1. Introduction

Tunable all-optical wavelength converters (AOWCs), with operation speed beyond the limits of electronic devices and agile output wavelength tuning will be essential for future flexible transparent WDM networks [1,2]. Among various kinds of AOWCs one-pump fiber parametric wavelength converters (1P-FPWCs) based on degenerated four-wave mixing (FWM) in optical fibers have some important advantages. First, because of the femtosecond response time associated with the third order non-linearity, 1P-FPWCs are bit-rate independent for any practical optical communication system [3,4]. Second, 1P-FPWCs may be applied to signals with new modulation formats, like the differential phase-shift keying (DPSK) that improves the receiver sensitivity and constitutes a good choice for long-haul transmission systems [5]. Third, the extinction ratio of the converted signal is very close to that of the input signal, allowing the converted signal to propagate for distances similar to the ones that could be achieved by the original signal [6]. Lastly, only one pump is required in a 1P-FPWC and for a fixed input signal wavelength, by tuning the pump, the converted wavelength can be tuned over a range two times larger than that of the pump [2,3], which is an important merit because the tuning range is often limited by that of the EDFA boosted high power pump source.

The key performance characteristics of 1P-FPWCs, such as conversion gain, tuning range, required fiber length and pump power depend on the properties of the media fiber. An ideal fiber for achieving broadband and high efficient wavelength conversion should have a high nonlinearity and a low group velocity dispersion (GVD) with a low dispersion slope (S). Today, with the development of soft glass optical fibers, the effective fiber nonlinear coefficient can be enhanced by a few orders of magnitudes [7–9]. But most of the soft glasses, such as lead-silicate, chalcogenide, tellurite and bismuth oxide glasses, have a very highly normal material dispersion at 1.55mm, thus not suitable for telecom band FWM devices [10]. Holey fibers (HFs) with flexible geometry allow the highly normal material dispersion to be overcome and GVD to be tailored [10]. Thus HF technologies combined with soft glasses offer great potential for satisfying the above requirements. To explore the possibilities, the design methods of soft glass HFs have been intensively studied recently [10–12]. But these methods are often very time consuming and not readily applicable to tunable 1P-FPWCs.

To resolve this problem we develop a hybrid approach combing downhill simplex algorithms (DHSAs) with FWM modeling. As far as we know, it is the first use of optimization algorithms to design HFs for tunable 1P-FPWCs. The DHSA is one of the most widely used methods for nonlinear unconstrained optimization [13]. This direct-search method attempts to minimize a scalar-valued nonlinear function of N real variables using only function values. It maintains at each step a simplex and iteratively updates the worst vertex by four operations: reflection, expansion, contraction and shrinkage. By repeating this series of operations, the method finds the optimal solution. While DHSAs are relatively mature algorithms, there is still one major problem to apply them to the specific problem. In DHSAs, the goodness or badness of the vertex is quantitatively described by the fitness function (FF) value. Thus a simple but appropriate definition of the FF is vital for a fast and effective optimization algorithm. In methods proposed before, the FF evaluation often takes a lot of time. For example, in Refs. [10]. and [11] the FF to be minimized is defined as the sum of the GVDs at N different wavelengths selected from the FWM band. Thus the evaluation involves calculation of the GVDs at the N wavelengths, which incurs hundreds of times of mode solving in every iteration, resulting in a heavy computation load, especially when the FWM band is broad. What is more, the key performance characteristics of 1P-FPWCs cannot be designated before or known after the optimization.

To overcome these obstacles we first explore the relations between the fiber properties and the characteristics of 1P-FPWCs through FWM modeling, and then use the DHSA to optimize the fiber structure exploiting these relations. The computation load is greatly reduced owning to this hybrid approach. More importantly, this approach also enables an inverse design of the HFs according to the pre-set conversion gain, tuning range, fiber length and pump power. Another important issue in the HF design is that the fiber dispersion properties are often sensitive to the fabrication errors [10–12]. But as far as we know, no study has been carried out to understand the sensitivity of the 1P-FPWC’s characteristics to the HF fabrication errors. So in the end of the paper we analyze this problem and find a method to mitigate the impact of the fabrication errors.

2. Four wave mixing modeling

In a previous work, we have systematically analyzed the effects of the fourth order dispersion on the tuning ranges of 1P-FPWCs and presented a method to fully utilize the fourth order dispersion to maximize the pump tuning range for a given pump power. We also deduced some useful formulas revealing the relations between fiber dispersion coefficients and the maximal pump tuning range [14]. But those formulas are obtained under an assumption that only highly nonlinear fibers (HNLFs) with one zero dispersion wavelength (ZDW) are used. Different from HNLFs, dispersion-flattened HFs with a low GVD and a low dispersion slope often exhibit two ZDWs in the telecom band [10]. This exotic dispersion property is beneficial to broadband and highly efficient FWM process, but makes the formulas in [14] not applicable any more. To solve this problem, we are first going to explore the relations between the properties of HFs with two ZDWs and the characteristics of 1P-FPWCs through FWM modeling.

For 1P-FPWCs, the wavelength conversion gain G can be derived analytically when the pump depletion and fiber loss can be neglected and is given by [3]

From Eq. (6), for a given input signal frequency ${\omega }_{\text{s}}$, when ${\omega }_{\text{c}}={\omega }_{\text{s}}$, $\Delta {\omega }_p$, equal to ${\omega }_{p\text{1}}-{\omega }_{p\text{2}}$, is maximized and given by

Note that for dispersion-flattened HFs with two ZDWs $-{\beta }_2/{\beta }_4>0$ is a typical phenomenon because only when $-{\beta }_2/{\beta }_4>0$ can the dispersion curve maintain a “u” or “n” type profile with two ZDWs [10]. When ${\omega }_{\text{c}}={\omega }_{\text{s}}$, ${\omega }_p$ can be tuned in the range of

The deduction process is similar as that in [14], thus omitted here for conciseness. From Eqs. (7) and (9) it is obvious that the center of the tuning range locates at ${\omega }_c$, while the tuning range $\Delta {\omega }_p$ depends on $\gamma P_0$ and ${\beta }_{2,4}^{}|{}_{\omega \text{=}{\omega }_c}$. As will be seen later, we can define a simple but effective FF based on these relations.

Note by Taylor expansion, ${\beta }_2(\omega )={\beta }_2+{\beta }_4{(\omega -{\omega }_c)}^2/2$. Thus the two ZDWs (${\omega }_{\text{ZDW1,2}}$) can be obtained from solving the quadratic equation ${\beta }_2(\omega )=0$ and are given by

Formulas (8) and (10) show that ${\omega }_c$ is not only the midpoint of ${\omega }_{ZDW1,2}$ but also the center of the pump tuning range. So by selecting ${\omega }_c$, instead of ${\omega }_{ZDW1,\text{2}}$, as the Taylor expanding center the truncation error of the formulas can be reduced to the maximal degree, and, as will be seen later, the formulas are accurate enough even when the FWM band is about 230nm wide, more than 1.5 times larger than the sum of S, C and L bands, thus sufficient for most practical WDM network applications. But it is noteworthy that the above formulas deduced from Eq. (3) are approximate and cannot be applied to any wavelength range. If the wavelength range is too broad to neglect the truncation error, the fifth (or even higher) order dispersion must be considered. For example, in recent experiments utilizing degenerated FWM to realize band translation between the near-infrared and visible spectral ranges, terms up to the eighth order dispersion have to be considered in order to provide an accurate description of the phase matching [16,17]. In this case no simple analytic expressions can be obtained for $\Delta {\omega }_p$ or ${\omega }_{ZDW1,2}$ and the dispersion curve may be asymmetric about ${\omega }_c$ and even has more than two ZDWs [16–18].

3. Inverse design method based on downhill simplex algorithm

Based on the above relations we seek to use the DHSA to design the fiber. We choose to define the FF as follows

Although L can be designated at will and a smaller L is often desirous to reduce the cost, mitigate the impact of the fiber longitude non-uniformity [21,22], refrain stimulated Brillouin Scattering [7] and achieve a compact footprint [7,8], it is noteworthy that, from formula (5), $P_0$ is inversely proportional to L for a given $G_{\min }$. Thus if L is too small, the corresponding $P_0$ will be very large. Thus an upper limit can be imposed on $P_0$ in the optimization. The required $P_0$ for a specific HF is $P_0$ $=\sqrt{G_{\min }}/\gamma L$, where $\gamma \text{=2}\pi n_2/\lambda A_{eff}$ ($n_2$ and $A_{eff}$ are the nonlinear refractive index and fiber effective area, respectively). Note that no extra time-consuming mode solving is needed after the limit is imposed because $A_{eff}$ and γ are obtained in the FF evaluation. Finally it is noteworthy that Eq. (10) is not the only definition for the FF that could be used. Other definitions may also be used as long as they are consistent with the relations and converge as the fitness decreases.

4. Fiber design using the hybrid approach

When using the DHSA, the choice of the initial fiber structure is as important as the definition of FF, which qualifies the suitability of the properties of the fiber for the pre-set device characteristics. The design of the initial structure should also take account of confinement loss (CL) and fabrication feasibility. From previous reports, we know that the lead silicate glass (Schott SF57) HFs can provide both a high nonlinearity and a flat low dispersion and the CL can be as low as 0.077dB/m [10]. Furthermore this commercial glass was also already employed in the fabrication of complex HF preforms showing up to 160 air holes [23]. Based on this knowledge we choose an initial fiber geometry similar to the one proposed in [10], as shown in Fig. 1 . There are five rings of hexagonally arrayed equally sized air holes to keep up the feasibility of their fabrication and a small CL. We assume that the input signal wavelengths are 1530, 1550 and 1570nm, respectively and set the frequency tuning range of the converted wave $\Delta {\text{f}}_{\text{i}}$ at 30THz (the corresponding converted wavelength tuning range is about 230nm at 1550nm band); the conversion gain $G_{\min }$ at –2dB (high conversion gain is not essential for wavelength converters); the fiber length at 3m. From formulas (5)–(9) the expected $\gamma P_0$, ${\beta }_{2,4}^{aim}$ are 0.265/m,$-2.02\times {10}^{-27}{s}^2/m$ and $1.65\times {10}^{-54}{s}^4/m$, respectively. ${\beta }_3^{aim}$ is set at a very small value $\text{0 .1}\times {10}^{-\text{42}}{s}^{\text{3}}/m$. For comparison, the typical value of ${\beta }_3^{}$ for dispersion-flattened HNLFs is $\text{48}\times {\text{10}}^{\text{-42}}{s}^3/m$(S = 0.032$\text{ps}/n{m}^2/km$), two orders of magnitudes larger than ${\beta }_3^{aim}$ [24].

 

Fig. 1 Left: the structure of the HF. Right: the dispersion curve against wavelength for F1 (blue), F2 (red), and F3 (green).

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The optimized HF structural parameters were obtained quickly after several tens of objective function evaluations. The dispersion formula we used for the lead silicate glass is the same as that in [10]. The nonlinear refractive index of the glass is 40$\times {10}^{-20}{m}^2/W$. The weighting factor in Eq. (11) is set at 0.1. The results and corresponding parameters of the optimized HFs are listed in Table 1 . The relative differences between the ${\beta }_{2,4}^{}$s and ${\beta }_{\text{2,4}}^{aim}$s are below 1%. The ${\beta }_3^{}$s are at the order of $\text{1}\times {10}^{-\text{43}}{s}^{\text{3}}/m$. The dispersion curves of the optimized HFs are shown in Fig. 1. As expected these HFs exhibit two ZDWs centering on 1530, 1550 and 1570nm, respectively. The changes of $\Delta \beta$ against the pump wavelength are shown in Fig. 2(a) . To get a physical insight into the relations between the fiber’s dispersion coefficients and the 1P-FPWC’s characteristics, two black solid lines representing $\Delta \beta =0,-4\gamma P_0$ are also drawn in Fig. 2(a). The region between the two lines represents the exponential gain region, where${(\gamma P_{0}L)}^2\le G\le {\sinh }^2(\gamma P_{0}L)$. By optimizing ${\omega }_c$ and ${\beta }_{2,4}^{}$ the curve representing $\Delta \beta ({\omega }_p)$ vs. ${\lambda }_p$ can be retained in this region as long as possible, resultingin the maximal pump tuning range. Note due to the slight deviation of ${\beta }_{2,4}^{}$ from ${\beta }_{\text{2,4}}^{aim}$ the minimum points of the curves are slightly below the lower black solid line $\Delta \beta =-4\gamma P_0$. As seen in Fig. 2(b), this results in a slightly lower gain than $G_{\min }$. For comparison, $\Delta \beta$ is also calculated by the numerical method. In this method the calculation starts with a calculation of $n_{eff}(\lambda )$ over the whole tuning range using the mode solver, then $\Delta \beta$ by its definition (see Eq. (2)). As seen in Fig. 2(a) the results agree very well with each other.

Table 1. Structural and characteristic parameters of the HFs. r and Λ are the hole radius and pitch, respectively. γ and CL are caculated at 1550nm. The units for ${\beta }_{2,3,4}^{}$ are ${10}^{-27}{s}^2/m$, ${10}^{-\text{42}}{s}^{\text{3}}/m$ and ${10}^{-54}{s}^4/m$, respectively.

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Fig. 2 Left: change of Δβ against pump wavelength for F1 (blue), F2 (red), and F3 (green), calculated by analytic (dotted) and numerical methods (solid). The two solid black lines represent the boundaries $\Delta \beta =-4\gamma P_0,0$. Right: Conversion gain against pump wavelength for = 1530nm (blue), 1550nm (red), and 1570nm (green)

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The nonlinear coefficients at 1550nm are also listed in Table 1. Noting that γ is related to the wavelength, so the maximal and minimal γs over the FWM band are also listed. The minimum γ is adopted for the calculation of $P_0$ to warrant inequality (4) is satisfied over the whole FWM band. The resulted$P_0$s are 0.549, 0.586 and 0.624W, respectively, well below practical power levels, owing to the ultra-high nonlinearities of the soft glass HFs. For comparison, in an early experiment based on dispersion-flattened HNLFs with $\gamma {\text{=10W}}^{\text{-1}}{\text{km}}^{\text{-1}}$ and L = 115m, a pump power larger than 1W is required to achieve a 24nm wide pump tuning range [25]. The change of G against the pump wavelength is shown in Fig. 2(b). As expected, the pump wavelength tuning range is approximately 115nm wide, corresponding to a 230nm wide converted wavelength tuning range which is about 177% larger than the sum of S, C and L bands. G varies from –2.1dB to –1.1dB over the hole tuning range. The gain ripple is 1dB over the whole tuning range. The 0.1dB difference between the minimum gain and the expected $G_{\min }$ comes from the slight deviation of the optimized ${\beta }_{2,4}^{}$ from ${\beta }_{\text{2,4}}^{aim}$.

Figure 3 shows the change of Δβ against the pump wavelength when the pitch Λ is changed from its optimal value by ± 0.5% to ± 1% while the ratio of r/Λ is kept unaltered. To mitigate the impact of the variations (keep the same $\Delta {\omega }_p$ and $G_{\min }$), ${\text{P}}_0$ should be increased by about 4 or 8 times when the change Δ = 0.5% or 1% (as indicated by the two vertical and three horizontal dashed lines). But when the changes are in the opposite direction, adjusting the pump power cannot keep the same $\Delta {\omega }_p$ and $G_{\min }$. More specifically, when Δ = −0.5% (red solid line), the tuning range is decreased by about 47% (from 115nm to 61nm). When Δ = −1% (red dashed line), the whole curve remains outside of the exponential gain region, thus the HF is not suitable for tunable 1P-FPWCs any more. So in order to mitigate the impact of fabrication errors one can set r and Λ at values a little larger than the optimum ones, so that $\Delta {\omega }_p$ and $G_{\min }$ can be kept the same by adjusting the pump power when r and Λ deviate from the optimum values because of fabricate errors.

 

Fig. 3 Change of Δβ against pump wavelength when the pitch is optimal (green line) and altered from its optimal value by ± 0.5% (solid blue and red lines), ± 1% (dashed blue and red lines). The three horizontal black dashed lines represent $\Delta \beta =-4\gamma P_0$, when $P_0$ is adjusted to keep the same pump tuning range.

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5. Conclusion and discussions

In this paper we presented a novel hybrid approach combining the DHSAs with FWM modeling to design the HFs that are optimally suited for 1P-FPWCs based on degenerated FWM. By this approach the fiber structure parameters can be directly deduced from the pre-set device characteristics in a full-automatic way. This approach greatly reduces the computation time and is accurate enough for most 1P-FPWCs used in future broadband transparent WDM networks. It is noteworthy that the advantage of the hybrid approach we proposed results from simplifying the fitness function by FWM modeling, thus reducing the computation load. The same definition of the fitness function can also be used in other similar optimization algorithms, like genetic algorithms, thus expediting the process too. The sensitivity of the 1P-FPWC’s characteristics to the variations of the HF structural parameters is also investigated. The high sensitivity is an issue and challenge for the fabrication techniques. The dispersion flattened HFs proposed previously [10,11,26] also have the same problem. The reason is that the hole diameters of the dispersion flattened HFs are very small (~500nm) which is essential to obtain a large waveguide dispersion. We found in some cases the impact of the fabrication errors can be mitigated by setting r and Λ at values larger than the optimum ones in the fabrication stage and adjusting the pump power in situ later. Another effective method is to employ the microfludic HF technique. In the microfludic HF the small air holes can be replaced by the liquid-filled ones with much larger diameters [26], thus reducing the requirements for the fabrication techniques. Furthermore, the microfludic HF can provide some tunability by using different liquids and adjusting the liquid refractive index through thermo-optic effects [26]. So it is possible to realize in situ adjustments of the dispersion properties after the fabrication stage and make the HF more robust to the variations of the structural parameters. Using the hybrid approach to design such microfludic HFs, more time can be saved compared to other optimization algorithms proposed before because the times of the FF evaluation increase as the degrees of freedom increase (the liquid index is another free parameter). This is out of the scope of this paper and could be the subject of future work.