In this paper, we use a genetic algorithm and pulse-propagation analysis to design and optimize optical parametric oscillators based on soft-glass microstructured optical fibers. The maximum parametric gain, phase-match, walk-off between pump (1560 nm) and signal (880 nm) pulses, signal feedback ratio and signal-pump synchronization of the cavity are optimized. Pulse propagation analysis suggests that one can implement a fiber optical parametric oscillator capable of generating approximately 200-fs pulses at 880 nm with 43% peak-power conversion, high output pulse quality (time-bandwidth product ≈ 0.43) and a wavelength tuning range that is limited only by the glass transmission windows.
©2010 Optical Society of America
Fiber optical parametric oscillators (FOPOs), especially those based on microstructured optical fiber (MOF), are attractive sources of tunable, coherent radiation [1–8]. FOPOs can be compact, they cost less than crystal-based OPO systems and, unlike most lasers, FOPOs are capable of generating coherent light with broad continuous wavelength tunability. The use of optical fiber ensures the beam quality of the output light and microstructuring provides significant control of dispersion for four-wave mixing (FWM) phase matching across wide wavelength ranges.
Previous FOPO work, to the best of our knowledge, was based on the use of optical fibers made from silica glass [1–8]. Over the past few years, MOFs based on non-silica glass (soft glass) have become a useful platform for nonlinear applications [9–14]. Soft glasses have two major advantages for this application. Firstly, their broad transmission windows extends the operation wavelength range into the mid-infrared . Secondly, their high linear and nonlinear refractive indices increase the gain per unit length by orders of magnitude compared to silica glass .
In this paper, we propose a soft-glass-based MOF that is designed for a pulsed 880 nm FOPO that is synchronously pumped at 1560 nm. The choice of wavelength serves to highlight a viable alternative source for the 880 nm femtosecond Ti:sapphire laser for many applications such as chemical or biological detection [16, 17]. Given an optimized fiber design, we simulated the pulse evolution in a FOPO cavity using the generalized nonlinear Schrödinger equation. By studying the pulse evolution, one can optimize cavity parameters such as feedback ratio and synchronization to obtain a globally optimized FOPO system for a particular application. To the best of our knowledge, neither the optimization nor system simulation approaches have been applied to FOPO systems previously.
To date, most work on fiber design and optimization has focused on controlling the wave-length(s) at which the group-velocity dispersion is zero (ZDW) and on minimizing the slope of the group velocity (with respect to wavelength) . The standard approach is to assume that the local dispersion of the optical fiber is described by a Taylor-series approximation of the propagation constant, β(ω). One can then model the FOPO using a set of coupled wave equations . The adaptive range of this combination of approaches is limited for two reasons. Firstly, truncating the Taylor series fails to follow the true dispersion when the frequency shifts between pump the signal/idler are large. For large frequency shifts, as is the case for state-of-the-art FOPOs, the importance of higher-order terms of β(ω) increases significantly, and each term must be incorporated into any optimization procedure. Secondly, the coupled wave equations, which usually only consider the fields at pump, signal and idler wavelengths, are not suited to simulating complex nonlinear processes with dynamic frequency shifts or large spectral broadening, such as supercontinuum generation (SCG).
In most cases, it is difficult to include Raman scattering into the coupled wave equations, especially when simulating short pulses where the convolution of the Raman response function and the field should not be simplified. To address these two points, we use an eigenvalue solver to obtain β(ω), and we use the generalized nonlinear Schrödinger equation  to model pulse propagation. The result is a model that accurately describes FOPO performance and works equally well for small or large frequency shifts.
Previous reports have shown that genetic algorithms (GAs)  are useful for tailoring the dispersion and nonlinearity [13, 21] in an optical fiber and for optimizing an optical parametric amplifier . We use a GA to converge on the design of a MOF for large pump-signal frequency shifts. The GA optimizes not only phase matching but also the parametric gain and walk-off between pump and signal pulses. The pump and signal wavelengths used here (1560 nm and 880 nm respectively) serve as an example of a large frequency shift.
In Section 2, we discuss the details of fiber design including initial fiber geometry, the choice of fitness function and the resulting GA-optimized structure. In Section 3, we describe the FOPO configuration and simulation and we discuss the optimization of synchronization and signal feed-back ratio. Finally, we summarize the results of this study.
2. Fiber design and genetic algorithms
To model the FOPO, we first consider the FWM gain, which is given by 
where γ is the nonlinear coefficient, P 0 is the pump power and κ = βs + βi − 2βp + 2γP 0 is the phase mismatch. The quantities βs,i,p are the mode propagation constants for the signal, idler and pump fields, respectively, within the optical fiber.
To achieve maximum gain, it is critical to maximize γ and minimize κ. Note that κ is also a function of γ. Besides maximizing the gain, one also must consider the walk-off between the pump and signal pulses as they propagate through the optical fiber. This is especially important for large frequency shift and ultrashort pulses. Thus, the group velocities β 1 −1 (where βi = ∂iβ/∂ωi) must also be included in the optimization. To design a fiber structure optimized for highest output efficiency based on these relations, one has to search the parameter space of β (ωs,i,p) and γ for an optimal point. However, it is not obvious how best to search for a structure because each structural parameter correlates to all β and γ. To overcome this obstacle, we use a GA to optimize the fiber structure.
2.1. GA Background
GAs simulate natural selection. They start with a group of randomly distributed parameters and evolve them according to a design condition usually called the fitness function . There are different types of GAs (details on GAs can be found in the literature ). In this paper, we use one that operates with real-valued parameters . In order to use a GA for fiber design, an initial fiber structure with a certain number of free parameters and a fitness function must be chosen. For our problem, we start with the structure shown in Fig. 1 which has flexible dispersion and nonlinearity control abilities and has been used previously for the design of MOFs for SCG . The free parameters of the structure are R1, R3, R4, r1, r2 and L1 as shown in the figure. The other parameters are functions of these free parameters. The actual fiber structure will be surrounded by cladding but here, since the cladding is far from the core region and the modelling shows it has insignificant influence on the core modes, it is simplified to thin struts to avoid the influence of cladding modes in the modelling.
The choice of the fitness function requires some analysis of the FWM process that underpins the operation of the FOPO. Firstly, the fitness function for our GA model needs to be continuous so that it can converge. Secondly, since we wish to maximize γ and minimize both κ and walk-off Δβ 1 where Δβ 1 = β 1,s − β 1,p, one obvious choice of fitness function F is
In this way, the larger the fitness the higher parametric gain and the smaller walk-off will be. However, from a practical perspective, this fitness function can converge to extremely small values of |κ|or |Δβ 1| since either of them will lead to infinite fitness value. To avoid this undesirable behavior in optimization, we set limits to both of them. As an example, if we target the optimized FWM gain to be larger than 97% of the maximum achievable gain (perfect phase matching):
Hence, we define κ′ as
and use it in the fitness function instead of κ.
Similarly, we define a walk-off length as the length after which two pulses completely separate (assuming the pulse width is T 0 which corresponds to the width where intensity drops to 1/e 2 and the pulses almost stop interacting) and we want the fiber length to be a quarter of this length (25% of total walk-off). For example, for a fiber length of 3 mm (corresponding to approximately π nonlinear phase change in one of our test fibers when pumping with pulses with 5-kW peak power), we have
We define Δβ′1 as follows and use it in the fitness function instead of Δβ 1:
With these choices and definitions, we can set the target condition for our GA to
In this paper, we use a FWHM = 881 fs (T 0 = 500 fs) hyperbolic secant pulsed pump source with 5 kW peak power, which gives F target ≈ 1 × 107 ms−1W−1.
It is important to note that the mode propagation constant, β(ω), cannot be freely chosen. The bulk material properties are fixed for any given glass type, whereas we have some freedom to modify the waveguide contribution to β(ω) through the design of the waveguide. The goals for phase matching and walk-off may not always be achievable. The idea of setting these targets is to explore how closely they can be achieved via the use of flexible fiber designs and GA modelling techniques.
The GA model uses 1000 samples as its initial population (corresponding to 1000 structures with different choices of values for the free parameters). It evaluates the fitness function for each by solving the eigenmodes at different wavelengths using a finite element method and calculating β and γ. A type of tellurite glass (70TeO2−10Na2O−20ZnF2)  is used, which has nonlinear refractive index n 2 about 25 times larger than that of silica glass at 1550 nm .
2.2. GA results
The GA modeling was manually stopped after 6 generations since the maximum fitness had saturated. The average fitness of the samples in each generation increases stably as the generation evolves but the maximum fitness among each generation saturates at 3.2 × 106 after just the second generation (shown in Fig. 2a). There are several possible reasons for this. Firstly, the bulk material properties of the glass limit the range of obtainable fitness parameters. Secondly, the size of the population is large, increasing the likelihood that the GA yields structures with fitness values reach the limit. Finally, there can be many localized maxima of fitness within the parameter space that prevents the population from reaching higher fitness structures. Note also that we observe different saturation behavior with the same fitness function for different optimization wavelengths. The structure with the highest fitness value in the last generation is shown in Fig. 2b. Its corresponding parameter values are: R1 = 1.82 µm, R3 = 2.95 µm, R4 = 6.56 µm, r1 = 0.885 µm, r2 = 0.253 µm and L1 = 1.15 µm.
By studying the statistics of the population, we have a basic understanding of the sensitivity of the structure to the fitness and we therefore know the required fabrication tolerances. We examined several fiber designs from the last GA generation whose fitness values are larger than 90% of the maximum fitness. For these designs the standard deviations of the various free parameters are: σR1 = 4.4%, σR3 = 8.4%, σR4 = 8.5%, σr1 = 27.3%, σr2 = 11.2% and σL1 = 6.0%. From these results, we conclude that if the distortion during fabrication is kept under 4.4%, the error in the final fiber will be less than 10% in terms of fitness values which is corresponding to 10% in the product of parametric gain and walk-off length.
Figures 3a and 3b show the fiber properties (β, β 1 and γ) and FWM gain map when the fiber is pumped at 1560 nm with a 5 kWsource. It is interesting that a broad gain bandwidth is obtained automatically even though it is not considered explicitly in the fitness function. While the infrared end of the gain map is not shown in the figure, the gain extends to 8 µm. We discovered that a broad and relatively flat gain bandwidth is likely to be obtained by the GA. When the pump wavelength is not too far from (within the range where structure dispersion can cancel the material dispersion) the ZDW of the material (ZDW ≈ 2.2 µm for our glass) and the pump power is relatively high (≈ 5 kW in this case), the GA tends to converge at the point where the ZDW is close to the pump wavelength. In this situation, phase matching is achieved mainly through high-order dispersion. In our specific case, one observes an s-shape in the gain map (see the boxed region in Fig. 3b), which is due to higher-order β terms beginning with β 6. If β 2 is positive, which is the situation here, one cannot achieve phase matching without negative-valued high-order terms. Furthermore, in order for 4th, 6th, and higher-order terms to be significant, one must reduce the magnitude of β 2, which, in turn, leads to a large bandwidth . Note that, due to the symmetry of the FWM process, odd orders of β terms do not contribute to phase matching.
The ultra-broad bandwidth indicates the FOPO based on this fiber can be and should be tuned via a band pass filter to narrow the seed signal in the cavity. In the next section, we discuss the detailed design and optimization of a FOPO.
From the modeling results, we find that dispersion and nonlinearity are insensitive to the features of the outermost ring. Also, the variation in size of the inner ring of holes is not necessary. Therefore, for future work, it may be possible to simplify the initial structure by removing the outer-most ring of holes and using a single parameter for all six inner holes. This would reduce the modeling time and simplify the fabrication process.
3. FOPO modeling
We simulate the parametric oscillation process through three major steps. Firstly, we simulate pump pulse propagation through the fiber. Secondly, we filter the spectrum of the output and keep only the signal pulse. Finally, we combine the filtered output with a new pump pulse and let them propagate again through the fiber. We repeat step two and three until the output becomes stable. To the best of our knowledge, this modeling method has not been used for simulating a FOPO. The pulse propagation model is time consuming but we are able to model a FOPO with 100 passes on a multi-core desktop PC within a few hours using our own fully parallelized pulse propagation model written in C/C++. The pulse propagation model is built on the nonlinear Schrödinger equation:
where A(z, t) is the envelope of the electric field of a propagating pulse. The propagation constant β(ω) of fiber modes for specific fiber geometries can be pre-calculated using the finite element method (FEM). The combined material and confinement loss is given by α, and γ is the effective nonlinear coefficient of the fiber. The Raman response function is given by R(T), where we assume a Raman fraction (fR) of 0.064 for the tellurite glass  used in this work.
The FOPO configuration is shown in Fig. 4. The tunable stage with mirrors M1 and M2 is for cavity-pump synchronization. The tunable band-pass filter (BPF) has a 20 nm bandwidth and a Tukey-window shape  to minimize effects caused by the frequency response of the filter. The short-pass filter (SPF) at the output is used to filter the pump and transmit the signal at 880 nm. A beam splitter (BS) is used to combine an incoming pump pulse with the filtered signal and direct it towards the fiber. The fiber length used in this simulation is 3 mm. The length is selected by considering the total loss and gain in the whole FOPO system and the spectral shape degradation of the pump due to self-phase modulation, which corresponds to a nonlinear phase change of ~ π. The fiber itself is assumed to be lossless because of its short length.
The simulation is initialized by feeding a pump pulse with a quantum level of noise into the fiber leading to the generation of signal and idler spectral sidebands. The first pass forms the initial seed for feedback via the BPF. The filtered seed is combined synchronously with the next pump pulse and re-coupled into the fiber. The process is repeated for 100 round trips, by which the output is usually stabilized. To study the conditions leading to the generation of optimized output pulses, we explore the variations in the FOPO output pulses with different feedback ratios and time offsets between the signal and pump. The feedback ratio is the fraction of the output signal that is coupled back into the fiber after each pass.
3.1. Optimization of synchronization
Figure 5 shows the predicted output pulse duration and time-bandwidth product (TBP) as a function of temporal offset (synchronization) for the proposed FOPO. Synchronization between signal and pump pulses can be easily achieved in the laboratory by adjusting the cavity length. However, the sensitivity of the system to synchronization and the best synchronization state for a FOPO system are not always obvious. In the following paragraphs, we discuss the impact of poor synchronization on the output of a FOPO.
Figure 5a shows the peak power of the output pulses averaged over the last 20 passes of the simulation. The feedback ratio of this set of data is fixed to 0.5. The curve in the figure is asymmetric, reflecting combined effects of frequency chirp, pump depletion and walk-off between pump and signal pulses. The pump pulses acquire frequency chirp under their own SPM during which the signal pulses start to deplete the pumps at certain points while they walk across the pump pulses. As the offset time changes, the signal pulses overlap with different frequency components of the pump due to the frequency chirp and lead to deviations in phase match. The output peak power reaches a maximum when the offset time is approximately 0.55 ps, consistent with the fact that signal pulses propagate slower than the pump pulses.
We use the TBP to qualitatively assess the quality of output pulses. For transform-limited Gaussian and hyperbolic secant pulses, TBPs are 0.44 and 0.315, respectively. However, it is important to keep in mind that the TBP is calculated using full width at half maxima (FWHM), and thus can neglect pedestals or features that do not rise above half of the maximum of the measurement. Figure 5b shows the TBP of the pulses corresponding to Fig. 5a also averaged over the last 20 passes. The dip in TBP at 0.5 ps is close to that of a transform limited secant pulse. However, in this particular case, the output pulses have large spectral pedestals below the half maxima. Apart from the dip, the overall TBPs for different offset times are fairly constant at ~0.45, which indicates the synchronization does not influence pulse quality over the range of oscillation from 0.48–0.7 ps.
From our simulations, we conclude that the best synchronization for this particular FOPO configuration is with a time offset of 0.55 ps, which corresponds to the peak of the signal pulse occurring slightly earlier than that of the pump pulse.
3.2. The effects of feedback ratios
Figures 6a and 6b show the peak power and TBP for different feedback ratios with the temporal offset fixed to 0.55 ps. This information can provide guidelines to help choose optical components as well as to understand the potential influences from other nonlinear mechanisms such as supercontinuum generation. In Fig. 6a, one observes a threshold of feedback ratio of 0.3, with the output peak power reaching a maximum at 0.5. For feedback ratios larger than 0.5, the peak power decreases slightly. This results from the temporal broadening of the signal pulse as the feedback ratio increases and the BPF filters more power beyond its pass band. For feedback ratios greater than 0.7, the average output peak power starts to change irregularly from pass to pass. This effect will be discussed later in this section.
The corresponding TBP curve is shown in Fig. 6b. As the feedback ratio increases, we see that the TBP reaches a local minimum of 0.42 at a feedback ratio of 0.4. The TBP then increases to a maximum of 0.77 at a feedback ratio of 0.75. After reaching a maximum it also starts to behave irregularly. This is explored further below.
To convey a deeper understanding of the variation in behavior of the output pulses, Fig. 7 shows a few samples of the temporal and spectral evolution of the pulses for several different feedback ratios. The results show the very interesting evolution of the pulses within the cavity as a function of the number of passes. The signal pulses start from noise with a slightly lower frequency for the first few passes compared with the steady state. This is because the phase-matching condition changes slightly once the signal achieves enough power to begin to deplete the pump power. For a feedback ratio of 0.35, we see in Fig. 7a that the initial increase in signal pulse power is slow, but the system converges to a stable steady-state output power. When the feedback ratio is slightly higher, as in Fig. 7b, the signal power rises rapidly, overshooting the eventual steady-state value, and then converges to that steady-state value after a period of fluctuation. Further increases in the feedback ratio lead to the situation where the output power fluctuates regularly (Fig. 7c) and then irregularly (Fig. 7d). This behavior is due to pulse spectral broadening followed by BPF filtering. This broadening-filter loop leads to the oscillations of the output power. The high-feedback case in Fig. 7d also corresponds to the situation where the signal pulses behave like supercontinuum generation. Due to the complex nature of SC generation, the spectrum transmitted by the BPF is different for each pass.
It is clear from Fig. 7 that for a stable FOPO system the feedback ratio should be kept at or below 0.7. The unstable FOPO may have some potential use since the maximum peak power is larger than the stable case. For our proposed system, the best choice of feedback ratio which combines high output peak power (43% conversion efficiency) and small TBP (≈ 0.43) is ~0.5.
We simulated our proposed system incorporating the optimized synchronization of 0.55 ps offset time, and the optimized feedback ratio of 0.5. The stable output pulse shape and spectrum of the fiber and the FOPO are shown in Fig. 8 and 9, respectively. The pulse is Gaussian on top of a small pedestal. The peak power exceeds 2 kW and the FWHM is ~200 fs.
There are some limitations of our analysis. Firstly, the Sellmeier equation used here for tellurite glass may not be valid for the entire wavelength range because the experimental data used to generate the coefficients is ranging from 600 ~ 1050 nm . The ultra-broad bandwidth is calculated based on the Sellmeier equation, which implies that the actual bandwidth of a real fiber may not be as large as predicted. Secondly, the loss of the fiber is not included in the simulation, although the transmission losses through 3 mm of fiber will be small. Even for the idler wavelength (6.8 µm), which is beyond the edge of the transmission window for tellurite glass (normally 4–5 µm ), we expect the transmission losses after 3 mm to be less than 10 dB. However, in practise, we should expect to see a difference in the generation of the signal and idler pulses. The purpose of this work is to combine GA with beam propagation simulation to design a FOPO. To achieve a fully optimized FOPO, a few iterations of design and fabrication will be required. Once fabricated, these fibers will allow us to gather more information about the fiber’s absorption spectrum, Raman spectrum and nonlinear response as a function of wavelength allowing us to refine our prediction. Note that simply due to the Fresnel reflection, the maximum coupling efficiency is around 89%. Other optics within the FOPO system, such as mirrors beam splitters and filters, contribute additional losses such that not all the feedback ratios are accessible. Our analysis is relevant for bulk cavities which exhibit high losses as well as all fiber configurations which can greatly reduce intra-cavity losses.
We have, for the first time, modeled a soft-glass MOF-based FOPO where the fiber was designed and optimized through a GA and the system was optimized through a systematic pulse propagation analysis. By optimizing parametric gain, phase mismatch, walk-off between pump and signal pulses, feedback ratio and offset time for synchronization, the FOPO is capable of providing approximately 43% peak power conversion with only relatively little pulse quality degradation (TBP ≈ 0.43). We also discovered as a result of our optimization that an ultra-broad gain bandwidth (approximately 7 µm wide) is expected. Such a system would be very attractive for many applications that require wavelength tunable femtosecond-pulses from the near- to mid-IR.
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