## Abstract

We report on detailed numerical investigation of stress-induced birefringence in micro-structured solid-core optical fibers. The stress is induced either by external forces or stress applying parts inside the fiber. Both methods lead to different stress distributions where screening as well as enhancement effects due to the air-hole micro-structuring could be observed. Furthermore, we discuss the potential of the realization of polarization-maintaining low-nonlinearity micro-structured fibers that are suitable for applications in ultrafast optics.

© 2005 Optical Society of America

## 1. Introduction

Currently, micro-structured optical fibers (also called photonic crystal fibers (PCF)) are under intensive research due to their superior properties compared to standard single mode fibers [1]. They consist of a regular micro-structured array of holes, where the core is formed by a solid- or air-filled defect (Fig. 1). The freedom in the design parameters allows for new properties like dispersion shift, endlessly single mode operation [2] and even low-index bandgap guidance, e.g. in air. In combination with the precision in the fabrication process, exceptional fiber parameters have been achieved. For instance, shifting the zero dispersion wavelength in a fused silica fiber into the visible spectral region [3], truly single-mode extended large-mode-area (LMA) fibers [4] and even low-loss propagation in air-core fibers have been demonstrated [5]. Beside these extended fiber parameters, completely new experiments in different areas of research have been developed using micro-structured fibers [6–8].

However, many experiments and applications require polarization-maintaining or single-polarization fibers [9]. The birefringence required for this purpose is usually achieved by geometrical effects (form birefringence, e.g. elliptical cores) or the elasto-optic effect [10]. Form birefringence has been successfully applied to solid- [11] as well as to air-core photonic crystal fibers [12]. Beside the fact that larger values of form birefringence can be obtained compared to step index fibers, this technique suffers from the disadvantage, that the birefringence decreases rapidly for larger cores. Thus, to obtain polarization maintaining large-mode-area fibers one can use the well-known technique of stress-applying parts (SAP) inside the fiber. This is done by elements next to the core, which have a different expansion coefficient compared to fused silica and therefore generate a permanent stress field during the drawing of the fiber. Due to the elasto-optical effect birefringence is introduced. If this technique is applied to photonic crystal fibers the dependence of the polarization properties on temperature and wavelength is coupled to the ratio of stress- and form induced birefringence, but for large cores the behavior is similar to standard fibers using SAP [13]. Thus, the technique of SAPs has been demonstrated in combination with LMA PCF [14] and single-mode fibers with core diameters up to 15 µm are commercially available [15]. However, the scaling of a single mode core is not restricted to that value. Recently, impressive results from a Yb-doped large-mode area single mode photonic crystal fiber have been obtained, which were only possible by further increased core sizes (~40 µm) [16]. There, no polarization-maintaining mechanism is applied in the fibers, but the experimental results indicate a drop in the degree of polarization with increased output power. Similar experiments in ultra-fast optics would therefore benefit from polarization-maintenance especially in such extended large-mode area fibers.

In this contribution, we show how the air-hole structure influences the birefringence generated by a stress field. This stress field is either induced by an external point force or stress-applying parts. In contrast to previous works on external forces [17], we emphasize on the dependence of the birefringence from the outer fiber diameter and the screening resulting from separated rings of the air-hole structure. Furthermore, screening effects in micro-structured fibers, where stress-applying parts induce the birefringence, are investigated. With these results we draw some general conclusions for values of birefringence that can in principle be obtained in low-nonlinearity photonic crystal fibers.

## 2. Theory

The stresses resulting from thermal expansion and/or external forces can be calculated according to the equilibrium equation (Eq. 1), where σ is the stress tensor, ε_{x/y} are the normal strain components, γ_{xy} is the shear strain component, * D* is the elasticity matrix describing an isotropic material using Young’s modulus

*E*and Poisson’s ratio

*ν, α*is the expansion coefficient,

*F*the force,

*T*the high temperature and

*T*

_{ref}the reference temperature, e.g. room temperature. In this equation small displacements and the plain-strain approximation, meaning zero axial strain in z-direction, are assumed [18].

The elasto-optical effect relates the stress σ to a change of the refractive index Δ*n* according to

If the photo-elastic tensor is known the refractive index distribution can be used to solve the full vectorial wave equation. From there, the modal birefringence is calculated to *B*_{modal}${\mathit{=}n}_{\mathit{\text{eff}}}^{x}$${\mathit{-}n}_{\mathit{\text{eff}}}^{y}$
, where *n*_{eff}
are the effective indices of the two polarized fundamental modes. A good estimation of this quantity can already be made without the modal calculation even for photonic crystal fiber [17]. Thus, for fused silica, the birefringence can be evaluated using equation 3 with the values of C_{1} and C_{2} given in table 1, where *B* is the birefringence in the core center and *B*
_{av} the average birefringence over a given core area.

$${B}_{\mathit{av}}=\mid \left({C}_{2}-{C}_{1}\right)\iint \left({\sigma}_{x}-{\sigma}_{y}\right)r\phantom{\rule{.2em}{0ex}}\mathit{dr}\phantom{\rule{.2em}{0ex}}d\phantom{\rule{.2em}{0ex}}\phi \mid $$

It is now possible to calculate the birefringence created by external forces like twists or lateral forces as well as birefringence induced by SAP [17,10]. For some cases, analytical solutions of equation 1 are available. For complex geometries, like in our case of photonic crystal fibers, one has to solve the problem numerically. One flexible and easy way is the use of the finite element method (FEM), where commercial products are available [19].

Because most of the calculations in this paper refer to photonic crystal fibers, all calculations are done using FEM. A finite element analysis solves a numerical problem by subdividing it’s objects into very small but finite-size elements, where each element is described by equations concerning physical and boundary properties as well as their behaviors. Solving these equations predicts the behavior of the whole object. The quality of the results clearly depends on the shape and number of elements used. For our calculations, a triangulation is done and the element size is decreased until the calculated results have been converged. To reduce calculation time and memory requirements, the object is reduced according to symmetries by means of the right boundary conditions.

Our investigation concerns two cases: firstly, lateral external forces are applied to a micro-structured fiber and the dependence on the outer diameter is evaluated. Secondly, SAPs are placed outside the micro-structured core region. For both stress fields, the screening of the stresses due to the air-hole structure is evaluated in the following sections.

A fiber with SAPs in a circular shape (stress rods) is shown in Fig. 2. This type of fiber is called PANDA type fiber. Different thermal expansion coefficients for the material of the fiber and SAPs generate a permanent stress field when cooling the fiber below the softening temperature during the drawing process. Beside the physical parameters of the materials (*E, ν, α, C*), which cannot be changed greatly, the geometry parameters (the diameter of the stress rods *R*, the distance from the center r_{1} and the diameter of the fiber *D*) influence the value of the birefringence in the fiber center *B* following the analytical expression of equation 4 [20].

Generally, the distance r_{1} has to be minimized without influencing the guided mode. In the case of photonic crystal fibers, the SAPs can only be applied outside the air-hole cladding. A hexagonal holey cladding is shown in Fig. 1. The guiding properties in such solid core PCFs are only determined by the structural parameters, where *d* is the air-hole diameter and Λ is the pitch [2,21]. Clearly, the minimum distance of the SAPs has to be at least *r*_{1}
=(*N+0.5*)*Λ*, where *N* is the number of air-hole rings surrounding the core. It has already been proven that 4 to 5 rings allow for low loss guidance [22].

The results for these micro-structured PANDA type fibers are presented in section 3.2 and general conclusions for actively doped low-nonlinearity polarization-maintaining fibers will be drawn in section 4.

## 3. Screening and enhancement of induced stresses

#### 3.1 Stresses by lateral forces

The schematic representation of our calculations for a lateral external force is shown in Fig. 3. A point force is acting in x-direction, which induces a stress distribution inside the fiber. As a result of the FEM calculation, all components of the stress tensor are available. From this, the birefringence *B* or *B*_{av}
is calculated using equation 3. A typical result of such a calculation is shown in Fig. 3 containing the principle stress components σ_{x} and σ_{y} as well as the birefringence *B*.

The air-hole structure influences the induced stress distribution. The relative size of the hole is characterized by the quantity d/Λ, which is also responsible for the guiding properties of such a fiber. It is clear that in the limit of d/Λ=0 the birefringence becomes that of a standard step index fiber. If d/Λ reaches 1 the stresses and birefringence are screened completely, which means that the core is isolated from the cladding. We calculated the exact behavior of this screening effect. For this calculation a large mode area fiber with a pitch of Λ=12 µm is chosen, which expands the holey cladding to a diameter of ~130 µm. The results of the birefringence *B*_{av}
(averaged over a radius of 6 µm) are shown in Fig. 4 for different outer diameters *D* of the fiber; 170 µm, 250 µm and 400 µm. The acting force *F* is chosen in a way that the birefringence obtained without holey cladding is the same for all three fibers. This is an easy task, because the problem scales linearly with the force *F* (Eq. 1). As the air-hole size increases, a continuous drop in the birefringence is expected, but the calculations show that the birefringence remains almost constant and even increases for the smallest outer diameter of 170 µm until a certain value of d/Λ is reached. Furthermore, the screening depends on the outer diameter. To give some quantities: the value d/Λ where a 10 % drop in the value of the birefringence is observed due to the screening of the holey cladding is ~0.65 for *D*=400 µm, ~0.75 for *D*=250 µm and ~0.9 for *D*=170 µm. This result does not change significantly for the birefringence *B* in the core center (inset of Fig. 4). From this it can be concluded that the air-hole cladding does not introduce any disadvantages in terms of screening a stress field in a PCF up to a value of d/Λ>0.65. Just to remember: the condition for endlessly single mode operation for a one-hole missing fiber, where most of the photonic crystal fibers usually work, is d/Λ<0.45. Even if d/Λ needs to be larger and a high stress difference in the core region is required, the outer diameter can be chosen to be as small as the holey cladding diameter.

The reason why the screening of the stress is strongly dependent on the outer diameter, especially if the diameter of the holey cladding is comparable to the outer diameter, can be explained as following. The distribution of the induced stresses is visualized in terms of the Von Mises stress σ_{v}. This quantity is usually used to summarize the stress tensor to estimate yield criteria for ductile materials [23]. In the case of plane stress it is defined as:

Beside the differential stress (σ_{x}-σ_{y})^{2} responsible of birefringence it also represents the absolute value of the stresses. Fig. 5 shows the distribution of σv for the fiber with D=170 µm (a) and *D*=400 µm (b) around the holey cladding. For the thinner fiber, it can clearly be seen that the main part of the stresses are influenced by a *smaller part* of the holey cladding. In the thicker fiber the stress distribution is more uniform across the *whole* holey cladding. Thus, if less holes effectively influence the stress distribution, a larger hole diameter is necessary to screen the birefringence significantly, which is indeed observed (d/Λ~0.9 for the *D*=170 µm fiber) and therefore indicates the dependence on *D*. Furthermore, for a force placed close to the air-holes, the stress is enhanced between the holes due to the decreased area, if d/Λ increases. These stresses are channeled further in direction towards the core, acting against the screening. This is the reason for the increase of the birefringence with increased air-hole size in the 170 µm fiber. For the other fibers this effect is probably hidden due to the effect of screening by the whole cladding.

#### 3.2. PANDA-type induced stresses

If stress rods are placed inside a fiber, a different stress distribution is obtained compared to a point force acting on the fiber as done in the last section. Beside the obvious forces along the axes connecting the two stress rods, perpendicular forces act due to the dimension and shape of the rods. For a given fiber diameter *D* and a minimum distance of the stress rods *r*_{1}
an optimum rods diameter *R* can be calculated according to equation 4 to maximize the birefringence in the core center. By varying all geometric parameters including the position of the rods we checked numerically that this law also holds if a holey cladding is placed between the rods, which means that a further optimization is neither necessary nor possible.

For our further calculations we set the outer diameter of the fiber to *D*=400 µm. The holey cladding is that of the previous section, meaning Λ=12 µm and 5 rings of air holes. The optimized stress rod diameter *R* is 110 µm using equation 4 with a minimum distance r1 of 65 µm. The temperature values used are *T*=1000°C and *T*_{ref}
=20°C. The materials used are fused silica as the fiber material and boron-doped silica for the stress rods with the parameters given in Tab 1. The result of the dependence of the birefringence on the relative air-hole size is shown in Fig. 6. Again, the birefringence stays almost constant until it is significantly screened for values of d/Λ>0.65. With the conclusions drawn for the screening of a point force in the previous section, it is now clear that this dependence is similar to the large outer diameter because the stresses induced by the rods act more uniform across the holey cladding. A movie showing the resulting birefringence in a PANDA type fiber when changing the air-hole diameter is shown in Fig. 7.

Two more investigations have been done at a point where screening effects are observable, meaning d/Λ=0.75. Firstly, keeping all other parameters constant and removing the air-hole rings from the outside of the cladding leads to the graph shown in Fig. 8 (corresponding movie: Fig. 9). Secondly, the rings have been removed from the inside with results in Fig. 10 (corresponding movie: Fig. 11). Interestingly, removing up to three rings from the outside does not change the birefringence in the core center. If the rings are removed from the inside, the birefringence increases almost linearly. With two outer rings left, the same birefringence is obtained compared to the fiber without holey cladding. If only the outer ring of air-holes is left, the birefringence is even higher than without the holey cladding. This observation proves the proposal made in the previous section, that the smaller area between the air-hole increases the stress and is guided into the fiber center. This investigation seems to be a more or less academic task, but one important result is obtained in this configuration of the holey cladding: the birefringence and mode-field area for a seven-missing hole core is higher and therefore favorable in terms of reduced nonlinearity and achievable birefringence over a one hole missing fiber. Even if the relative hole size of such seven-missing hole fibers have to be much smaller for endlessly single mode operation (d/Λ<0.15, [22]), a more uniform distribution of birefringence across the core is obtained. Furthermore, seven-missing air-hole cores in combination with four ring holey cladding to confine the light have already proven to work experimentally up to a diameter of 50 µm in the 1 µm wavelength region. The combination of this holey cladding with the PANDA design to obtain highly birefringent fibers will be discussed in the next section.

## 4. Birefringence in rare earth doped, low nonlinearity photonic crystal fibers

Many experiments have proven the potential of power scaling when using rare earth doped fibers as a gain medium in lasers and amplifiers [24]. In these fibers, the actively doped core is usually surrounded by a second highly multimode waveguide, which is called inner cladding or pump core. This has the advantage, that low brightness high power diode lasers can be launched into this core. The pump light is then gradually absorbed over the entire fiber length and is converted into high brightness high power laser radiation. For power scaling it is necessary to have reduced nonlinear interaction of the laser light with the fiber material. In principle, the nonlinear effects scales with the length of the fiber and the intensity of the laser light. The intensity can be lowered using larger core diameters. The length of the fiber is given by the absorption length. Thus, assuming a fixed rare-earth doping concentration the overlap of the pump light with the active core has to be maximized, meaning the ratio of core diameter to outer diameter (here: ~pump core). On the one hand, the scaling of the laser core is limited by low loss single-mode conditions. On the other hand, for polarization maintaining fibers the size of the pump core is limited as it might include SAPs and the micro-structured core.

In Fig. 12, a conventional LMA step index fiber is compared to a micro-structured LMA fiber in terms of achievable birefringence for different outer diameters. Both fibers exhibit the same core diameter of 30 µm. For the step index fiber the minimum distance is set to *r*_{1}
=40 µm, which is a common distance for stress rods to not disturb the guided mode. Due to the expansion of the holey cladding in the PCF (d/Λ=0.25 and Λ=12 µm) *r*_{1}
has to be as high as *r*_{1}
=66 µm. A simple scaling of the geometry will change neither the birefringence nor the nonlinearity as defined above. The blue line in Fig. 12 indicates this fact, where the fiber with an original diameter of 400 µm has been scaled. To compare the achievable birefringence, the outer diameter is changed and the stress rods parameters are optimized, whereas the core region is kept constant. The main difference in achieved birefringence is only attributed to the difference in *r*_{1}
- no screening is expected for that holey cladding. As a result, the birefringence is decreased by more than 50%. To compensate for that, one has to use almost twice the outer diameter, which also means four times the length and therefore nonlinearity compared to a step index fiber. For actively doped low-nonlinearity photonic crystal fibers, the application of SAP to introduce birefringence seems to be questionable. Anyway, although the birefringence is lowered, photonic crystal fiber can provide true single-mode operation even at this large mode field diameter. On the other hand half of the birefringence could be enough to achieve polarization maintenance. Furthermore, several options are available to increase the birefringence in the fiber, which have not been included in the calculation. For instance, it is well known that pre-treatment of the fiber pre-form, e.g. preheating, is beneficial to maximize the birefringence. Additionally, the actual value of the expansion coefficients of the fiber materials will define the birefringence and changes are expected in a linear manner [25].

## 5. Conclusion

We analyzed the influence of a holey cladding to stress distributions inside a fiber induced by a point force acting on the fiber and stress applying parts inside the fiber.

For the first case, we found that the induced birefringence depends on the ratio of holey cladding diameter and outer fiber diameter, e.g. the distance of the force from the holey cladding. As a result, it could be shown that the closer the force is acting, the less screening is observed by the air-holes even at high air-hole diameters. This could be explained by the smaller effective number of holes influencing the stress distribution significantly. For thin fibers, the screening does not occur before a relative air-hole size d/Λ of ~0.9. In any other case the holey cladding does not influence the birefringence in the core center unless a value of d/Λ of above 0.65 is reached, which is well above the condition for endlessly single mode operation of d/Λ<0.45. Furthermore, we explained the fact that an enhancement of birefringence can be observed if the air-hole diameter is large and the fiber is made thin compared to the holey cladding. This could be interesting for sensor applications.

Secondly, we showed, that the birefringence induced by stress applying parts is significantly screened at value of d/Λ>0.65. We explained this by the fact, that again the whole holey cladding influences the stress distribution.

Finally, with the results obtained during our analysis we discussed the birefringence that can be achieved in LMA PCFs when stress-applying parts are placed outside the holey cladding. In comparison to LMA step index fibers, the birefringence is only half that value. The main reason is found to be the distance of the stress rods to the core.

## Acknowledgments

This work was supported by the Bundesministerium für Bildung und Forschung (BMBF) under contract number 13 N 8336.

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