Effective index numerical modelling of microstructured chalcogenideglass fiber for frequency conversion to the mid-infrared band Pierre Bourdon*a, Anne Durécua, Claire Alhenc-Gelasa, Laura Di Biancaa, Guillaume Canata and Frédéric Druonb a
Département Optique Théorique et Appliquée, Onera, The French Aerospace Lab, BP 80100, 91123 Palaiseau cedex, France b Laboratoire Charles Fabry de l'Institut d'Optique, CNRS, Université Paris-Sud, RD 128 campus Polytechnique, 91127 Palaiseau, France ABSTRACT Chalcogenide glass fibers offer broad transparency range up to the mid-infrared and high nonlinear coefficients making them excellent candidates for four wave mixing frequency conversion. However, the use of microstructured airchalcogenide fibers is mandatory to achieve phase-matching in such a fiber. Numerical modelling of the phase matching condition can be done using the simplified effective index model, initially developed and extensively used to design airsilica fibers. In this paper, we investigate the use of the effective index model in the case of microstructured As2S3 and As2Se3 fibers. One essential step in the method is to evaluate the core radius of a step-index fiber equivalent to the microstructured fiber. Using accurate reference results provided by finite-element computation, we compare the different formulae of the effective core radius proposed in the literature and validated for air-silica fibers. As expected, some discrepancies are observed, especially for the highest wavelengths. We propose new coefficients for these formulae so that the effective index method can be used for numerical modelling of propagation in air-chalcogenide fibers up to 5 µm wavelength. We derive a new formula providing both high accuracy of the effective core radius estimate whatever the microstucture geometry and wavelength, as well as uniqueness of its set of coefficients. This analysis reveals that the value of the effective core radius in the effective index model is only dependent on the microstructure geometry, not on the fiber material. Thus, it can be used for air-silica or air-chalcogenide fibers indifferently. Keywords: Fiber lasers, chalcogenide glasses, mid-infrared, nonlinear optics
1. INTRODUCTION Mid-infrared (MWIR) laser light in the 3 – 5 µm wavelength transparency window of the atmosphere is essential for many remote sensing techniques. However, this MWIR range is difficult to address with lasers, and most of the available sources are still bulky and power limited. The four-wave mixing (FWM) process, often known as "parametric amplification" in fibers, could be an appropriate technique to convert wavelength from the near-IR (1.5 µm or 2 µm for instance) to the mid-IR (between 3 and 4 µm), especially in chalcogenide fibers as they offer wide transparency range covering most of the mid-IR domain, and high nonlinearity. However, it can be proven that the phase matching condition for this nonlinear process in a classical step-index fiber is very challenging to achieve. Hopefully, efficient FWM can be achieved in microstructured fibers1, 2. The design of an efficient FWM fiber component necessitates accurate numerical models to assess the fiber dispersion curves, from which the phase matching conditions in the fiber can be derived. The most accurate numerical models rely on finite-elements computation. The main issue of these finite-element models (FEM) is that they are very timeconsuming. Anyway, due to their unequalled accuracy and versatility, we can use these FEM computations results to reference less accurate results from simplified numerical models.
*
[email protected]; phone (+33) 1 80 38 63 82; fax (+33) 1 80 38 63 45; www.onera.fr
In this paper, we present the improvements we have made to the effective index model so that we can apply it to the case of air - chalcogenide-glass fibers (As2S3 and As2Se3) designed for the purpose of frequency conversion to the mid-IR. In a first part of the paper, the effective model developed for air – silica fibers is described and applied directly to air – chalcogenide-glass fibers. The parameters of the model are then adapted conveniently to improve the results and describe more accurately the microstructured air – chalcogenide fiber properties.
2. THE EFFECTIVE INDEX MODEL OF A MICROSTRUCTURED FIBER 2.1 Principle and basic equations of the effective index model Among the numerous simplified models available to compute the dispersion of an air – chalcogenide-glass fiber, we use a semi-vectorial effective-index method (EIM). This method is first used to obtain the dispersion curves of microstructured chalcogenide optical fiber (MCOF). We limit our study to periodic air-hole hexagonal arrays, as EIM is not really applicable to non periodic microstructures. The basic principle of the effective-index method is to replace the microstructured fiber by an equivalent step-index fiber, as illustrated in Fig. 1. The equivalent fiber clad index can be obtained analytically and the refractive index of the chalcogenide material is used as the core-index.
neff FSM
nchalco
nchalco
Figure 1. Radial structure of a microstructured fiber (left) and the equivalent step-index fiber used in the effective index model (right). d is the diameter of an air hole (in white on the left-hand part of the figure) and Λ is the pitch of the hexagonal array of air holes.
Figure 2. The hexagonal array of unit cells the MCOF clad is comprised of. Each hexagonal cell is centered on an airhole (gray-filled circle of radius a) and the size of the hexagonal cell is b = Λ
3 . 2π
The hexagonal lattice of the MCOF, supposed to be infinite, is decomposed in hexagonal unit cells centered on an air hole (Fig. 2). Each hexagonal unit cell is approximated by a circular cell of outer radius b having the same air-filling fraction b = Λ
3 , where Λ is the hole-to-hole distance of the microstructured cladding (i.e. the pitch of the
2π
heaxagonal array of air-holes). Applying electric and magnetic field boundary conditions on each cell, E z ( ρ = b) = H z ( ρ = b) = 0 , leads to an eigenvalue equation for the fundamental LP01 mode3: 2
n2 I 2 (W ) 1 W P1' (U ) nc2 1 P1' (U ) nc2 1 1 1 n2 1 + 2 − W 1 − 2 + 2 2 + 2 a2 + c2 , =− − I 1 (W ) W 2 UP1 ( ρ = a) na 4 UP1 ( ρ = a) na na U W W U
where a = d/2 is the radius of an air hole, nc is the refractive index of the chalcogenide glass, na is the refractive index of Uρ Uρ Ub Ub 2 2 air (na = 1), , and W = k 0 a n FSM − na2 , U = k 0 a n c2 − n FSM P1 ( ρ ) = J 1 ( )Y1 ( ) − Y1 ( ) J 1 ( ) a a a a . a ∂P1( ρ ) ' P (U ) = 1 U
∂ρ
ρ =a
Solving the eigenvalue equation gives the so-called fundamental space filling mode index nFSM that we will consider as the equivalent fiber clad index from now on. The MCOF is now replaced by an equivalent step-index fiber with a core index nc and a clad index nFSM. Propagation in this equivalent step-index fiber can now be treated solving the classical eigenvalue equation for the J 0 (U eff ) K 0 (Weff ) 2 2 fundamental mode of a step-index fiber: − = 0 where U eff = k 0 rc nc − n eff , U eff J 1 (U eff ) Weff K 1 (Weff ) 2 2 and rc is the equivalent step-index radius. Weff = k 0 rc neff − n FSM
As the core of a MCOF has no real delimited boundary, the electric field penetrates somehow between the air holes into the fiber clad. So, in this effective index model, the accurate value of one final parameter of the equivalent step-index fiber is still unknown: the effective core-radius rc. The proper value for this effective core-radius has been calculated in the case of air-silica fibers. For regular hexagonal array of air-holes, rc = 0.64 Λ is often chosen as a good first approximation of this effective radius. Refinements of this rc expression have been proposed by Park and Lee4 and Li et al.5. They both take into account the diameter of an air-hole. Li et al. added the wavelength parameter to the formula. Both formulae have been validated over the transparency window of silica, up to 2.5 µm, but not for higher wavelength. Park and Lee formula for rc:
Li et al. formula for rc: r c
rc
= Λ
, 3 empirical coefficients. c1 d − c3 Λ 1 + exp c2
5 λ d = Λ ∑ nij , 36 empirical coefficients. Λ Λ i=0 i
j
j =0
As FWM frequency conversion to the mid-IR involves different wavelengths ranging from 1 µm to 5 µm, we need to validate these formulae of the effective core-radius over a broader range of parameters than for air – silica fibers. Not only do we have to consider a different material, the chalcogenide glass, but also laser wavelength ranging from 1 µm up to 5 µm.
2.2 Application of the effective index model to the case of air – chalcogenide-glass fibers We started by applying directly the effective index model to the case of a MCOF, using the same parameters for the model as for air – silica fibers. More specifically, we used the rc formulae derived by Park and Lee and Li et al. without modifying the empirical coefficients optimized for air – silica fibers up to 2.5 µm wavelength. In parallel, we modelled the microstructured air – chalcogenide-glass fibers using a FEM and considered the FEM calculations as reference results. The relevance of this approach was confirmed by a preliminary study where we applyed the FEM to derive dispersion curves for air – silica fibers: the FEM allowed to retrieve the same exact results obtained by Park and Lee and Li et al. in the case of silica fibers.
Effective index neff
Dispersion (ps/nm/km)
Fig. 3 presents sample results obtained in the case of a Λ =5 µm air – As2S3 fiber. Similar results can be obtained for different pitch values ranging from 3 µm to 10 µm.
FEM
FEM
Park and Lee
Park and Lee
Li et al.
Li et al.
Wavelength (µm)
Wavelength (µm)
Figure 3. Computed values of the effective index of the fiber for various d/Λ values (left) and of the dispersion of the fiber for d/Λ = 0.8 (right). The FEM results are compared with results from the effective index method using Park and Lee or Li et al. values for rc. The fiber considered here has Λ = 5 µm.
It can be observed that over the wavelength range of optimization of the effective index model parameters, the effective index calculated using Park and Lee or Li et al. values for rc are very close to the reference FEM results. But beyond 2.5 µm wavelength, there are some discrepancies between the calculated and reference values of the effective index of the fiber. These discrepancies are even amplified the fiber dispersion curve (20 % maximum discrepancy on the dispersion value for λ = 5 µm), as this curve is obtained as the result of the second derivative of the effective index. In order to be able to use the effective index model in our case, we had to improve substantially its accuracy. We chose a similar approach to Park and Lee's and Li et al.'s, modifying the rc formulae coefficients to fit the reference FEM results over the entire wavelength range of interest better.
3. IMPROVEMENT OF THE EFFECTIVE INDEX MODEL APPLIED TO THE CASE OF CHALCOGENIDE-GLASS FIBERS 3.1 Derivation of new coefficients for the rc formulae Through a very long process of computation, using the finite-element model, we calculated the effective-index of the fundamental mode propagating in the fiber core for multiple couples (d/Λ,Λ), covering the entire range of variations of those parameters. Comparing with the EIM, we were able to obtain, for each couple (d/Λ,Λ), the value of rc delivering the FEM-computed effective-index value with minimal error. The process of computation is described in Fig. 4. For each set of values (d/Λ,Λ), we assessed the discrepancy between the EIM and FEM results as a function of rc. The value of rc minimizing the least-squares difference between the EIMassessed effective index curve over our entire wavelength range of interest (1 – 5 µm) and the FEM-assessed effective index curve was considered optimal. The final result is, for each Λ value, the best value for rc as a function of d/Λ. Through standard least-squares fitting analysis, we derived new values of the 3 empirical coefficients for the Park and Lee rc formula and of the 36 empirical coefficients for the Li et al. rc formula.
rc / Λ
Error
Optimal rc / Λ ratio Least-squares fit
rc / Λ
d/Λ
Figure 4. On the left, process of finding the optimal values for rc/Λ, with Λ = 5 µm. For each d/Λ value, we search for the optimal value of rc minimizing the least-squares error between the EIM and FEM assessed effective index vs. λ curve. On the right, the resulting values of rc/Λ vs. d/Λ for Λ = 5 µm.
Effective index neff
Dispersion (ps/nm/km)
Using these new coefficients, we were able to fit more accurately the effective index vs. wavelength and dispersion curves for an As2S3 MCOF. Fig. 5 presents some sample results obtained using the effective index model with these modified formulae for rc.
FEM Original Li et al. coef. Improved Li et al. coef.
FEM Improved Park and Lee coef. Improved Li et al. coef.
Wavelength (µm)
Wavelength (µm)
Figure 5. Computed values of the effective index of the fiber for various d/Λ values (left) and of the dispersion of the fiber for d/Λ = 0.8 (right). The FEM results are compared with results from the effective index method using modified Park and Lee or modified Li et al. values for rc. The fiber considered here has Λ = 5 µm.
As could be expected (as most mathematical functions can be fitted using polynomial laws, as long as the order of the polynomial fit is high enough), the polynomial Li et al. formula gives the best results. On the other hand, the modified Park and Lee formula is far less accurate, especially for the highest wavelength. However, this was expected as the Park
and Lee formula does not vary with λ. Thus, there is little chance for such a "spectrally-static" formula to perform well in our attempts to fit the FEM results over a very broad spectral range. Note that there was an issue with the Li et al. polynomial fit: the set of optimal coefficients was not unique. Moreover, a different set of coefficient was obtained when we considered another chalcogenide glass, for instance As2Se3. Consequently, we tried another approach to obtain a more robust and universal formula.
3.2 Towards a more robust and universal formula for rc We went through another very long process of computation, using the finite-element model: we calculated the effectiveindex of the fundamental mode propagating in the fiber core for multiple sets of parameters Λ, d/Λ and λ, covering the entire range of variations of those parameters. Comparing with the EIM, we were able to obtain, for each set of parameters, the value of rc delivering the FEM-computed effective-index value with minimal error.
rc,optimal / Λ
This time, we obtain a 3D surface describing the variations of this optimal rc value vs. the variables d/Λ and λ (see Fig. 6).
Wavelength (µm)
Figure 6. 3D surface giving the optimal values of rc/Λ vs. wavelength and microstructure geometrical parameter d/Λ. For such optimal value of rc, the difference between the EIM and FEM results is minimal.
We observe almost linear variations of this surface with λ. The variations of the surface with d/Λ are very similar in curvature and shape to a Park and Lee formula. Consequently, the most universal formula we propose is written as a Park and Lee formula with wavelength dependent coefficients:
rc
=
Λ 1 +
. c1 (λ ) d − c 3 (λ ) exp Λ c 2 (λ )
A linear dependence of the 3 coefficients on λ gives good results, but introducing a mild curvature through a quadratic dependence of c3 on λ delivered the optimum accuracy of the model. Fig. 7 shows that this new version of the EIM gives very accurate fit of the effective index over a very broad wavelength range. The use of this optimized Park and Lee formula for rc is not as accurate as the modified Li et al. formula (approximately one order of magnitude less accurate), but it delivers results that are very close from the FEM results anyway and the set of coefficients is now unique.
2,32
2,32
Finite-element model EIM+best rc formula
Finite-element model EIM with Park and Lee EIM with Li et al.
EIM+improved Li et al. 2,30
Effective index
Effective index
2,30
2,28
2,26
2,28
2,26 2,5
3,0
3,5
4,0
4,5
5,0
2,5
Wavelength (µm)
3,0
3,5
4,0
4,5
5,0
Wavelength (µm)
Figure 7. Left: comparison of the effective-index value for the fundamental mode propagating in a Λ = 5 µm, d/Λ = 0.8 air-As2S3 fiber, calculated using a finite-element model (blue solid line), the EIM with Park and Lee formula (red circles) and the EIM with Li et al. formula (green dashed line). Right: the EIM results are obtained with our new optimized formula (red circles) and with the improved Li et al. formula (green dashed line).
We also observed that the formula operates whatever the material considered. It works for silica, As2S3 and As2Se3 indifferently. This is a very interesting result we always observed during this study: the rc value does not depend on the material used, only on the geometry of the microstructure (i.e. d and Λ values). It means that the improved formulae can be used indifferently for air-silica and air-chalcogenide fibers, and even more, in this last case, they can be used whatever chalcogenide glass is involved.
4. CONCLUSION In this paper, we applied the effective index model to the analysis of wave propagation inside a microstructured air – chalcogenide-glass fiber for the purpose of frequency conversion to the mid-infrared. As we observed some discrepancies between a reference finite-element model and the effective index model, especially at the highest wavelength, we modified the formulae for the effective core radius of the fiber, rc, in order to improve the accuracy of the model in the mid-infrared. We obtained very good results with these refined coefficients for rc formulae, especially with a 5th order polynomial formula. Unfortunately, the set of values of improved coefficients we obtained for this polynomial fit were not unique and depended on the fiber material. Finally, we managed to derive a new formula for the effective core radius of the fiber. With this optimized formula, we are able to achieve very good accuracy of the effective index model in the mid-infrared. Furthermore, this formula is very universal and can be used whatever the material considered. Future investigations will focus on comparing these numerical results with experimental measurements. Exploration of new chalcogenide materials will also be performed to check if the extended model is still valid for other families of glasses.
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[2] Nodop, D., Jauregui, C., Schimpf, D. D., Limpert, J. and Tunnermann, A., "Efficient near-infrared light conversion to visible and mid-infrared radiation in an endlessly single-mode photonic crystal fiber," Photonics West 2010 SPIE Conference 7580, Talk 7580-13 (2010). [3] Midrio, M., Singh, M. P. and Someda, C. G., “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol 18(13), 1031-1037 (2000). [4] Park, K. N. and Lee, K. S., "Improved effective-index method for analysis of photonic crystal fibers," Opt. Lett. 30(9), 958-960 (2005). [5] Li, Y., Yao, Y., Hu, M., Chai, L. and Wang, C., "Improved fully vectorial effective index method for photonics crystal fibers: evaluation and enhancements," Appl. Opt. 47(3), 399-406 (2008).
