1. Introduction

Four-wave mixing (FWM) in highly nonlinear optical fibers can be used for a wide range of applications in future high-capacity transparent optical networks such as optical regenerators, wavelength converters, parametric amplifiers and optical demultiplexers [1]. The key parameters for achieving a broadband and highly efficient FWM process are a high effective nonlinearity, a low group velocity dispersion with a low dispersion slope and a short fiber length. Combining photonic crystal fiber (PCF) technology with highly nonlinear glasses offers great potential for meeting these requirements. The effective fiber nonlinearity (the nonlinear coefficient) is written as [2]

where n ₂ is the nonlinear refractive index, λ the wavelength, and A _eff the effective mode area, which is determined by the PCF design. Thus, the fiber nonlinearity γ can be enhanced by reducing A _eff or/and by using a glass material whose n ₂ is high. The combination of small hole-to-hole-pitch and large air-filling fraction results in tight mode confinement due to the high glass/air index contrast and thus in a small effective mode area. Most nonlinear glasses have a high normal material dispersion at 1.55µm, which dominates the overall dispersion of fibers with conventional solid cladding. Fortunately, the PCF geometry affects drastically the waveguide dispersion, allowing the highly normal material dispersion to be overcome and the sign and magnitude of the group velocity dispersion can be tailored. Examples of suitable glasses that have been used to make PCFs include chalcogenide [3], tellurite [4], bismuth oxide [5], and lead-silicate glasses [6, 7, 8]. Among different kinds of signal processing applications in high capacity wavelength division multiplexed networks wavelength conversion has a major role. Four-wave-mixing based wavelength conversions at 1.55µm in a 2.2 m-long dispersion-shifted lead-silicate holey fiber [9] and in a 1-m long dispersion shifted bismuth-oxide holey fiber [10] were recently demonstrated.

The fabricated nonlinear glass PCFs have mostly a small core supported by fine struts produced using the extrusion technique [5] or the inflating method [10]. These so-called suspended core PCFs offer very large nonlinearities but allow for only relatively limited dispersion control around 1.55µm. Recently, a new technique for fabrication of nonlinear glass PCFs, the structured-element-stacking (SEST) technique that combines the advantages of extrusion and stacking was developed [9].

In this work, using an inverse simulation, we optimize dispersion properties of hexagonally arranged PCF sructures to be fabricated from selected nonlinear glasses by a conventional “stack and draw” technique. In papers [11, 12] a genetic algorithm is used to design photonic crystal fiber structures with user-defined chromatic dispersion properties. Employing Nelder- Mead simplex method for optimised designs of the lead silicate SEST fiber is mentioned in Ref. [13] without further details. We have explored the use of the Nelder-Mead method [14] for dispersion optimization of the nonlinear PCFs structures we simulated.

2. Inverse design method

In order to calculate the wavelength dependence of the effective index (n _eff) of the modes of PCF required for chromatic dispersion calculations, we employ a commercial full-vector mode solver based on the Finite Element Method (FEM). Taking into account that the PCF fiber with hexagonally arrayed air holes has C_6v symmetry (six-fold rotation symmetry with a reflection axis) and its fundamental polarization doublet modes belong to mode classes p = 4 and 3 we can reduce the computational window to the corresponding minimum sector [15] including only one-quater of the fiber cross section. Figure 1 shows the computational window together with a perfect electric and perfect magnetic conductor (PEC and PMC) boundary conditions applied along symmetric planes valid for the mode class p = 4. Anisotropic perfectly-matched layers (PML) [16] are positioned outside the outermost ring of holes in order to further reduce the computational window and to evaluate the confinement loss of PCF with a finite number of rings of holes. The advantage of flexible finite-element meshing of the multiscale geometry of PCF is also obvious from Fig. 1. The dispersion parameter D of a PCF is calculated from the n _eff values versus the wavelength using [16]

where c is the velocity of light in a vacuum and Re stands for the real part. The confinement loss CL is deduced from the value of n _eff as [16]

in dB/m, where Im stands for the imaginary part.

The material dispersion of nonlinear glasses given by Sellmeiers formulas (see Appendix A) is directly included in the calculation. Selected nonlinear glasses with the values of their refractive indices n, nonlinear indices n ₂ and material dispersions MD at 1.55µm are listed in Table 1. The dispersion of the glasses was calculated using Eq. 2 with n _eff replaced by n.

For FWM-based telecom applications a zero dispersion wavelength around 1.55µm is desirable, with a dispersion magnitude and slope designed to be as small as is possible. For the dispersion optimization we use as the objective function to be minimized with the Nelder-Mead simplex method (Appendix B) the function defined as [12]

where D is the dispersion parameter calculated at a wavelength λ _i and the sum is performed over 5 (uniformly spaced) points in the interval. The free parameters are the pitch of the triangular lattice, Λ, and the diameter d _i of the holes in the ith ring normalized by the pitch Λ.

 

Fig. 1. Computational window with applied boundary conditions and finite element mesh.

Download Full Size | PDF

3. Optimized fibers and discussion

The structural parameters of selected PCFs resulting from their dispersion optimization through minimizing the objective function (Eq.4) using the downhill simplex method are listed in Table 2. The corresponding confinement loss CL is given in Table 2, the effective area A _{e f f} [2] and nonlinear coefficient γ in Table 3. We employed the modification [19] of the usual definition of A _eff given in Ref. 2 for the calculation of γ. This modification eliminates the contribution of any field located in the holes to the effective nonlinearity considering that air has a very low nonlinearity. Our calculation shows that the percentage fraction of fundamental mode power located in the holes ranges from 1% to 5% for fibers F8, F1, F2, F3 and F4 and is negligible for fibers F5, F6 and F7, therefore using the modified A _{e f f} results in slight changes (less than 0.5 %) in the calculated γ only for fibers F8, F1, F2, F3 and F4. Dispersion of the optimized PCFs in the 1.5–1.6µm range is given in Table 3 and illustrated by Figs. 2–9. PCFs F1, F2, F3, and F4 have smaller single-mode cores whereas PCFs F5, F6, F7 and F8 have larger multi-mode cores (it can be related to the existence of two solutions for a diameter of nonlinear glass rod in air which yield a zero dispersion at 1.55µm [20]).

Table 1. Selected nonlinear glasses.

View Table | View all tables in this article

It is obvious from Table 3 and Figs. 2–5 that the optimized lead silica glass PCF (F1), bismuth glass PCF (F2) and teluritte glass PCF (F3) provide near-zero, flat dispersion at 1.55µm. Dispersion characteristics of chalcogenide PCF (F4) are a bit worse owing to the necessity to compensate a large normal dispersion of chalcogenide glass (see Table 1). Fibers F1, F3 and F4 were optimized with two free parameters: the equal air hole diameter and the pitch. Five airhole rings for fibers F1 and F3 and only four rings for F4 were needed to reduce confinement loss down to an acceptable level when compared with a higher attenuation of bulk nonlinear glasses. However, five rings of air holes of equal diameter were not enough for bismuth glass PCF. So we also experimented with the design approach based on varying the hole diameter of each air-hole ring. Keeping outer air holes large improves the confinement loss which enables to decrease the number of rings [21, 16]. Fiber F2 has only three air hole rings (but four free parameters) and still an acceptable confinement loss. The optimized single-mode PCF structures are highly nonlinear, however, their dispersion characteristics are very sensitive to variations in structural parameters as demonstrated in Figs. 2–5 when the pitch is changed from its optimal value by ±2% while all d _i/Λ are kept unaltered (red and blue curves).

Fibers F5, F6, F7 and F8 have fundamental mode’s zero dispersion wavelengths around 1.55µm with non-zero moderate dispersion slopes (see Table 3) which are less sensitive to structural variations as illustrated in Figs. 6–9 again through varying the pitch (red and blue curves). Larger cores of the PCFs lead to negligible fundamental mode’s confinement loss even for the PCF structures with only two rings of air holes of equal diameter, and also to lower, but still high, effective nonlinearity (see Tables 2 and 3). However, the larger PCFs cores support higher order modes. We have found that TE₀₁, HE₂₁, and TM₀₁ modes have low confinement losses at 1.55µm. Therefore, careful mode-matching has to be used to prevent the excitation of higher order modes. The next higher order modes are much more lossy, e.g. HE ₃₁₁, HE₃₁₂ [15], and EH₂₁ modes have confinement losses around 1 dB/m and HE₁₂ mode even 100 dB/m in fiber F5.

Using the downhill simplex method for minimizing the objective function (Eq.4) seems to be more effective than using the genetic algorithm. Less than a hundred evaluations of the objective function was needed with the simplex method, compared to hundreds of the function evaluations involved in the latter approach [12]. Constraints imposed on structural and optical parameters of an optimized PCF (e.g. on air-hole diameter, air-filling ratio and confinement loss) were additionally introduced into the objective function as penalty terms for constraint violation. For a given set of free structural parameters only one minimum of the objective function has been found providing the single-mode PCFs F1, F2, F3 and F4 with zero dispersion and zero dispersion slope around 1.55µm. In the case of multi-mode PCFs F5, F6, F7 and F8 with zero dispersion and non-zero dispersion slope at 1.55µm there is a continuum of such solutions (in accordance with Ref. [22], Fig. 7). However, when minimizing the objective function is not stopped while confinement loss is still low and continues further, fundamental mode’s confinement loss in the two-ring PCFs grows up fast, overcoming an acceptable value before an additional significant decrease of dispersion slope is achieved. The second multi-mode solution of the nonlinear glass rod mentioned above was used as an rough estimate of PCF’s core diameter from which a starting set of free structural parameters was derived. Low-loss multimode dispersion optimized PCFs F5, F6, F7 and F8 were obtain quickly after several tens of objective function evaluations.

Table 2. Geometrical parameters and confinement loss CL (calculated at 1.55 µm) of the optimized fibers.

View Table | View all tables in this article

Table 3. Dispersion D, and dispersion slope S, effective area Aeff and nonlinear coefficient γ (calculated at 1.55 µm) of the optimized fibers.

View Table | View all tables in this article

 

Fig. 2. Chromatic dispersion curve (black) as a function of wavelength for the optimized single-mode PCF with lead silicate glass named in Tables 2 and 3 as Fiber F1.

Download Full Size | PDF

 

Fig. 3. Chromatic dispersion curve (black) as a function of wavelength for the optimized single-mode PCF with bismuth glass named in Tables 2 and 3 as Fiber F2.

Download Full Size | PDF

 

Fig. 4. Chromatic dispersion curve (black) as a function of wavelength for the optimized single-mode PCF with tellurite glass named in Tables 2 and 3 as Fiber F3.

Download Full Size | PDF

 

Fig. 5. Chromatic dispersion curve (black) as a function of wavelength for the single-mode optimized PCF with chalcogenide glass named in Tables 2 and 3 as Fiber F4.

Download Full Size | PDF

 

Fig. 6. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for optimized multi-mode PCF with lead silicate glass named in Tables 2 and 3 as Fiber F5.

Download Full Size | PDF

 

Fig. 7. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for the optimized multi-mode PCF with bismuth glass named in Tables 2 and 3 as Fiber F6.

Download Full Size | PDF

 

Fig. 8. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for the optimized multi-mode PCF with tellurite glass named in Tables 2 and 3 as Fiber F7.

Download Full Size | PDF

 

Fig. 9. Fundamental mode’s chromatic dispersion curve (black) as a function of wavelength for the optimized multi-mode PCF with chalcogenide glass named in Tables 2 and 3 as Fiber F8.

Download Full Size | PDF

4. Conclusion

Dispersion optimization of photonic crystal fibers made from the selected highly nonlinear glasses was performed using a fully-vectorial mode solver based on the finite element method and the Nelder-Mead simplex method. Firstly, the PCF structures with air holes hexagonally arrayed in 5,4 and 3 rings made from lead silica, bismuth, tellurite and chalcogenide glasses were optimized to achieve a small and flattened dispersion in the 1.5–1.6µm range suitable for four-wave mixing-based telecom applications. The optimized PCFs are single-mode with high nonlinearity but high sensitivity of their zero dispersion wavelengths to variations of structural parameters is an issue and a challenge for further enhancement of highly nonlinear glasses-based PCFs’ fabrication techniques. Efficient coupling into PCFs with a very small effective ares might be achieved through tapering. Secondly, two-ring PCFs with larger multimode cores and their dispersion less sensitive to structural variation have been obtained through dispersion optimization performed again for the selected nonlinear glasses. It is supposed that these alternative PCF designs will be easier to fabricate. On the other hand, a selective coupling to a fundamental mode without an excitation of higher order modes demands special attention.

Appendix A: Sellmeier formulas for selected highly nonlinear glasses

The refractive index as a function of wavelength for each selected glass is first obtained by the following Sellmeier (or a similar) equation and accompanying coefficients. Wavelength λ is in µm in Eqs. 5, 6, 7 and 8.

For lead silicate glass (Schott SF57) [23]:

(5)$$n^2=1+\frac{B_1{\lambda }^2}{{\lambda }^2-C_1}+\frac{B_2{\lambda }^2}{{\lambda }^2-C_2}+\frac{B_3{\lambda }^2}{{\lambda }^2-C_3},$$

where B ₁=1.81651371, B ₂=0.428893641, B ₃=1.07186278, C ₁=0.0143704198, C ₂=0.0592801172, C ₃=121.419942.

For bismuth glass (55Bi₂O₃·45B₂O₃) [24]:

(6)$$n^2=A_0+A_{1}x+A_2{x}^2+\frac{B_{1}x}{{\lambda }^2-C_0-C_{1}x}-D_0{\lambda }^2,$$

where x = 0.55, A ₀=1.90598, A ₁=5.78900, A ₂=-2.22010, B ₁=0.16995, C ₀=0.02116, C ₁=0.09230, D ₀=0.01857.

For tellurite glass (75TeO₂·25ZnO) [25]:

(7)$$n^2=A+\frac{B{\lambda }^2}{{\lambda }^2-C}+\frac{D{\lambda }^2}{{\lambda }^2-E},$$

where A = 2.4843245, B = 1.6174321, C = 0.053715551, D = 2.4765135, E = 225.

For chalcogenide glass (As₂S₃) [26]:

(8)$$n^2=1+\frac{B_1{\lambda }^2}{{\lambda }^2-C_1^2}+\frac{B_2{\lambda }^2}{{\lambda }^2-C_2^2}+\frac{B_2{\lambda }^2}{{\lambda }^2-C_3^2}+\frac{B_4{\lambda }^2}{{\lambda }^2-C_4^2}+\frac{B_5{\lambda }^2}{{\lambda }^2-C_5^2},$$

where B ₁=1.898367, B ₂=1.922297, B ₃=0.87651, B ₄=0.11887, B ₅=0.95699, C ₁=0.15, C ₂=0.25, C ₃=0.35, C ₄=0.45, C ₅=27.3861.

Appendix B: Downhill simplex method

The Nelder-Mead “downhill simplex” algorithm is one of the most widely used methods for nonlinear unconstrained optimization [14]. This direct-search method attempts to minimize a scalar-valued nonlinear function of N real variables using only function values, without any derivative information.

The Nelder-Mead method maintains at each step a nondegenerative simplex, a geometric figure in N dimensions of nonzero volume that is a polyhedron with N+1 vertices. Each iteration of simplex-based direct search method begins with a simplex specified by its N+1 vertices and the associated function values. One or more test points are computed, along with their function values, and the iteration terminates with a new (different) simplex such that the function values at its vertices satisfy some form of descent condition to the previous simplex. Among such algorithms, the Nelder-Mead algorithm is particularly parsimonious in function evaluations per iteration, since in practice it typically requires only one or two function evaluations to construct a new simplex.

The “N+1” points is chosen and defined an initial simplex. The method iteratively updates the worst point by four operations: reflection, expansion, contraction and multiple contraction (shrinkage). Figures 10 and 11 illustrate these operations in two-dimensional variable space where a simplex is a triangle. Reflection (see Fig. 10) involves moving the worst point (vertex x ₃) of the simplex (where the value of the objective function is the highest) to a point (x r) reflected through the centroid of the remaining N points (x g). If this point is better than the best point then the methods attempts to expand the simplex along this line (to x _e). On the other hand, if the new point is not much better then the previous point, then the simplex is contracted along one dimension from the highest point (see Fig. 11) to the point between x _g and the better of x ₃ and x _r (x _c or x _cc). Moreover, if the new point is worse then the previous points, the simplex is contracted along all dimensions towards the best point (x ₁) and steps down the valley. By repeating this series of operations, the methods finds the optimal solution.

The Nelder-Mead algorithm is implemented by employing the solver “fminsearch” included in the MATLAB Optimization Toolbox [27] that minimizes the objective function written as a MATLAB script. The objective function is calling a fully-vectorial FEM solver provided by COMSOL Multiphysics software (with the RF modul) [28] for the PCF eigenvalue analysis. Within the script PCF’s cross-section geometry to be analyzed is built, relevant parameters for the FEM solver are set, and a parametric (wavelength) scan with fundamental mode tracking and postprocessing (namely the calculations of effective area A _eff and dispersion D) are also performed. Constrains can be imposed on PCF’s structural and optical parameters in the objective function script as well.

 

Fig. 10. Nelder-Mead simplices after a reflection and an expansion step. The original simplex is shown with a dashed line.

Download Full Size | PDF

 

Fig. 11. Nelder-Mead simplices after an outside contraction, an inside contraction, and a shrink. The original simplex is shown with a dashed line.

Download Full Size | PDF

Acknowledgments

This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic under grant numbers 1P05OC002 and OE08021.

References and links

1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002). [CrossRef]  

2. G. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995).

3. T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibers,” Electron. Lett. 36, 1998–2000 (2000). [CrossRef]  

4. V. V. R. Kumar, A. K. George, J. C. Knight, and P. S. J. Russell, “Tellurite photonic crystal fiber,” Opt. Express 11, 2641–2645 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2641. [CrossRef]   [PubMed]  

5. H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. M. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5082. [CrossRef]   [PubMed]  

6. K. M. Kiang, K. Frampton, T. M. Monro, R. Moore, J. Tucknott, D.W. Hewak, D. J. Richardson, and H. N. Rutt, “Extruded singlemode non-silica glass holey optical fibres,” Electron. Lett. 38, 546–547 (2002). [CrossRef]  

7. V. V. R. Kumar, A. K. George, W. H. Reeves, J. C. Knight, P. S. J. Russell, F. G. Omenetto, and A. J. Taylor, “Extruded soft glass photonic crystal fiber for ultrabroad supercontinuum generation,” Opt. Express 10, 1520–1525 (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-25-1520. [PubMed]  

8. P. Petropoulos, H. Ebendorff-Heidepriem, V. Finazzi, R. C. Moore, K. Frampton, D. J. Richardson, and T. M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568. [CrossRef]   [PubMed]  

9. S. Asimakis, P. Petropoulos, F. Poletti, J. Y. Y. Leong, R. C. Moore, K. E. Frampton, X. Feng, W. H. Loh, and D. J. Richardson, “Towards efficient and broadband four-wave-mixing using short-length dispersion tailored lead silicate holey fibers,” Opt. Express 15, 596–601 (2007) http://www.opticsexpress.org/abstract. cfm?URI=OPEX-15-2-596. [CrossRef]   [PubMed]  

10. K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, “Four-wave mixing based widely tunable wavelength conversion using 1-m dispersion-shifted bismuth-oxide photonic crystalfiber,” Opt. Express 15, 15418–15423 (2007), http://www.opticsexpress.org/abstract.cfm? URI=OPEX-15-23-15418. [CrossRef]   [PubMed]  

11. E. Kerrinckx, L. Bigot, M. Douay, and Y. Quiquempois, “Photonic crystal fiber design by means of a genetic algorithm,” Opt. Express 12, 1990–1995 (2004), http://www.opticsexpress.org/abstract.cfm? URI=OPEX-12-9-1990. [CrossRef]   [PubMed]  

12. F. Poletti, V. Finazzi, T. M. Monro, N. G. R. Broderick, V. Tse, and D. J. Richardson, “Inverse design and fabrication tolerances of ultra-flattened dispersion holey fibers,” Opt. Express 13, 3728–3736 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3728. [CrossRef]   [PubMed]  

13. J. Leong, S. Asimakis, F. Poletti, P. Petropoulos, X. Feng, R. Moore, K. Frampton, T. Monro, H. Ebendorff- Heidepriem, W. Loh, and D. Richardson, “Nonlinearity and dispersion control in small core lead silicate holey fibers by structured element stacking,” presented at OFC/NFOC 2006 Anaheim, California, USA, 2006, paper OTuH1.

14. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998). [CrossRef]  

15. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm? URI=OPEX-11-11-1310. [CrossRef]   [PubMed]  

16. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003), http://www.opticsexpress. org/abstract.cfm?URI=OPEX-11-8-843. [CrossRef]   [PubMed]  

17. J.-K. Kim, “Investigation of high-nonlinearity glass fibers for potential applications in ultrafast nonlinear fiber devices,” Ph.D. thesis (Virginia Polytechnic Institute and State University, Blacksburg, VA., 2005), http://en.scientificcommons.org/1530271.

18. M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997). [CrossRef]  

19. V. Finazzi, T. M. Monro, and D. J. Richardson, “Small-core silica holey fibers: nonlineariry and confinement loss trade-offs,” J. Opt. Soc. Am. B 20, 1427–1436 (2003). [CrossRef]  

20. E. S. Hu, Y. L. Hsueh, M. E. Marhic, and L. G. Kazovsky, “Design of highly-nonlinear tellurite fibers with zero dispersion near 1550 nm,” in Proc. European Conference on Optical Communication 2, 1–2, (2002).

21. G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28, 989–991 (2003). [CrossRef]   [PubMed]  

22. F. Poletti, K. Furusawa, Z. Yusoff, N. G. R. Broderick, and D. J. Richardson, “Nonlinear tapered holey fibers with high stimulated Brillouin scattering threshold and controlled dispersion,” J. Opt. Soc. Am. B 24, 2185–2194 (2007). [CrossRef]  

23. Schott Optical Glass Catalogue 2007, http://www.schott.com.

24. I. I. Oprea, H. Hesse, and K. Betzler, “Optical properties of bismuth borate glasses,” Opt. Mater. 26, 235–237 (2004). [CrossRef]  

25. G. Ghosh, “Sellmeier-coefficients and chromatic dispersions for some tellurite glasses,” J. Am. Ceram. Soc. 78, 2828–2830 (1995). [CrossRef]  

26. A. V. Husakou and J. Herrmann, “Supercontinuum generation in photonic crystal fibers made from highly nonlinear glasses,” Appl. Phys. B 77, 227–234 (2003). [CrossRef]  

27. http://www.mathworks.com.

28. http://www.comsol.com.