In this paper, we present a numerical analysis of the coupling coefficients in dual-core air-silica microstructured optical fibers with π/6 symmetry. The calculations are based on an especially fitted application of the coupled mode theory for microstructured optical fibers. This method is compared with three other techniques, the supermode method, the beam propagation method and the equivalent fiber model, and is shown to be very computationally efficient. Our studies enable us to derive a formula linking the coupling coefficients to core separation according to the wavelength, the pitch and the hole diameter of the fiber structure.
©2009 Optical Society of America
Microstructured Optical Fibers (MOFs) have attracted a lot of interest in the past few years. Indeed, their unique structure leads to novel optical properties [1,2]. MOFs are characterized by wavelength-scale air holes running parallel to the propagation axis. Light is guided by modified total internal reflection between the solid pure silica core and the holey cladding.
The “stack and draw” technique flexibility of such fibers enables to easily design multiple core MOFs by substituting capillaries by rods that act as coupled cores after drawing . When high coupling is needed, these dual-core or multi-core fibers (DC-MOFs and MC-MOFs, respectively) provide improvement in the size and performance of the fiber-based devices such as directional couplers [4,5], wavelength division multiplexers  and polarization splitters . MC-MOFs can also be used when weak coupling is sought [7,8]. In all these applications, the coupling coefficient C between two cores must be evaluated. This parameter is involved in the inter-core power exchange efficiency and is proportional to the overlap integral between the mode fields in each independent core. The knowledge of C is useful for the description of DC-MOFs as well as MC-MOFs guiding properties. Indeed, the latter case can be solved by introducing the set of coupling coefficients of all the existing pairs of cores into the coupled mode equations .
Several publications already report techniques to calculate C. As for examples, the full vectorial beam propagation method , the full vectorial finite element method [6, 11], or the equivalent fiber model  have been proposed to analyze DC-MOFs.
In this paper, we investigate the coupling properties of DC-MOFs using another approach, not considered so far in the literature to our knowledge, based on the coupled mode theory (CMT).
First, after a short description of the geometry and the operating conditions of the studied DC-MOFs, we present three different methods already published to calculate C. Then we detail our CMT-based method and we discuss its advantages. We compare these methods in terms of precision and computing performances. Finally, we introduce a simplified model allowing a precise evaluation of the coupling coefficients versus the core separations.
2. Studied microstructure definition
We consider a microstructured optical fiber with π/6 symmetry described by its two main geometrical parameters: the air hole diameter d and the pitch Λ (Fig. 1). They define the ratio d/Λ and the air-filling fraction equal to (π/2√3) (d/Λ)2 which are involved in many MOF guiding properties. The cladding lattice is a triangular arrangement of air holes in silica background material. The guiding properties are due to modified total internal reflection between the solid core and the holey cladding.
The confinement of the optical field in each core is dependent on the relative size of the wavelength λ, the air hole diameter d and the pitch Λ of the periodic microstructure. These parameters act on the field extension outside the core and consequently influence the coupling between the guided modes of DC-MOFs. The versatility of core arrangements and relative positions in the microstructure allowed by the “stack and draw” manufacturing process provide additional degrees of freedom for the design of coupled core MOFs. Indeed, the cores can be spaced by one or several air holes in geometry #1 such as described in Fig. 1(a). They can also be connected by one or several silica bridges as shown in geometry #2 (Fig. 1b).
All along the paper, we develop our analysis of DC-MOFs with two cores arranged according to the two different geometries, in which the second core takes the place of the mth hole next to the first core, denoted hm. Thus, the distance between cores S is defined relatively to the pitch size of the microstructure Λ by the relation S=mΛ with m≥2 in geometry #1 or S=2mΛ cos(π/6) with m≥1 in geometry #2.
We arbitrary limit our study to circular holes. However, the exact shape of the holes in MOFs is extremely dependent on the reached air-filling fraction and the drawing conditions such as temperature and inner and outer air pressure. Nevertheless, there are no major difficulties to apply the described numerical methods to the real cross-section of MOFs.
3. Numerical methods for coupling coefficient calculations
In this section, we describe four methods that can be used to calculate the coupling coefficients in two-core microstructured optical fibers. We first present the three already published methods that will be compared to our CMT-based one.
3.1 Supermode method (SM)
The supermode theory relies on the study of the composite waveguide as a whole, i.e. the DC-MOF including its two cores. For the fundamental modes, the resolution of Maxwell’s equations in the DC-MOF results in two sets of symmetric and antisymmetric modes, so-called fundamental supermodes, for both x- and y-polarization orientations. We name E+ x, E- x the symmetric and antisymmetric x-polarized fields, and E+ y and E- y the y-polarized ones, respectively.
Any x-/y-polarized guided field in the structure can be decomposed as a linear combination of the two x-/y-polarized supermodes respectively (Fig. 2). As their propagation constants β+ x/y and β- x/y are different, they propagate at different phase velocities. These phase mismatches induce along the fiber an evolution of the global field distribution. This can be interpreted as power coupling between the two cores as well as the spatial beating of the two corresponding supermodes. The coupling coefficients Cx and Cy corresponding to each polarization can be inferred from  as
Using any mode solving software such as those based on the Finite Element Method (FEM), the Finite Difference Method (FDM), the Multipole Method (MM) [14,15,16] or the Plane Wave Expansion Method (PWEM), it is straightforward to calculate C as they provide eigenvalues which are the propagation constants. Thus, the calculations of the field distribution is not required, making this method very efficient in terms of computation time and memory space.
3.2 Beam Propagation Method (BPM)
A simple mean to characterize multiple core optical fiber coupling behaviors is the Beam Propagation Method. This powerful tool simulates the propagation of an electromagnetic field in any structure, whatever the index profile (Fig. 3). In this paper, we used a commercial software based on FDM and the simulations were performed either in scalar mode or in semi-vectorial or full-vectorial modes.
Assuming that both cores are strictly identical, the coupling coefficient can be deduced from the evolution of the power flow in each core along propagation :
where Lcx/y is the coupling length for the x-axis or the y-axis polarized modes respectively, i.e. the length over which 100% of the energy is transferred from one core to another.
3.3 Equivalent Fiber Model (EFM)
This technique consists in replacing a MOF by an equivalent step-index fiber  using the improved effective index method . The two cores of the equivalent step-index fiber are made of pure silica while the cladding index is the effective index of the space filling mode which is the fundamental mode of the infinite periodic cladding of the MOF. Then, after field calculations by analytical or numerical means, the coupled-mode theory is applied to the step-index dual-core structure.
3.4 Coupled Mode Theory (CMT)
We propose to extend the usual coupled mode theory from conventional optical fibers to microstructured optical fibers in order to model their coupling behavior and to compute the coupling coefficient C between the two cores of a DC-MOF.
Our model relies on the coupled mode theory which assumes that the two-core MOF can be described as the sum of a single-core MOF said unperturbed MOF, and an index spatial perturbation. This latter takes the form of a silica rod filling the corresponding air hole, thus constituting the second core surrounded by a six air holes ring (Fig. 4).
e⃗1(x, y) and e⃗2(x,y) are the transverse electric fields of the guided modes of each optically independent core (i.e. the unperturbed MOF) constituting the dual-core MOF (i.e. the perturbed MOF). Thus, the transverse field e⃗2(x,y) in the second core exhibits the same spatial distribution as e⃗1(x, y) but is translated with respect to e⃗1(x, y) by the value of the core separation S (Fig. 1) along the x-axis, leading to e⃗2(x, y)=e⃗1(x-S,y).
are normalization terms, k is the wave number in vacuum, n(x,y) and n̄(x,y) are the refractive index profiles of the two-core MOF and the single-core MOF, respectively. The upper integral of Eq. (3) has only to be calculated over one air hole area (Aperturbation) because the difference between a two-core MOF and a single-core MOF defined by the expression Δn2(x,y)=n2(x,y)-n̄2(x,y) is non-null only in the air-hole corresponding to the second core position (Fig. 4).
Considering the identity e⃗2(x, y)=e⃗1(x-S, y), Eq. (3) can also be written:
Therefore, to apply Eq. (4), only one field calculation is needed (e⃗1(x, y)). In our case, the field distribution e⃗1 of the guided mode in a single-core MOF is computed using a FEM based software (Fig. 5) but all other methods (FDM, MM, PWEM) giving field distribution can be utilized. One can note that the electric field distribution in the different holes (m=2 to 4) are very similar to each other while the amplitude decreases as m is increased.
The last step is a simple overlap integral computed over a limited spatial domain (Aperturbation is the hole area). The main advantage of this technique is that the calculation time is independent of C even for very low coupling coefficient values for which the coupling length is very long thus requiring long computational time up to several days with beam propagation methods. However an upper limit exists since this method is based on the assumption that the field of a two-core MOF can be decomposed as the sum of two modes evolving in independent single-core MOFs which becomes incorrect in the case of very close cores.
Our technique only needs the knowledge of the transverse distribution of the guided mode field in the core area and in the mth hole corresponding to the position of the second core. Using our model, it is straightforward to determine the coupling coefficients in two-core microstructured optical fibers in both geometries and for the two polarization states.
As an illustrating example, we investigate the coupling characteristics of a DC-MOF with Λ=4µm as a function of the ratio d/Λ in the case of two cores separated by one hole (m=2). The operating wavelength is λ=1.55µm. The numerical results presented in Fig. 6 show the evolution of the coupling coefficients C for both polarization states and for both DC-MOF geometries. One can note the extreme sensitivity of C with respect to the ratio d/Λ, putting into evidence the difficulty that may be encountered when designing and drawing DC-MOFs with controlled coupling features. The geometry #2 leads to higher C values compared to geometry #1 because of higher easiness for the optical power to cross silica bridges rather than air-holes and despite the greater separation distance between the two cores in this geometry. Polarization effects are also observed. They are shown to be more important as d/Λ is increased. This is a consequence of the d/Λ-linked growing difference between the x- and y-polarized field distributions induced by the field boundary conditions at silica-air hole interfaces. As shown by Reichenbach  and in the next section, these results agree with those found with other methods.
Figure 7 presents the influence of the core relative positions on C values. As expected, C decreases as S increases, and one can note that very low values can be achieved with reasonable d/Λ ratios and core separations. Calculations could be made over more than twelve decades due to the precision and the dynamic of the FEM for field calculations, even far in the cladding as it will be shown in section 5 (Fig. 9). Thus, coupling lengths considerably greater than thousand kilometers are achievable. Such results may be very useful for designing multi-core microstructured optical fibers devoted to the increase of the data rate in long range telecommunication links .
4. Comparison of the different methods
In order to compare the different methods mentioned previously, we present calculation results of the coupling coefficients in a DC- MOF (Table 1) where the cores are separated by one air hole (geometry #1). The pitch of the structure is Λ=4 µm. Calculations are made for x-polarized fields.
In general, the calculations based on the supermode method, the CMT and the semi-vectorial BPM provide reliable results for the coupling coefficients. Their values only differ by 5%. The difference may be due to the mesh geometry, rectangular for the BPM software and triangular for the FEM software used for CMT and supermode methods. The latter describes more precisely the hole geometry and thus leads to lower numerical error. The full-vectorial BPM gives similar values for C as the semi-vectorial BPM but is much more time consuming. Because of the strong index contrast at air-silica interfaces, the scalar BPM is less adapted to the study of dual-core MOFs (≈10-20% error) since d/Λ is greater than 0.25. The results with the EFM are far from the results of the other simulations. This can be understood as the EFM is an approximation of the two-core MOF. Therefore, field distributions of the step-index fiber are noticeably different from those of the microstructured fiber. One can notice that the discrepancy between the results from the CMT and those from the supermode method and the semi-vectorial BPM increases with C. This is due to the fact that the method is based on the assumption of two highly optically isolated cores, which become inaccurate for high values of C, i.e. low inter-core distance or low ratios d/Λ. Concerning computer resources, the memory space allocated by each program is reasonably low. However, the calculation time of the BPM is much longer and depends on the coupling lengths which are inversely proportional to C.
Results presented in Fig. 8 are performed with another MOF in order to test the four techniques when calculating small values of C. The structure of the microstructured optical fiber is the same as previously but the cores are now separated by three air holes (m=4), resulting in a great reduction of C.
Almost the same conclusions as for Table 1 can be drawn from Fig. 8. However, a drawback appears for the supermode method. Indeed, this latter is well adapted for high coupling coefficients and becomes obsolete for small values of C due to the lack of precision of the FEM software in calculating the propagation constants. Yet, it should be noted that when considering a two-core waveguide, the mesh may cause the cores to be perceived as asymmetric. Therefore, the computed energy transfer is not 100% between the two cores and C must be calculated from more complex formulas  taking into account the difference in propagation constants of each individual core. This phenomenon is more obvious as the coupling coefficients become weaker. This problem is overcome with the CMT method because only one core is considered. Only two points have been calculated for the semi- and full-vectorial BPM because the simulation times are very long (several weeks). Then, such methods are not suitable for low coupling coefficient calculations.
5. Simplified model
Achieving of very low coupling DC-MOFs and MC-MOFs leads to design fibers with both high air-filling fraction and large core separations. The unperturbed MOF must be composed of a number of hole rings great enough in order to include the position of the second core and ensure high precision calculations of the mode field and the coupling coefficient C. Thus, the structure is very complex because of its high number of holes linked by very tiny silica bridges. To properly represent such a fiber and so to limit numerical errors during simulation, the mesh drawing should be very precise, including a very high number of nodes. Then, computational limitations may be reached, that may lead to inefficiencies or impossibilities in the simulation process. To overcome these difficulties, we propose a simplified model for computing low coupling coefficient values. It is based on the simple simulation of a DC-MOF with only one hole between the two cores. Such a structure is easy to simulate, precise and time efficient with any available commercial FEM software.
To support our simplified model development, we study and present in details coupling behaviors in MOFs with geometry #1 and with an x-polarized electric field. Same deductions can be made for polarization along y axis and for the second geometry. The analysis of the field in the different holes along x-axis shows two very interesting features. First, the spatial distributions of the mode field e⃗1(x, y) in these holes are very similar to each other (Fig. 5). This is confirmed by the results of the field overlap integral calculations between the electric fields present in the different air holes (Table 2) that are all very close to the unity. There are no more than 5% residual errors, from the first hole neighboring the core (m=1) to the fourth one (m=4).
Then, the ratio γ between the amplitude of the field e⃗1 (x,y)|m in one particular hole hm and its immediately following one hm+1 (field e⃗1(x,y)|m+1) is shown to be constant whatever the hole under consideration in the studied domain (m=1 to 4). This can be explained by the fact that such a microstructured optical fiber guides the light by modified total internal reflection and consequently, the electric field in the silica cladding has a decreasing exponential form corresponding to an evanescent field. An illustration is given in Fig. 9 showing the field evolution of the guided mode in a logarithmic scale (dB) as a function of the horizontal distance along the x-axis. A linear decreasing of the amplitude of the field at the different air-hole position is clearly visible. Obviously, the slope depends on the ratio d/Λ, as well as on the pitch Λ and on the wavelength λ.
These observations allow to write the following expression of the constant ratio γ:
We can extend this result in order to establish a simple relationship between the fields in each air-hole:
where m1 and m2 are integer values representing the positions of the two considered air holes in the microstructured cladding.
We now define and express the coupling coefficient between the unperturbed MOF core (m1=0) and a second one placed at the mth air-hole position (m2=m):
Expressing e⃗1(x, y)|m with the field in the second hole e⃗1(x, y)|2 and γ:
which can be finally and simply written as :
where C2 is the coupling coefficient between two cores separated by one hole (m=2).
This expression shows that the calculation of any coupling coefficient Cm between one core and a second one placed at the mth hole position can simply be obtained from the knowledge of γ and C2. Such a result is confirmed by the results presented in Fig. 7 on which we can verify that for a particular ratio d/Λ the distance between the curves of identical polarization remains constant.
The exponential decay of the field in the cladding ensures that this model is valid and presents low relative error even for values of m greater than those presented here. Thus, Eq. (9) enables to calculate very low C values with very good approximation while requiring low computer time and resources.
In order to numerically validate this affirmation, many comparisons between CMT and simplified model results have been carried out (Table 3). The values obtained by both methods are identical until CMT method was unable to give valid results (e.g. for m=5). This is mainly due to the limitation of available field calculation methods to provide correct values far from the core for great values of d/Λ. Furthermore, for high values of m (great number of holes between cores), some difficulties appear in numerically simulating MOF structures with great number of hole rings (too many nodes in the mesh geometry). The simplified model overcomes these drawbacks and is able to provide reliable results in these conditions.
The coupled-mode theory (CMT) has been investigated and fitted to calculate the coupling coefficients (C) of dual-core microstructured optical fibers. This method only requires the preliminary calculation of the mode field profile in the single core microstructure that can be obtained by the mean of numerous numerical methods. It provides precise results and is shown to be efficient in particular in terms of time of simulation and of required computing resources.
The principle of our method was numerically validated by the comparison with several already published methods, such as supermode, equivalent fiber model and BPM (scalar, semi- and full-vectorial) methods. Furthermore, CMT has shown its ability to calculate very low values (down to 10-10 m-1) while the other methods present precision or computing time limitations below 0.1 m-1 values. The influence of the ratio d/Λ, of the microstructured geometry, of the polarization state and of the core separation was presented.
The mode field profile in the cladding area was analyzed, pointing out firstly the similarity of the field distribution in the different holes of the microstructure, and secondly the exponential decrease of their amplitude from one hole to the other as function of the core distance. This contributed to validate our results for very low C values when no other tested calculation methods could give a comparison. Furthermore, this led to establish a simplified model allowing C evaluation even for much lower values.
Our model, precise and efficient, especially for weak coupling, could be applied to multi-core microstructured optical fiber, and then, may show its utility in their various applications such as very high bandwidth telecommunications systems, multi-core fiber lasers, couplers, WDM multiplexers and sensors.
Authors are very grateful to Professor Suzanne Lacroix and Professor Nicolas Godbout from Ecole Polytechnique de Montréal for very useful discussions on the coupled mode theory. This study was supported by the Délégation Générale à l’Armement which provides the research grant of Nicolas Mothe.
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