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A highly birefringent photonic crystal fibre has been characterised as a function of temperature. The modal birefringence has been found to be independent of temperature from -25 to 800 °C.
©2004 Optical Society of America
1. Introduction
A single mode optical fibre that has a significant difference (≥ 1×10-4) in the effective refractive indices for the two polarisation modes (eigenstate) is described as a highly birefringent fibre (HiBi). The modal (or phase) birefringence B is defined as [1]
where nx and ny are the effective refractive indices for each polarisation mode. The modal birefringence is often presented in the form of a beat length given by LB = λ/B [1], where λ is the free space wavelength. The modal birefringence found in conventional HiBi optical fibres can be attributed to either form or stress modal birefringence [2,3]. Form modal birefringence is usually introduced by having an elliptical shaped core and/or cladding region. Stress modal birefringence arises through the stress-optic effect [3]. By placing regions with different thermal expansion coefficients from the host material on either side of the core, differential stress for the x and y polarisation modes is introduced.
Photonic crystal fibres (PCF) with modal birefringence comparable to and greater than conventional highly stressed bow-tie and PANDA fibre have been demonstrated previously [1,4–7]. Modal birefringence in highly birefringent photonic crystal fibres (HiBi-PCF) usually arises from the fibre waveguide geometry i.e., form modal birefringence, where two-fold rotational symmetry in the arrangement, position and/or shape of both the holes and the core region causes modal birefringence. The increased refractive index contrast available with PCFs also offers greater flexibility in design. Since guidance properties are determined by the modes interaction with the core region and the surrounding rings of holes, it is expected that the wavelength dependence of these fibres is also different. Polarising HiBi-PCFs with useful properties such as operation over a very large wavelength span are possible [8,9].
Since modal birefringence arises from thermal stresses in bow-tie and PANDA fibres they are highly temperature dependent. HiBi fibres with low temperature dependence would be of significant interest for a number of applications, particularly those involving sensing where temperature variations are present. For example, temperature effects in the highly stressed conventional HiBi fibre limit the accuracy of fibre optic gyroscopes, as well as current sensors based on spun elliptically polarising optical fibre [10,11].
The temperature dependency of the modal birefringence of fibres based on waveguide geometry promises to be much less than that of stress induced modal birefringence. The dominant mechanism that has been suggested to contribute to the temperature dependence of the modal birefringence in HiBi-PCFs is thermo-optic effects in the waveguide material [12–14]. This temperature dependence is predicted to be very low for waveguides in silica materials which have a thermo-optic refractive index coefficient of dn/dT = 1.1×10-5K-1, but is approximately an order of magnitude larger for HiBi-PCFs fabricated in a polymer host material such as PMMA which has a thermo-optic coefficient of dn/dT = 1.1×10-4K-1 [12–14] The thermo-optic coefficient of the air in the holes is negligible, dn/dT = 1×10-6K-1.
In this paper, we study the temperature dependency of the group birefringence of a HiBi-PCF. Results are presented over a temperature range from -25 to 800°C, and for comparison, a conventional bow-tie HiBi fibre is studied over the range from -25 to 175°C. In addition we introduce a method that allows the modal birefringence to be determined from the experimentally measured group birefringence.
2. Methods
2.1 Experiments
Figure 1 shows the cross-section of the fibre used in this study. The air holes are arranged in a triangular lattice. The average hole diameter is 4.4μm and the pitch 7.4μm. The elliptical shaped fibre core has an approximate dimensional ratio of 2:1 (10.9μm: 5.6μm), with position and geometry of the holes of the innermost ring structure also slightly distorted from a perfect six fold rotationally symmetric structure. The highly birefringent photonic crystal fibre was manufactured from stacked capillaries using a proprietary method.
The crossed polariser low coherence interferometric method was used to determine the group birefringence [1]. To implement this technique, the interferometer shown in Fig. 2 was constructed. Broadband depolarised light was launched into a fibre pigtailed polarising beam splitter (PBS). The PBS was pigtailed with polarisation maintaining linearly birefringent fibre such that each output arm contained light polarised in one linear polarisation state only. One of these output arms of the beam splitter was spliced to the HiBi-PCF under test with a rotational orientation such that both polarisation modes were equally excited, i.e., at 45 degrees. By reflecting light (approx 4%) from the cleaved end face of the HiBi-PCF the accumulated phase shift between the two polarisation modes was effectively doubled. The wavelength dependent output intensity that arises from unbalanced path lengths in the HiBi-PCF within the interferometer was recorded on an optical spectrum analyser. The periodic modulation imposed on the output spectrum, caused by the total retardance of the test section of fibre, can be considered as an interference profile or interferogram for brevity.
To improve the accuracy of the measurements, the interferogram was recorded over a large range of wavelengths by illuminating the test fibre with a broadband light source whilst recording the output spectrum using an optical spectrum analyzer. In this case both a white light source and an erbium doped fibre - amplified spontaneous emission (EDF-ASE) light source were used. The observed period of the spectral ripple is a function of the path imbalance between the two polarisation modes arising from the total retardation of the test section. By heating this section and observing the phase and period of the spectral ripple any change in the total retardance can be detected. The signal processing was automated and determined both the phase and average period of the interferogram by applying a fast Fourier transform.
2.2 Group and modal birefringence analysis
The group birefringence is the difference in the group indices for the two polarisation modes; it contributes to the polarisation mode dispersion (PMD) of the fibre. The average group birefringence of the test section of a fibre can be determined from the spectral ripple period, Δλ, seen in the interferogram. The method is detailed in the following analysis.
The phase difference, ϕ=4πLB/λ, between the two polarisation modes after two trips through the test section of fibre of length, L, needs to be evaluated [1]. A single period of the spectral ripple, Δλ, corresponds to a 2π phase shift between the two polarisation modes. Differentiating the phase with respect to wavelength and substituting dϕ=2π,and dλ=Δλ leads to the following relationship between the group birefringence, Bg, and wavelength [5],
The modal birefringence B (see Eq. (1)) is often of more interest as it determines the polarisation holding power of the fibre. Polarisation maintaining properties are important in most sensing and interferometric applications. To calculate the modal birefringence for the HiBi-PCF from the typical interferogram shown in Fig. 3, the wavelength dependence of the modal birefringence must be determined. If for example, the modal birefringence increases with wavelength this would have the effect of compressing (or reducing the period) of the spectral ripple seen in the interferogram. To determine the modal birefringence an assumption concerning the wavelength dependence of the modal birefringence is necessary. This is simplified for conventional stress HiBi fibre, where the modal birefringence is largely independent of wavelength [3], and therefore the derivative term in Eq. (2) vanishes and the group and modal birefringence are equal. However, in the case of HiBi-PCFs the wavelength dependence cannot be ignored when evaluating the modal birefringence. The assumption that the wavelength dependence of the modal birefringence in HiBi-PCFs follows an empirical power law dependence of the form B=αλk, where α and k are constants to be determined, is strongly supported in the literature [1,7,15]. Substituting this assumption into Eq. 2 it follows that
As a result, the constants α and k can be determined by plotting group birefringence versus wavelength and numerically fitting a power law function B=αλk. The modal birefringence can then be calculated from the group birefringence using
Equation (4) shows that the group and modal birefringence differ not only in magnitude but for typical k values for HiBi-PCFs of between 2 and 3 [1,7,15] they are also opposite in sign. Consequently, the fast and slow polarisation axes are reversed when considering either modal or group birefringence [12,18].
3. Results
3.1 Temperature dependence of HiBi-PCF
A 1.2 m sample of the HiBi-PCF was placed in an environmental chamber and temperature tested from -25 to 95°C. Interferograms were recorded at approximately 25-degree intervals between -25 and 95°C. The test sample was then shortened to 0.53m to fit it inside a tube furnace where the fibre was tested over the temperature range from 95 to 800°C and interferograms were recorded at 50-degree intervals. Over the entire test temperature range the modal birefringence only varied between 1.11 ×10-4 and 1.14 ×10-4, which corresponds to a range of approximately +/- 1.5%. This is within the range of the experimental error. The modal birefringence, at the central wavelength (1540nm), as a function of temperature is represented by the crosses in Fig. 4. For comparison, the temperature dependence of the modal birefringence of conventional bow-tie HiBi fibre is also shown on the same graph. A slope of 7.05 ×10-4 K-1 (ΔB/B) is evident for the bow-tie HiBi fibre, whereas the calculated modal birefringence of the HiBi-PCF is independent of temperature.
3.2 Determining the wavelength dependence of the HiBi-PCF
The group birefringence of the HiBi-PCF was measured using the crossed polarizer low coherence method. The recorded interferogram, using a white light source, was divided into sections and the group birefringence calculated for the centre wavelength of each section. A broadband EDF-ASE source was also used to further extend the range over which the group birefringence was calculated. These values were then plotted (see Fig. 5) and a power law function of the form B=αλk was numerically fitted to the data to determine the k value. This k value was then used to calculate the modal birefringence of the HiBi-PCF using Eq. (4) (also shown in Fig. 5). When the two curves are plotted they appear to diverge with increasing wavelength. However, the modal birefringence is a constant fraction of the group birefringence as defined by Eq. (4). The resulting modal birefringence measured at 1540nm for this HiBi-PCF is 1.13 × 10-4.
The group birefringence of a sample of conventional bow-tie HiBi fibre was also characterised using the same methods. In this case the fibre was designed to operate around 800nm and was too lossy at 1550nm to use the EDF-ASE source and the white light source only was used. The relatively flat wavelength dependence can be seen in Fig. 5.
Fig. 5. Measured group birefringence and calculated modal birefringence as a function of wavelength for a bow-tie fibre and HiBi-PCF (measurements made @ 25°C).
4. Discussion and conclusion
The modal birefringence of the HiBi-PCF studied here is, within experimental error, independent of temperature over the range -25 to 800°C. These results not only reinforce the assumption that modal birefringence based purely on waveguide geometry is largely temperature independent, they also suggests the absence of residual thermal stresses in this fibre. The large temperature range for these tests suggests a possible usefulness beyond the normal operating range for optical fibres. During these tests the acrylate coating material was completely burnt away above 400°C leaving behind bare uncoated fibre. The challenge may be in finding coating materials that could operate over an equally extended temperature range.
Sensing and interferometric applications requiring polarisation maintaining fibre with little or no temperature dependence may benefit greatly from using such HiBi-PCFs. HiBi-PCFs with low temperature dependence could also be designed to incorporate other features such as special dopants for sensing and amplification such as in a singly and strongly polarised high power fibre laser, for example. In addition polarizing fibre with large and thermally stable polarizing windows could be manufactured.
It has also been shown that by studying the wavelength dependence of the group birefringence, the modal birefringence can be determined. The experimentally determined value of k for this fibre was found to be 2.82. Substituting this k value into Eq. (4) gives a group and modal birefringence that are opposite in sign. This means that the fast and slow axes of the fibre are reversed when considering the phase and group velocities of the polarisation modes.
Acknowledgments
A. Michie acknowledges the financial support of ABB and Transgrid. Useful discussions with Nader Issa, Ian Bassett, and John Haywood are also acknowledged. The Australian Research Council is also thanked for financial support of this work through an ARC Discovery Project.
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