1. Introduction

Single polarization fibers (SPF) and polarization maintaining fibers (PMF) have found many applications ranging from telecommunications, sensor (gyroscope), and lab measurements. In recent years, we have proposed, implemented, and commercialized a single polarization fiber with dual air holes [1-2]. In order to achieve single polarization operation, the fiber must possess both high birefringence and fundamental mode cutoff. In the dual air hole fibers [1], both features are achieved with the air holes in conjunction with high refractive index in the core. Such fibers can be used with conventional fibers by properly splicing the fibers for a wide range of applications that require polarization management.

However, recent developments such as those in the area of high power fiber laser also created the demand for fibers with larger mode area so that the fiber laser can operate at higher power without suffering from various nonlinear optical effects. It is also desirable that a single polarization fiber can work in a wider wavelength range so that one fiber can accommodate more applications. Different designs of single polarization fibers have emerged in the literature that are based on photonic crystal fibers with [3] and without using stress components [4]. To accommodate a wider range of needs both in terms of larger mode field and high single polarization bandwidth, in this paper we propose a novel fiber design that utilizes both stress rods and air holes. The addition of stress rods, in addition to air hole structure, allows more flexible adjustment of different parameters for desired fiber properties. For example, this new fiber structure can have a core with lower refractive index, resulting in large mode area. The fiber can be designed as either polarization maintaining fiber or single polarization fiber depending on which wavelength window is used. In Section 2, we describe the fiber structure and define the key fiber parameters. In Section 3, we describe the modeling method that addresses the fiber design problem involving multiple physical effects. In Section 4, we present the detailed analysis of various aspects of the fiber design, which carries the main results of this paper. The analysis includes the birefringence properties of the fiber from two different sources, the dependence of the fiber properties on all major fiber parameters and design examples illustrating desired performance. Finally, in Section 5 we present a brief conclusion.

2. Fiber structure

The proposed fiber consists of a core and a cladding containing two air holes adjacent to the core and two stress rods aligned in the perpendicular direction of the air holes as shown in Fig. 1. The parameters used to specify the fiber configuration include geometric parameters and refractive indices that are related to the dopants and doping levels in the core and the stress rods. The fiber core can be specified by the core radius R_core when its shape is circular or by the semi-axial dimension in the x-direction (“a”) and in the y-direction (“b”) when its shape is elliptical. The location and the dimension of the stress rod are specified by the distance between the center of the stress rod and the center of the fiber core, D_rod, and the stress rod radius R_rod, respectively. The air holes are always placed next to or very close to the core for the reason to be elaborated in Section 4, and both air holes have a radius R_hole. The refractive index of each part of the fiber is specified by the relative refractive index change, or delta, which is defined below,

There are two delta values to be specified, which are delta of the core Δ_core, and delta of the stress rod, Δ_Rod so that “i” in Eq.(1) can either represent the fiber core or the stress rod.

It should be noted that air holes and stress rods have also been used in a different fiber design in a recent paper [5]. In [5], four air holes have been placed around a pure silica core with two additional stress members placed on each side of the fiber core for the purpose of achieving single mode and polarization maintaining performance through the control of leakage loss of fundamental mode and higher order modes. Even with tight bending, a merely 2-3 dB differential loss between two fundamental polarization modes was achieved when the overall fiber suffers from 20dB/m loss. In contrast, the current paper discusses how the use of air holes in conjunction with two stress rods can result in novel single polarization properties, which are intrinsic to the fiber even without bending.

 

Fig. 1. Schematic of fiber design with air holes and stress rods.

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3. Finite element modeling

In this paper, all the numerical modeling was conducted by using a Finite Element Method (FEM) involving different physical mechanisms, which describes the buildup of anisotropic stress during the fiber making process, the stress-optic effect and electromagnetics dealing with the waveguide effect. In the first step, we calculate the stress distribution due to differential thermal expansion and translate that into anisotropic index distribution via stress-optic effect [6, 7]. The plain strain approximation, which takes into account only the stress and strain on the x-y, is applied to obtain the stress distribution. Note that the length in z direction along the fiber is very large and the stress varies little over z-position. The stress-optic effect causes an anisotropic change of the refractive index at each position at the cross section of the fiber,

where C₁ and C₂ are the stress-optical coefficients, σ_x and σ_y are the thermal stresses at a given position of the fiber cross section, and Δn_x and Δn_y are the refractive index changes at a given position of the fiber cross section due to the stresses. In the fiber design, the core is doped with GeO₂ and the stress rods are doped with B₂O₃. The material parameters including the coefficients of thermal expansion (CTE) for GeO₂ doped and B₂O₃ doped silica follow those from Refs.[6, 7].

In the next step, the stress induced refractive index changes are superimposed with the refractive indices due to the use of different dopants at different positions of the fiber so that we obtain new refractive index distributions along the x and y directions,

The effective indices are then calculated for both polarizations of the fundamental mode similarly to that in [8] for different fiber designs. For a polarization maintaining fiber that operates below the cutoff wavelengths of the fundamental polarization modes, the fiber birefringence is determined by the difference between the effective indices n_x,eff and n_y,eff of the x- and y-polarization modes,

We note that this is a more rigorous way to calculate the birefringence than the simplified method, which calculates the birefringence from refractive index difference in x- and y-direction at the center of the fiber using Eqs.(4, 5) [4, 9]. Also Eq.(6) includes both the stress-induced and form birefringence. On the other hand, for a single polarization fiber, the cutoff wavelength of a particular polarization mode can be calculated by searching for the wavelength at which the effective index is equal to the refractive index of the cladding. The single polarization bandwidth is calculated from the difference between the cutoff wavelengths of the two fundamental polarization modes.

The procedure described above can be used to conduct modeling using different parameters. It is difficult to scan all the parameters altogether as too many parameters are involved. To make the study more efficient, we obtain the dependence of the fiber properties on a particular parameter by fixing other parameters and scanning this parameter only.

4. Detailed modeling results

In this section, we present the detailed modeling results using the modeling method described in Section 3, which include the understanding of birefringence mechanism, the roles of various fiber parameters, and design examples.

4.1 The birefringence mechanism

First, we want to analyze and understand the birefringence contributed from the stress rods and the air holes. Note that the birefringence in the proposed fiber structure comes from two different mechanisms. The birefringence contributed from the stress rods is called stress-induced birefringence while the birefringence from the air holes is called form birefringence. The key question here is whether or not the birefringence from the two sources is added constructively.

To answer the question, we look at an example fiber that has the parameters Δ_core= 0.35%, R_core=4.2μm, Δ_rod = -0.75%, R_rod=15μm, R_hole=10.2μm. With the presence of both the stress rods and dual air holes, the calculated total birefringence is 5.34×10^-4. We then model a fiber with the same structure except that we take out the air holes. The modeling shows the birefringence of 4.31×10^-4. We subsequently model the fiber by taking out only the stress rods and find that the birefringence is 1.12×10^-4. It can be found that the sum of the individual birefringence is approximately equal to the total birefringence with both the air holes and stress rods in place. This indicates that the birefringence from both structures contributes to the total birefringence constructively.

The additional birefringence mechanism is critical in achieving high overall birefringence of the fiber and large bandwidth for the single polarization operation. Note that another mechanism that is critical to the single polarization fiber design is the mechanism to have fundamental mode cutoff, which is largely contributed from the low index air holes in conjunction with a proper choice of core delta. The single polarization operation is achieved at a wavelength range above the cutoff of one fundamental polarization mode but below the cutoff of the other fundamental polarization mode. In the previous design [1-2], air holes plays both the roles of generating high birefringence and inducing fundamental mode cutoff. With the additional birefringence from stress members, fiber parameters can be adjusted in a wider range allowing more flexibility in design, such as to have a larger mode area while keeping a large single polarization bandwidth.

4.2 Role of core parameters

Now, we look at the dependence of the fiber properties on the core parameters. We first fixed all other fiber parameters except the fiber core delta. We have chosen Δ_rod = -0.75%, R_Core=4.2μm, R^Rod=15μm, D_Rod=25μm, R_hole=6μm. In Fig. 2, we show the dependence of the cutoff wavelengths of both fundamental polarization modes on the core delta. Due to larger birefringence, the two polarization modes reach the cutoff at different wavelengths. It can be found that the cutoff wavelengths of the two polarization modes can be tuned by changing the core delta. The higher the core delta, the higher the cutoff wavelength for each polarization mode will be.

To understand the role of the core radius, we changed the core radius and locked all other parameters to examine the effect of core radius on the cutoff wavelengths. The parameters used in the calculation are: Δ_core=0.35%, Δ_Rod=-0.75%, R_Rod = 15μm, D_Rod=25μm, and R_hole=6μm. The results plotted in Fig. 3 show that the core radius is another parameter that can be used to tune the cutoff wavelengths of both polarization modes. We also calculate the birefringence. For the two examples in Fig. 2 and Fig. 3, the total birefringence normalized to 1550 nm is around 4.0×10^-4 to 5.0×10^-5, a birefringence that is comparable to the commercial Panda type polarization maintaining fiber [10].

Core parameters are also important to determine the fiber mode area in addition to their roles in affecting the fundamental mode cutoff. For a standard single mode fiber, the core delta is typically around 0.34% while the core radius is around 4.2 μm. For certain applications when a larger mode area is desired, core delta should be lowered to allow for larger core radius while keeping the fiber single-moded. Therefore, the desired fiber mode area is another factor affecting how the fiber core parameters are chosen. The choice at a particular design is a balance of different considerations.

 

Fig. 2. Dependence of the cutoff wavelengths of two polarization modes on the core delta.

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Fig. 3. Cutoff wavelengths of both fundamental polarization modes as a function of the core radius.

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4.3 Roles of the stress rod parameters

We also study the dependence of fiber birefringence level and cutoff wavelengths of both polarization modes on the doping level of the stress rods while keeping all other fiber parameters unchanged. In Fig. 4, we have chosen Δ_core = 0.35%, R_core=4.2μm, R_Rod = 15μm, D_Rod = 25μm, R_hole = 6μm. In Fig. 4(a), we show the dependence of the total birefringence contributed from the two the stress rods and the form birefringence from the dual air holes on the delta from Boron doping. The effect of Boron doping level is seen clearly. Boron doping decreases the refractive index of glass, contributing to a negative delta value. As more Boron is added, a more negative delta is reached. The total birefringence increases essentially linearly with the Boron doping level. On the other hand, the cutoff wavelength of each polarization mode is affected differently by the Boron doping level in the stress rods. In Fig. 4(b), while the first cutoff wavelength associated with one polarization mode is basically flat, the second cutoff associated with another polarization mode moves closer to the first cutoff when Boron doping level is reduced. This behavior can be explained by the smaller birefringence related to lower Boron doping level, which is understandable since higher birefringence will make the two polarization modes more different, leading to higher single polarization bandwidth.

Other parameters for the stress rods include the size of stress rods and their distance to the fiber center. As the stress rods are primarily used to provide birefringence, the roles of these parameters are well understood [10]. In general, larger stress rod size and shorter distance between the stress rod center and the fiber center lead to higher birefringence.

 

Fig. 4. (a). Birefringence at 1550nm as a function of stress rod Delta. (b) Cutoff wavelengths of both fundamental polarization modes as a function of Boron Delta.

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4.4 Roles of the air hole parameters

The size of the air holes and their positions can affect also the performance of the fiber. In Fig. 5, we show the dependence of the cutoff wavelengths of both fundamental polarization modes on the air hole radius. We have chosen the fiber parameters to be Δ_core=0.3%, Δ_rod=-0.75%, R_core=4.2μm, R_Rod=15μm, and D_Rod=25μm. The air holes are positioned in direct contact to the core while other parameters are kept unchanged. As the air hole size increases, the cutoff wavelengths decrease. The single polarization bandwidth remains basically unchanged. The air hole size can be used as a parameter to control the location of the cutoff wavelengths in conjunction with other parameter choice.

 

Fig. 5. Dependence of the cutoff wavelengths of both polarization modes on the air hole size. The air holes are positioned right next to the core.

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In all of the above examples, the air holes are placed next to the core without separation. The separation between the core and the air holes can also affect the fiber properties, in particular the fundamental mode cutoff. In Fig. 6 we show the dependence of the cutoff wavelengths on the separation between the core and the air holes. We have chosen the fiber parameters to be Δ_core=0.3%, Δ_rod=-0.75%, R_core=4.2μm, R_Rod=15μm, D_Rod=25μm, and R_hole=6μm. It can be found that as the separation increases, the cutoff wavelengths shift to higher values, but the single polarization bandwidth does not change significantly. As we also learned from the studies of other parameters in previous subsections, a number of factors, such as the increase of the fiber core delta, core radius, can cause the increase of the fiber fundamental mode cutoff. The choice of those parameters is mostly dictated by other fiber properties for the fiber to be single-moded and to have the desired mode area. There are limited additional parameters to keep the fundamental mode cutoff or the single polarization operating window down. The air holes should essentially be in contact with the core in order to keep the single polarization operating window low enough to be within those of typical interest, such as 1310nm and 1550nm.

 

Fig. 6. Dependence of the cutoff wavelengths of both polarization modes on the separation between the core and the air holes.

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4.5 Additional factors affecting the fiber properties

 

Fig. 7. Dependence of the cutoff wavelengths of both polarization modes on the core minor dimension.

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In many of the examples above, we have chosen circular cores. Circular cores are preferred in many situations. The core can also be elliptical and it can bring in interesting features. Fig. 7 shows effects of core minor dimension on cutoff wavelengths while other parameters are fixed. We have chosen the fixed parameters to be Δ_core=0.3%, Δ_rod=-0.75%, a=4.2μm, R_Rod=15μm, and D_Rod=25μm and R_hole=6μm. The air holes are positioned right next to the core with no separation. It can be found that the cutoff wavelengths depend strongly on the minor core dimension. This feature is quite useful as it offers a way of fine-tuning the location of the cutoff wavelengths.

4.6 Design examples

A single polarization fiber is typically designed to work at a certain wavelength window, for example, around 1550nm. As shown before, adjustment to core radius, core delta, stress rod delta can change the optical performance. It is interesting to know when the single polarization operating window is locked in a wavelength of interest, how the single polarization operating bandwidth changes with the core delta. Once the core delta is chosen the core radius can only be chosen within a narrow range for the fiber to be single moded. The choice of core delta also largely determines the fiber mode area. Fig. 8(a) illustrates the results of single polarization bandwidth versus core delta while keeping the single polarization operating window around 1550nm by adjusting the core dimension. In Fig.8(b) we show the relation between the effective area for the fundamental polarization mode operated at 1550 nm. We have chosen the fixed fiber parameters to be Δ_rod=-0.75%, R_hole=6μm, R_Rod=15μm, and D_Rod=25μm. It is shown that the lower the core delta is, the higher the single polarization bandwidth will be. For the fiber with core delta of 0.2%, the single polarization bandwidth is 178 nm. Similar trend follows for the effective area with higher effective area reached at lower core delta. This means that both large mode area and high bandwidth can be achieved with this new fiber design.

The fibers illustrated in Fig. 8 can also function as PM fibers. Our calculations show that they have the birefringence in the order of 3-5×10^-4, similar to that of conventional PM fibers In addition, although in the above example, the single polarization operation occurs around 1550 nm, a wavelength of typical interest, by proper adjusting the fiber parameters, the operating window can be shifted to other wavelength of interest such as 1060 nm and 1310nm.

 

Fig. 8. (a) Dependence of the single polarization bandwidth as a function of the core delta with core radius adjusted to keep single polarization window at 1550nm. (b) Effective area of the corresponding fiber in (a).

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5. Conclusion

In this paper, we propose a new fiber design that is based on the use of air holes and stress rods to perform as single polarization fibers as well as polarization maintaining fibers. The stress rods and air holes both contribute to the overall birefringence of the fiber although through different mechanisms. We have established a finite element model that is capable of studying the design problem by accommodating both the thermal stress effect and waveguide effect. Effects of the fiber parameters to various aspects of the fiber properties are revealed. The modeling results indicate that the fiber structure allows flexible adjustments of fiber parameters for desired properties. A fiber with single polarization bandwidth as high as 178 nm around 1550nm is illustrated with a relative large effective area of 93 μm².