1. Introduction

The design of waveguides and optical fibers is crucial for achieving high performance in applications such as dispersion compensation [1], parametric amplification [2], wavelength conversion [3,4], and supercontinuum generation in both anomalous [5,6] and normal dispersion regimes [7,8]. One of the most important properties controlling the physical behavior of these optical systems, in both the linear and nonlinear regimes, is chromatic dispersion. The waveguide dispersion not only depends on its constituent materials, but also displays a high sensitivity to changes in geometry (see, e.g., [9]).

Different approaches have been followed to control dispersion properties. They range from the use of empirical relations involving the $V$-parameter, obtained for PCFs that consist of triangular grids of air holes on a pure silica matrix [10,11] to guided test-and-trial procedures based on, e.g., genetic algorithms [12] or topology optimization [13,14], which can become time-consuming if the parameter space to be explored is not restricted judiciously [15]. In general, the design process requires the minimization of a merit function. When the first derivatives of the merit function are available, gradient-based algorithms are certainly the first choice in multidimensional minimization, as they can dramatically reduce the number of evaluations of the merit function [16,17]. Some of those methods rely on iterative strategies that produce a series of local approximations of the merit function. It is worth noting here that the above methods are compatible with the use of approximate analytical expressions for the waveguide dispersion [9], as well as with any particular mode solver (for instance, [18,19]). Moreover, such theoretical results offer opportunities to calculate the merit function in larger neighborhoods of given values for the set of parameters and thus, to further reduce the number of iterations. In particular, the geometrical scaling symmetry exhibited by guiding systems under certain conditions [20] opens the possibility to adjust their optical properties, and has been essential in the design of PCFs [9,16].

In this paper, we generalize the geometrical — or external — scaling transformation presented in Ref. [20] by introducing a material — or internal — transformation by which the numerical aperture is modified, taking advantage of invariants of the scalar wave equation in non-dispersive media [21]. Next we derive a series of approximated analytical expressions that describe the propagation constant, the group index and the group velocity dispersion in terms of their counterparts in the unscaled system and the corresponding external and internal scale factors. Thus, the calculation of these properties for a given waveguide suffices to obtain analytically those corresponding to scaled versions of the initial waveguide, including both geometrical and material scalings, as we show in conventional fibers and PCFs. Finally, we prove the power of our new results to design photonic crystal fibers featuring different targeted dispersion curves, such as flattened profiles or featuring a high slope, in four iterations or less. These results show up the efficiency of our approach compared to pure brute-force strategies.

2. External and internal scaling symmetries in guiding systems

We start with the 2D wave equation in the weak-guidance approximation [22],

The solution for a more general transformation given by $n^2(\textbf {x}_{\mathrm {t}})\mapsto n^2_{[sr]}(\textbf {x}_{\mathrm {t}}) = r\,n^2(\textbf {x}_{\mathrm {t}}/s)$, namely, a simultaneous external and internal scaling transformation, is obtained just considering the variable changes $\textbf {x}_t\mapsto \textbf {x}_{\mathrm {t}}/s$ and $\omega \mapsto s\sqrt {r}\omega$:

In practice, the internal scaling transformation typically keeps unchanged the refractive index of one of the constituent materials, the reference material, $n_{\mathrm {ref}}$ (e.g., the refractive index of the cladding of a step-index fiber). This scaling transformation can be managed in the wave equation subtracting the term $(\omega /c)^2 n^2_{\mathrm {ref}}(\omega ) \Psi$ from both sides of Eq. (1),

For guiding systems described within the weak-guidance approximation, Eq. (5) can be simplified. According to the weak-guidance approximation, we can assume that $\beta (\omega )$ is similar to $\beta _{\mathrm {ref}}(\omega )$ [i.e., $\Delta \beta ^2(\omega ) \approx 2\beta _{\mathrm {ref}}(\omega ) \Delta \beta (\omega )$], and, in the same order of approximation, we can consider that the reference propagation constant is weakly dispersive [$\beta _{\mathrm {ref}}(a\omega )\approx a\beta _{\mathrm {ref}}(\omega )$], in such a way that the following expression is reached,

For the sake of completeness, we can also show the equivalent quadratic expression for the effective refractive index, $n_{\mathrm {eff}}$,

 

Fig. 1. (a) Effective refractive index of the fundamental mode of a SMF-28 fiber (thin dashed line), refractive index of the core (thin solid line) and the cladding (thick solid line). Variation of $n_{\mathrm {eff}}$ (b), $n_{\mathrm {g}}$ (c) and $\beta _2$ (d) when several scalings (keeping the refractive index of the cladding as the reference for internal scalings) are applied to the fiber: $(s,r) = (0.9,1)$ [blue], $(1.1,1)$ [orange], $(1,0.9)$ [green], and $(1,1.1)$ [red]; in all cases, both solving completely the mode problem (solid lines) and using the approximated analytical expressions given by Eqs. (10), (11) and (8) (dotted lines). The subscripts sc and unsc refer to scaled and unscaled, respectively.

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It is worth mentioning that Eq. (12) proves theoretically and generalizes the expression for global scaling in Ref. [16], and ultimately in Ref. [9]. What is more, the successful application of these expressions to the design of photonic crystal fibers [9,16] reveals that our results might be even valid beyond the weak-guidance approximation, in line with the dispersion estimate considered in Ref. [23] for a high-index-contrast integrated waveguide.

3. Scaling in photonic crystal fibers

To illustrate our approach in PCFs, here we will focus on triangular PCFs with circular holes, although, unlike other methods [10,11], the procedure can be easily adapted to other geometries with no extra computational effort. A triangular lattice is characterized by the hole radius, $a$, and the lattice period, or pitch, $\Lambda$. A direct application of the external and internal scalings parameterized by $s$ and $r$ to PCFs would lead to new PCFs with pitch $s\,\Lambda$, hole radius $s\,a$, and the material refractive indices — typically, silica, air or both — scaled by a factor $r$ [21]. However, PCFs allow controlling both external and internal scalings only by means of their geometry, which is in contrast to conventional fibers. This approach requires a careful application of the formulas derived in the previous section. In particular, an effective step-index-fiber model for PCFs [24,25] can be introduced to define the scaling parameters $s$ and $r$ properly. For such a purpose, we parametrize the PCF by the pitch and the hole-filling fraction of the photonic crystal cladding, $f = (2\pi /\sqrt {3})(a/\Lambda )^2$.

Now, the changes considered in the PCF are a global geometrical scaling, $s_g$, and a filling fraction scaling, $s_f$, in such a way that $(\Lambda , f)\mapsto (s_g \Lambda , s_f f)$. The relation between $(s_g, s_f)$ and $(s,r)$ can be determined by the homogenization of the photonic crystal cladding [24]. The core radius of the equivalent step-index fiber can be approximated by $r_0 = \Lambda - a$, with its core index being the same of the PCF matrix, $n_{\mathrm {co}}$, and its cladding index being a simple spatial average of the photonic crystal cladding materials, $n^2_{\mathrm {cl}} \approx (1-f) n^2_{\mathrm {co}} + f n^2_{\mathrm {ho}}$ [24,25], where $n_{\mathrm {ho}}$ is the refractive index of the material filling the holes. In this way, the square of the numerical aperture, $\mathrm {NA}^2 = n^2_{\mathrm {co}} - n^2_{\mathrm {cl}} \approx f (n^2_{\mathrm {co}} - n^2_{\mathrm {ho}})$ is proportional, at first order, to $f$ and, therefore, the material scale is just given by the change in the filling fraction:

 

Fig. 2. (a) Effective refractive index of the fundamental mode of a PCF with $\Lambda =2.3\,\mu$m and $a=0.3\,\mu$m (thin dashed line), refractive index of the solid core — and matrix of the cladding — (thick solid line) and average index of the cladding (thin solid line). Variation of $n_{\mathrm {eff}}$ (b), $n_{\mathrm {g}}$ (c) and $\beta _2$ (d) when several scalings (keeping the refractive index of the solid core as the reference for internal scalings) are applied to the fiber: $(s_g,s_f) = (0.9,1)$ [blue], $(1.1,1)$ [orange], $(1,0.9)$ [green], and $(1,1.1)$ [red]; in all cases, both solving completely the mode problem (solid lines) and using the approximated analytical expressions given by Eqs. (10), (11) and (8) and the auxiliary Eqs. (13) and (14) (dotted lines). The subscripts sc and unsc refer to scaled and unscaled, respectively.

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4. Design of PCFs

The optimization of the chromatic dispersion of a family of fiber designs requires the definition of a merit function evaluating the mismatch between the behavior that a particular structure presents and a target behavior. For instance, if we are interested in the optimization of the group velocity dispersion (GVD), a simple expression is the mean square difference in a certain frequency range ($\omega _1,\ldots ,\omega _{N_\omega }$),

To reach that minimum, a series of approximations of $\chi ^2$ can be sequentially built and minimized [16,17]. To be more precise, in the case of optimizing PCFs with regular cladding, and given a certain configuration defined by its pitch and hole radius, $(\Lambda ,a)$, its GVD in the frequency range of interest must be calculated using any mode solver available (which is, in our case, the one reported in [19]). Next, Eq. (8) provides excellent estimations of the GVD for scaled configurations, $\beta _{2[sr]}$, around the unscaled one, and allows to define an approximation of the merit function,

 

Fig. 3. (a) Dispersion of five optimized PCF designs [colored solid lines] achieved considering as target dispersions five flattened dispersion profiles with $\beta _2=0$, $\pm 30$ and $\pm 60$ ps$^2$km$^{-1}$, respectively, in the wavelength range $[1.4,1.7]\,\mu$m [black dashed lines]. (b) Evolution of the geometrical parameters during the optimization procedure of the examples shown in (a) [marked with the same colors] (see details in the text). (c) Chromatic dispersion curves [colored solid lines] obtained along the free-of-constraints optimization towards the target $\beta _2=-60\,$ps$^2$km$^{-1}$ [black dashed line]. Initial estimates based on Eq. (8) restricted to $s_f=1$ (i.e., only $s_g$ is adjusted) [green dotted line] and unrestricted (i.e., both $s_g$ and $s_f$ are adjusted) [orange dotted line].

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In order to illustrate our approach, we plot, in Fig. 3(a), five different examples of the design processes of PCFs with flattened dispersion profiles over $300$ nm, in the range $1.4\,\mu \mathrm {m} \leq \lambda \leq 1.7\,\mu \mathrm {m}$ [$\beta _2(\omega )=0$, $\pm 30$ and $\pm 60$ ps$^2$km$^{-1}$; dashed lines in Fig. 3(a)]. For all five cases, we start from the same inital configuration, hence showing this choice is not particularly relevant. In our calculations, we have used the Sellmeier coefficients for silica reported in Ref. [28]. In these examples, $7$ wavelengths in the above range are considered as the points used for evaluating the approximation of the merit function [Eq. (16)]. In order to ensure that the procedure converges properly, it is sometimes convenient to moderate the speed of convergence. In Fig. 3(b) [solid lines], the evolution of the design parameters at each step along the procedure is shown, in which we have allowed variations up to $30$ % in each step and we consider the process has converged when the predicted scalings $(s_f,s_g)$ would imply changes for the next iteration smaller than $0.1$ %. Even so, in those cases with $\beta _2(\omega )=0$, $\pm 30$ and $+60$ ps$^2$km$^{-1}$ as target, convergence is achieved in four iterations or less.

Figure 3(b) also shows that, when $\beta _2(\omega )=-60\,$ps$^2$km$^{-1}$ is targeted, the convergence process do not initially approach the optimal configuration, which is in contrast to the other optimizations. Although after eight iterations good convergence is still achieved, the evolution has not been as fast as in our other designs. However, if no restrictions are imposed on the scale factors at each iteration, convergence is achieved in less than four iterations, as shown in Figs. 3(b) [blue dashed line] and 3(c) [colored solid lines]. To highlight the importance of the material scaling transformation, we also compare in Fig. 3(c), for the first iteration, the estimated dispersion profiles obtained through only geometrical scaling [green dotted curve] and both geometrical and material scalings [orange dotted curve]. This example also points out the usefulness of our scaling transformations in any waveguide design procedure for finding appropriate approximations.

Finally, we also apply our approach to triangular PCFs with a high and negative dispersion slope, namely, one order of magnitude larger than in standard fibers. The potential applications of these chromatic dispersion profiles range from third-order-dispersion compensation [29] or spectral shaping [30] to dispersive wave emission in the normal [8,31] or anomalous dispersion regime [32], or novel bound states in mode-locked fiber lasers [33]. Accordingly, we set $\beta _2=\pm 30$ ps$^2$km$^{-1}$ at $1.55\,\mu$m and $\beta _3 = -1$ ps$^3$km$^{-1}$ in the range $1.53\,\mu \mathrm {m} \leq \lambda \leq 1.57\,\mu \mathrm {m}$ as our new targets and repeated the process explained previously. The optimized designs present, for the solutions with negative and positive dispersion at $1.55\,\mu$m, a pitch and a hole radius $(\Lambda ,a) = (1.212,0.555)\,\mu$m and $(1.207,0.467)\,\mu$m, respectively. The results are plotted in Fig. 4.

 

Fig. 4. Dispersion of two optimized PCF designs [colored solid lines] achieved considering as target two linear dispersion profiles in the wavelength range $[1.53,1.57]\,\mu$m [black dashed lines] with $\beta _2=\pm 30$ ps$^2$km$^{-1}$ and $\beta _3 = -1$ ps$^3$km$^{-1}$ at $1.55\,\mu$m (see details in the text).

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5. Conclusions

The development of approximated analytical expressions for the mode effective index, the group index and the chromatic dispersion of scaled fibers is a powerful tool. Once a rigorous solution is known for a given fiber, the reported expressions provide an accurate evaluation of the mode parameters in the neighborhood defined by geometry and material scaling. Our results can push forward the efficiency of the designing tools of special optical fibers and even integrated waveguides, as we illustrate here with the design of a flattened dispersion PCF. In addition, our analytical expressions can benefit other applications areas such as the comparison of specific experimental results with a theoretical model that may require adjusting the geometry and materials values of a given guiding system.