulation and dispersion occur in them.
The propagation of two orthogonally linearly polarized optical pulses in the MOF is described by coupled modified nonlinear Schrödinger equations 
where β 2 is the group velocity dispersion, γ is the nonlinear coefficient, κ=π/LB is the birefringence coefficient, σ=(β 1x-β 1y)/2, and are the group velocities along the main axes of the fiber. We can rewrite the system of coupled equations (1) using the transform
where the last term rapidly oscillates for large κ and is averaged out. The dispersion can be neglected, since the useful length of the MOF is limited by walk-off due to polarization mode dispersion and is much shorter than the dispersive length for a low-dispersion, high-birefringent fiber. We finally arrive at the system of equations
This system of coupled equations has the analytical solution 
If the output polarizer selects the x-component, the XPM FROG signal at the output of the MOF of length L, ESIG (t,τ)=Ex (L, t), can be written as
where τ is delay between the orthogonally polarized pulse replicas, r is the real-valued coefficient that is introduced as Ex (0, t)=rE(0, t), Ey =(1-r 2)1/2 E(0, t), and E(0, t) is the measured pulse. The mathematical spectrogram is the squared modulus of the Fourier transform of the XPM FROG output signal
The measured spectrogram is the convolution of the mathematical spectrogram with an optical spectrum analyzer instrumental function K(ω) 
3. Simulated annealing method
In 1953, Metropolis developed a method for solving problems that mimics the way thermodynamic systems go from one energy level to another . A generalization of the Metropolis Monte-Carlo method, Simulated Annealing, is a minimization technique which has given good results in avoiding local minima; it is based on the idea of taking a random walk through the space at successively lower temperatures, where the probability of taking a step is given by the Boltzmann distribution . The algorithm can be described as follows:
Step I Initialize an arbitrary configuration E 0(tk ) for the system at temperature T, where T is reasonably large.
Step II Starting from the configuration E 0(tk ), sample N configurations E 1 …EN looking for points with low energies; in these random walks, the probability of taking a step is determined by the Boltzmann distribution,
where 𝓔i is a real-valued energy function for the configuration Ei (here it is a measure of the difference between the measured and reconstructed spectrogram).
Step III Go to Step II, replacing E 0 by EN and T by a lower temperature.
Step III is repeated on the subsequent configuration until the temperature of the system is equal to 0. It can be shown that the configuration that corresponds to the global minimum of 𝓔 can be reached according to such an algorithm whenever the temperature decreases at a logarithmic rate . In practice, however, a linear or even exponential temperature decrease schedule can often be implemented. In this work the role of energy is played by the measure of difference between the measured and reconstructed spectrograms, the so-called FROG error function, and exponential cooling in an appropriate temperature interval was chosen. In the core of the program, functions from the GNU Scientific Library were used . For faster convergence, an adaptive step was implemented. With the used 2 GHz-Athlon PC, the reconstruction time was approximately 30 minutes for 64×64 grid. Simple pulses like a chirped sech-pulse could be reconstructed within minutes from a noise-free spectrogram. The parameters of the MOF used in the following simulations are taken from the experiment : the MOF length L=10 m, the nonlinear coefficient γ=14 W-1km-1, the polarization mode dispersion σ=36 ps/km.
The program for reconstruction of pulse from the spectrogram was extensively tested using the generalized pulse trains . The pulses are generated in the time-domain using the expression
with a global phase
where the normalized coefficients a, b, c represent the chirp and higher order phase distortions and q is the measure of self-phase modulation. Each of the np pulses is centered at ti with a pulse FWHM of T. Then E ′(t) is Fourier transformed into the frequency domain and spectral the phase distortion ξ(ν) is applied to it before the inverse Fourier transformation to the time domain
The pulse #1 represents a self-phase modulated pulse with cubic phase distortion, the pulse #2 is a double pulse with a complicated structure, the pulse #3 is a burst of five overlapping pulses and the pulse #4 is a pulse with cubic spectral phase distortion that leads to appearance
of satellites in the time domain. Table 1 also lists the final FROG errors G . It can be seen that the pulse amplitude and phase are extracted reasonably well even for pulses with complicated structures.
The robustness of the reconstruction algorithm to the spectrometer noise was also verified. A Gaussian noise was independently generated for each spectral trace. A noisy spectrogram and results of pulse retrieval are shown in Fig. 3 for the first pulse from Tab. 1. The noise floor was approximately 20 dB bellow the spectral peak of the signal as can be seen on Fig.3a. Pulse was reconstructed with the error G=0.00676 that is one order of magnitude worse than the value for the noiseless spectrogram but the retrieved field mimics well the original pulse (Fig. 3b). An experimental error related to inaccuracy in the time delay τ can be neglected for picosecond pulses when a high resolution delay line is used and therefore we did not simulate this kind of error.
The simulated annealing method allows us to easily investigate effects of the resolution of spectrometer and of walk-off between the probe and cross-polarized pulses, and also to eliminate these effects. For demonstration purposes, a spectrogram was numerically generated with an exaggerated polarization mode dispersion of σ=0.15 ps/m and a rectangular 2-nm-wide instrumental function. The pulse was reconstructed both by taking into account aforementioned effects and by neglecting them. The results are shown in Fig. 4a for a negatively chirped pulse. It can be seen that the omission of the finite resolution of the analyzer leads to a narrower reconstructed pulse compared to the original pulse. Also the phase of the reconstructed pulse is deformed. This can be easily understood since the lower resolution of the spectrum analyzer widens the measured spectra and a broad spectrum corresponds to a narrow and/or chirped pulse. When the polarization mode dispersion is omitted during reconstruction, the reconstructed pulse is broader and has a lower peak power since lower power is necessary to achieve comparable cross phase modulation when walk-off is not present. When the actual resolution and polarization mode dispersion is taken into account, the reconstructed pulse resembles the original one quite well with the FROG error G=0.000109. Figure 4b shows how the polarization mode dispersion inflicts upon the mean frequency of the spectrogram. The zero-crossing point is shifted from the origin by an amount that is close to σ L for a small splitting ratio r. It was shown that in an isotropic medium the mean frequency is the time derivative of the pulse autocorrelation . In birefringent fibers this feature vanishes.
The method of simulated annealing was implemented in an algorithm for retrieving the amplitude and phase from spectrograms acquired using the fiber FROG with XPM geometry and highly birefringent measurement fiber. It was shown that linearly polarized states should be used in this case and that the walk-off caused by the polarization mode dispersion limits the useful length of the measurement fiber and hence the sensitivity, and also breaks the link between the autocorrelation of the pulse and the mean frequency of the spectrogram. The resolution of the spectrum analyzer and walk-off were included into algorithm and their importance for correct reconstruction was demonstrated. While this algorithm is inherently slow, it can be used to include various experimental parameters like the walk-off due to birefringence and the spectrum analyzer instrumental function or even the dispersion and higher order effects when it is combined with the beam propagation method. The development of this technique is the part of ongoing work and it is tested with experimental spectrograms acquired by MOF FROG method that will be the subject of another paper.
This project was supported by the Grant Agency of the Academy of Sciences of the Czech Republic (A1067301) and the Czech Science Foundation (GA102/02/0779).
References and links
1. R. Trebino ed., Frequency-resolved optical gating: The measurement of ultrashort laser pulses, (Kluwer Academic Publishers, Boston/Dodrecht/London2002). [CrossRef]
2. R. Trebino, K.W. DeLong, D.N. Fittinghoff, J.N. Sweetser, M.A. Krumbuegel, and D.J. Kane, “Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using Frequency-Resolved Optical Gating,” Review of Scientific Instruments 68, 3277–3295 (1997). [CrossRef]
3. K.W. DeLong, R. Trebino, J. Hunter, and W.E. White, “Frequency-resolved optical gating with the use of 2nd-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994). [CrossRef]
4. M.D. Thomson, J.M. Dudley, and L.P. Barry, et al., “Complete pulse characterization at 1.5 mu m by cross-phase modulation in optical fibers,” Opt. Lett. 23, 1582–1584 (1998). [CrossRef]
5. J.M. Dudley, L.P. Barry, and J.D. Harvey, et al., “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999). [CrossRef]
6. 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, 490–492 (1999). [CrossRef]
7. P. Honzatko, J. Kanka, and B. Vrany, Manuscript in preparation. Some details were presented in P. Honzatko, J. Kanka, B. Vrany, “Alignment-free CPM FROG based on a microstructure optical fiber,” ETOS 2004, Cork, Ireland.
8. V.V. Bryskin and M.P. Petrov, “Passive mode locking in a birefringent fiber laser,” Tech. Phys. Lett. 22, 153–155 (1996).
9. M. Horowitz and Y. Silberberg, “Nonlinear filtering by use of intensity-dependent polarization rotation in birefringent fibers,” Opt. Lett. 22, 1760–1762 (1997). [CrossRef]
10. A.S. Kaminskii, E.L. Kosarev, and E.V. Lavrov, “Using comb-like instrumental functions in high-resolution spectroscopy,” Meas. Sci. Technol. 8, 864–870 (1997). [CrossRef]
11. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087–1092 (1953). [CrossRef]
13. S. Geman and D. Geman, “Stochastic relaxation, gibbs distributions, and the bayesian restoration of images,” IEEE Trans. on Pattern Analysis and Machine Intelligence 6, 721–741 (1984). [CrossRef]
14. M.A. Franco, H.R. Lange, J.-F. Ripoche, B.S. Prade, and A. Mysyrowicz, “Characterization of ultra-short pulses by cross-phase modulation,” Opt. Commun. 140, 331–340 (1997). [CrossRef]
15. See pages of the GNU Scientific Library Project: http://www.gnu.org/software/gsl