Numerical analysis and experimental design of tunable birefringence in microstructured optical fiber

doi:10.1364/OE.10.000246

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Numerical analysis and experimental design of tunable birefringence in microstructured optical fiber

C. Kerbage and B. Eggleton »View Author Affiliations

Optics Express, Vol. 10, Issue 5, pp. 246-255 (2002)
http://dx.doi.org/10.1364/OE.10.000246

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Abstract

We present detailed experimental and numerical results for birefringence tuning in microstructured optical fibers. Index tunable polymer is infused into specific air-holes to obtain birefringence whose tunability is achieved by temperature tuning the polymer index. We also study the symmetry properties of the modes for different waveguide structures.

[Optical Society of America ]

1. Introduction

Microstructured photonic crystal optical fibers (MOFs) [1

P. V. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica”, The Bell System Technical Journal 53, 1021–1039, (1974).

] have generated increased interest recently because they provide extra degrees of freedom in maniplating optical propeties of light such as dispersion, nonlinearity and birefingence of optical fibers [2–4

T. A. Birks, D. Mogilevstev, J. C. Knight, and P. S.J . Russell, “Dispersion Compensation Using Single-Material Fibers,” IEEE Phot. Tech. Lett. 11, 674–676, (1999). [CrossRef]

]. For example, strong waveguide dispersion and enhanced nonlinearity can be obtained in a MOF that comprises a small silica core surrounded by closely spaced air-holes [3

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air-silica microstructure optical fibers,” Opt. Lett. 25, 796–798, (2000). [CrossRef]

]. Similarly, enhanced birefringence can be achieved in such fibers [5

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T.A. Birks, and P.S.J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325–1327, (2000). [CrossRef]

], particularly in fiber designs that incorporate ellipitcal air-holes [6

M. Steel and J. R. M. Osgood, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231, (2001). [CrossRef]

] or asymmetrical distributions or air-holes [6

M. Steel and J. R. M. Osgood, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231, (2001). [CrossRef]

, 7

M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490, (2001). [CrossRef]

MOFs also provide a platform for a new class of optical devices and a number of functionalities have been demonstrated. In particular, the air-holes allow for the infusion of active materials yielding novel hybrid all-fiber optical devices that exhibit desirable properties, such as enhanced tunability, compactness, and intrinsic low insertion loss [8–11

B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713, (2002),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698. [CrossRef]

]. These devices have exploited index tunable materials to manipulate and switch mode propagation. Several examples of tunable devices have been reported including, tunable resonant filters and variable optical attenuators [10

C. Kerbage, A. Hale, A. Yablon, R. S. Windeler, and B. J. Eggleton, “Integrated all-fiber variable attenuator based on hybrid microstructure fiber,” Appl. Phys. Lett. 79, 3191–3193, (2001). [CrossRef]

,11

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. Burdge, “Control of waveguide properties in hybrid polymer-silica microstructured optical fiber gratings,” in OFC conference 2000, 3, pp. 134–136.

]. We recently presented an approach for introducing strong birefringence into MOF by infusing index tunable materials into specific air-holes [12

C. Kerbage, P. Steinvurzel, A. Hale, R. S. Windeler, and B. J. Eggleton, “Birefringent tunable hybrid microstructured optical fiber,” in CLEO conference 2002.

]. Filling specific air-holes creates strong waveguide asymmetry that can be temperature tuned. The birefringent MOF was studied in two different contexts: birefringence in the cladding region, which is characterized using a long-period grating [13

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Tech. 155, 1277–1294, (1997). [CrossRef]

]; and birefringence in the core using a tapered MOF [14

C. Kerbage, P. Steinvurzel, P. Reyes, P. S. Westbrook, R. S. Windeler, A. Hale, and B.J. Eggleton, “Highly tunable birefringent microstructured photonic crystal optical fiber,” Opt. Lett, in press , (2002).

]. As well as being of general interest, investigations of these novel waveguide structures is motivated by the their potential applications as polarization controlling devices in lightwave systems.

Following our initial report, in this paper we present a comprehensive study of birefringence tuning in a hybrid MOF for a range of different waveguide geometries and compare experimental results to numerical simulations using the beam propagation method (BPM). The paper is structured as follows: In Section 2 we present a detailed summary of the fiber design and methodology for introducing waveguide asymmetry. In Section 3 we describe the BPM approach for numerical simulation of birefringence in MOF. In Section 4 we describe experimental setup for fabricating MOF with tunable birefringence and characterizing polarization properties. In Section 5 we present detailed experimental results, which are compared to numerical simulations using BPM and examine basic properties of these fibers, such as polarization dependent loss, using Jones Matrices.

2. MOF waveguide design and characteristics

2.1 Design

Fig. 1 summarizes the principle of the birefringent tunable MOF where selected air-holes are infused with index tunable polymer creating strong waveguide asymmetry. The MOF has a photosensitive germanium (Ge) core with diameter ~8μm and Δ =(n₁- n₂)/n₁ ~0.35%, where n₁ and n₂ are the refractive indices of the Ge core and the silica respectively. Fig. 1(a) shows a single layer of air-holes, each of diameter ~40μm, incorporated in the cladding of the MOF in a hexagonal geometric distribution [15

C. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–123,(2000), http://www.opticsexpress.org/oearchive/source/22997.htm [CrossRef] [PubMed]

]. These air-holes, which form an inner cladding region of ~34μm in diameter, are large enough to allow for the infusion and positioning of polymer at desired regions along the length of the fiber.

Fig. 1. (a) Scanning electron micrograph of the MOF with single layer of air-holes in the cladding and polymer infused in one of the holes. (b) Schematic cross-section of the asymmetric waveguide.

At wavelengths around 1.5μm, the core mode is unaffected by the presence of the airholes. Interaction between the propagating mode and the tunable cladding region can be achieved in two different ways: excitation of a cladding mode that is confined by the air-hole region [12

C. Kerbage, P. Steinvurzel, A. Hale, R. S. Windeler, and B. J. Eggleton, “Birefringent tunable hybrid microstructured optical fiber,” in CLEO conference 2002.

], or by tapering the MOF to a reduced outer diameter so that the fundamental mode leaves the core region and expands into the cladding [9

J. K. Chandalia, B. J. Eggleton, R. S. Windeler, S. G. Kosinski, X. Liu, and C. Xu, “Adiabatic Coupling in Tapered Air-Silica Microstructured Optical Fiber,” IEEE Phot. Tech. Lett. 13, 52–54, (2001). [CrossRef]

,10

]. The latter approach provides a robust device architecture for manipulating light propagation in the MOF. In particular, this MOF offers low splice loss to conventional fibers that is achieved by collapsing the air-holes first and then fusing the fibers together [10

Fig. 2 shows a schematic diagram of the tapered MOF. In the waist of the tapered MOF, the length is 1 cm, the outer diameter size is 30μm, the inner cladding region is ~8 μm and Δ =(n₂- n_air)/n₂ (~30%) is large where n₂ and n_air are the refractive indices of the silica-cladding and air, respectively [10

]. Light, which initially propagates in the Ge core, spreads into the cladding as it propagates into the taper region and is guided by total internal reflection at the silica/air-holes interface in the waist of the MOF. Infusion of tunable index polymer into the air-holes in the waist changes the boundary conditions at the above mentioned interface [10

]. The inset in Fig. 2 shows the large temperature dependence of the polymer index, which is an order of magnitude greater than that of silica (dn_pol/dT~-4×10^-4/°C).

When no polymer is present in the air-holes, light is guided by total internal reflection and experiences minimal loss as it propagates through the taper [9

]. When polymer with an index close to silica is introduced into the waist the mode field refracts and penetrates into the polymer filled air-hole region. This is illustrated in the bottom of Fig. 2 which shows the calculated mode field profile of the fundametal mode along the length of the waist of the tapered MOF for a fixed polymer index. Light, which is lost in the polymer, is not guided through the taper, manifesting as attenuation of the optical signal, which is also exacerbated by the inherent loss of the polymer. On the other hand, when the index of the polymer is much lower than that of silica, the mode is guided by total internal reflection and propagates through the taper with minimal loss (0.2dB).

Fig. 2. Schematic of all-fiber variable attenuator based on tapered MOF with polymer infused in the waist, the inset shows refractive indices of the polymer and silica dependence on temperature. Also the mode field profiles are shown at different positions along the length of the waiste region of the MOF, with polymer (n_pol=1.434) at 0cm, 1cm, and 2cm along the waist.

2.2 Tunable birefringence

Birefringence in optical fibers arises from the difference in the effective indices (n_x-n_y) of the two orthogonal polarization modes of an optical fiber. In MOFs, strong birefringence has been demonstrated in fibers that incorporate elliptical air-holes [6

M. Steel and J. R. M. Osgood, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231, (2001). [CrossRef]

], or an asymmetric distribution of air-holes in the fiber [5

]. It has also been shown that in general, MOFs that exhibit greater than two-fold rotational symmetry reveal no birefringence [7

M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490, (2001). [CrossRef]

]. In particular, waveguides that exhibit six-fold symmetry, such as the MOF shown in Fig. 1, comprising a hexagonal array of air-holes surrounding a circular Ge-doped core, are not birefringent [7

M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490, (2001). [CrossRef]

]. To break the symmetry of this waveguide, we selectively fill certain air-holes with index tunable materials, such as acrylate based polymer. Since the index of the polymer is temperature dependent, we can also tune the birefringence by heating this structure.

Fig. 3 shows the calculated mode profile in the waist of the tapered MOF with outer diameter 30μm. In Fig. 3(a), the mode profile is confined in the inner cladding when no polymer is present in the air-holes and exhibits the geometry of the fiber. Figs. 3(b) and (c) show respectively the mode profiles in the waist with polymer present in two opposite holes with refractive indices n_pol= 1.420 and 1.434. The mode profile extends into the polymer, changing the effective index of the corresponding polarization and manifesting in birefringence, of the order of 5×10^-4 in Fig. 3(b) and 10^-3 in Fig. 3(c). Moreover, the leakage of the mode into the polymer will result in losing light of one polarization, manifesting as polarization dependent loss (PDL), discussed further below.

Fig. 3. Mode field profile in the waist of the MOF with (a) no polymer in the air-holes, with polymer (b) n_pol=1.420 , and (c) n_pol=1.434 .

3. Beam propagation method

BPM is a numerical tool for the study of wave propagation in media and solves either the Helmholtz or the paraboloic wave equation [16

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Tech.. 18, 1084–1100, (2000). [CrossRef]

,17

M. D. Feit and J. A. Fleck, ”Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” App. Opt. 19, 2240–2246, (1980). [CrossRef]

]. The launched (excited) field, which is expressed as a function of its known transversal coordinates, is propagated through the waveguide where the field components are computed at two planes separated by a small but sufficient propagation step Δz. Moreover, the method contains all information necessary for a complete desciption of the field, such as the mode propagation constants and the eigenfunctions.

To calculate the birefringence in the MOF, a structure was created that has a similar index profile as that of the real fiber. The launch condition is a Guassian beam of width 8 μm and the full vector analysis by means of the iterative method is used to perform the simulations. The effective indices of both orthogonal polarizations (TE and TM), which are defined as slow and fast axes in Fig. 4, of the mode is then calculated at 1550 nm. The axis with the higher index value is called the slow axis and the other is called the fast axis. The magnitude of the birefringence is obtained by taking the difference between the effective indices of the different polarizations

B = ∣ n_{x} - n_{y} ∣ = ∣ n_{TE}^{eff} - n_{TM}^{eff} ∣

(1)

where

n_{TE}^{eff}

and

n_{TM}^{eff}

are the effective indices of both TE and TM polarization modes, respectively.

Fig. 4 shows different geometries of birefringent MOFs in which specific air-holes are infused with polymer (n_pol=1.43) and the corresponding calculated mode field patterns . To break the symmetry of the waveguide, specific air-holes can be chosen in such a way that the two orthogonal polarization modes have different refractive indices. As shown in Fig. 4(a–c), the mode profile exhibits the waveguide symmetry. In Fig. 4(d–f), the mode profiles exhbit asymmetry of the waveguide.

In addition, PDL, which is the variation of the transmitted power as the state of signal polarization taken on all values, can be calculate using BPM. Launching the same conditions as previously mentioned, the power evolution along the waveguide can then be monitored (calculated) for both TE and TM modes.

Fig. 4. Schematic of the cross sections of the waveguide showing different birefringent waveguide geometries Calculated mode profiles corresponding to the different geometries shown in Fig. 3.

4. Experiment

Fig. 5 shows the experimental setup. The fiber is tapered to 30 μm over a length of 1cm and the tapered regions are 1cm long. The mode field propagates in the Ge doped core and extends into the cladding as it propagates in the tapered region, where it is guided by total internal reflection at the interface. The mode field propagates back in the core region in the up-tapered region, with minimum loss of 0.1 dB. The fiber structure exhibits a six-fold symmetry and low PDL.

To introduce spatial asymmetry, certain air-holes are sealed with epoxy, so that only the open air-holes can be filled with polymer. A monomer blend is infused in to the open airholes in the waist at a speed of 0.01 cm/sec for a section of 1cm. The monomer is then UV cured over sections of 0.3 cm for about 15 min to form a polymer of refractive index 1.434 at room temperature at 1550 nm.

The tapered fiber is then inserted in a capillary heater, which provides temperature tuning and packaged protection. The maximum power consumption is ~300 mW, corresponding to about 3V of applied voltage and switching speed of about 1 sec. A laser source at 1550 nm is launched in the fiber. The output light is detectd by a HP lightwave polarization analyzer, which resolves the output light into the Stokes parameters that are plotted on the Poincaré sphere [18

H. G. Jerrard, “Transmission of light through birefringent and optically active media: the Poincaré sphere,” J. Opt. Soc. Am 44, 634–640, (1954). [CrossRef]

] . The Poincaré sphere maps all state of polarization to the surface of a sphere. Each point on the sphere represents a unique polariztion form or state. The north and south poles of the sphere represent right-hand circular and left hand-cicular, respectively, while the equator corresponds to linear polarizations.

Fig. 5 Experimental setup showing light form a laser source incident on the MOF, which is placed in a capillary heater, and detected by a polarization analyzer.

Fig. 6. Poincaré sphere measured as a function of temperature for the different waveguides geometries. (a) No phase shift detected without polymer in any of the air-holes, (b)&(c) symmetric waveguides show very small pahase shift. (d–f) Asymmetric waveguides show rotations on the sphere, which correspond to strong birefringence The red line correspond to the polarization states in the near part of the sphere and the blue line to those on the oppositre side of the sphere.

Fig. 6 shows the evolution on the Poincaré sphere of the output light measured by heating the MOF for the different geometries. The temperature ranges from 20°C to 150°C which correspond to polymer index change (Δn_pol) of ~0.04. Any change in phase shift between the orthogonal polarizations in the fiber correspond to rotations on the sphere. One rotation on the sphere corresponds to a 2π phase shift. The birefringence can then be deduced from the phase by the following relationship

ϕ = (β_{x} - β_{y}) L = 2 π (n_{x} - n_{y}) \frac{L}{λ} = \frac{2 πBL}{λ}

(2)

where B, L and λ are the birefringence, length of the waist, and wavelength. β _x and β _y are the propagation constants related to the effective indices n_x and n_y, respectively. In Fig. 6(a)–(c) the evolution detected on the sphere is minimal due to the symmetry exhibited by the waveguides. On the other hand, the MOFs with asymmmetrically filled air-holes show strong birefringence tuning. The polarization undergoes a phase shift of 2π in Fig. 6(d), more than 2π phase shift in Fig. 6(e), and three rotations (6π) in Fig. 6(f). The small change in the polarization state observed when all the holes or three alternate holes are filled with polymer can be attributed to the infusion and curing scheme of the polymer.

5. Experimental details

Fig. 7 shows the plot of birefringence as a function of temperature for the asymmetric geometries shown above in Fig. 6(d)–(f). The calculated and measured values of birefringence are plotted for the three cases: (a) two opposite holes filled, (b) one hole filled, and (c) two adjacent holes filled with polymer. The right axis shows the phase change between the orthogonal polarizations as detected on the Poincaré sphere for the different geometries. The calculated and measured data tend to agree at high temperature, while at low temperature scattering and non-uniformity of the polymer affect the measurements. The error bars reveals the error in repeating the experiment under the same conditions of temperature and input reference.

Fig. 7. Plot of the birefringence and phase shift as a function of temperature and polymer index for (a) two opposite holes filled, (b) one hole filled, and (c) two adjacent holes filled with polymer. The solid lines correspond to the numerical simulations and the squares with error bars represent the experimental results.

The structures shown above also exhibits PDL, which arises from the refraction of the mode into the polymer region. Fig. 8 shows the measured and simulated PDL as a function of tempetature for two different waveguide geometries: (a) two opposite holes filled with index tunable polymer; and (b) only one hole is filled with polymer. The PDL was measured using the Jones matrix method, which extracts the Jones matrix elements and solves the matrix analysis utilizing three orthogonal input polarization states and measuring the polarization response of the device at 1550 nm. First, the tranmission is measured and a three point reference frame (0°,58.9°, 118.1°) is calculated with a short fiber cable. Care must be taken not to disturb the fiber during the measurements to avoid altering the reference frame and polarization state in the fiber. Once the Jones matrix elements (J) are known, the PDL can be calculated by

PDL = 10 \log \frac{T_{max}}{T_{min}} = 10 \log \frac{s_{2}^{2} (J)}{s_{1}^{2} (J)}

(3)

where T is the optical transmittance (or power) taken over the entire polarization state space and s₁ and s₂ are the singular values of the eigenvalue equations [19

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices,” IEEE Phot. Tech. Lett. 4, 451–454, (1992). [CrossRef]

]. The PDL is large when the index of the polymer is high because, as mentioned before, the field leaks into the polymer. As a result, one polarization experiences loss with respect to its orthogonal polarization. Other sources can contribute to PDL such as the pigtails and interconnections of the instrument. The right axis of Fig. 8 shows the corresponding phase shift between the orthogonal states and indicates that there is a tradeoff between birefringence and PDL.

Fig. 8. Calculated and measured PDL as a function of temperature and refractive index of the polymer. The right axis shows the corresponding phase shift. As in Fig. 7, the solid lines correspond to the numerical simulations and the squares with error bars represent the experimental results.

7. Conclusion

We have presented different geometries of MOF that exhibit tunable birefringence. Birefringence is achieved by selectively filling the air-holes with polymer and tunability is attained by thermally changing the refractive index of the polymer. The experimental results are compared with the simulations performed using the beam propagation method. Birefringence and PDL have been discussed in the context of temperature and refractive index of the polymer.

Acknowledgments

The authors would like to thank R.S. Windeler for the fabrication of the fiber, A. Hale for the preparation of the polymer, and P.S. Westbrook and P. Reyes for useful discussions.

References and links

1.	P. V. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica”, The Bell System Technical Journal 53, 1021–1039, (1974).
2.	T. A. Birks, D. Mogilevstev, J. C. Knight, and P. S.J . Russell, “Dispersion Compensation Using Single-Material Fibers,” IEEE Phot. Tech. Lett. 11, 674–676, (1999). [CrossRef]
3.	J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air-silica microstructure optical fibers,” Opt. Lett. 25, 796–798, (2000). [CrossRef]
4.	T. P. Hansen, J. Broeng, E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Phot. Tech. Lett. 13, 588–590, (2001). [CrossRef]
5.	A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T.A. Birks, and P.S.J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325–1327, (2000). [CrossRef]
6.	M. Steel and J. R. M. Osgood, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231, (2001). [CrossRef]
7.	M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490, (2001). [CrossRef]
8.	B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713, (2002),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698. [CrossRef]
9.	J. K. Chandalia, B. J. Eggleton, R. S. Windeler, S. G. Kosinski, X. Liu, and C. Xu, “Adiabatic Coupling in Tapered Air-Silica Microstructured Optical Fiber,” IEEE Phot. Tech. Lett. 13, 52–54, (2001). [CrossRef]
10.	C. Kerbage, A. Hale, A. Yablon, R. S. Windeler, and B. J. Eggleton, “Integrated all-fiber variable attenuator based on hybrid microstructure fiber,” Appl. Phys. Lett. 79, 3191–3193, (2001). [CrossRef]
11.	P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. Burdge, “Control of waveguide properties in hybrid polymer-silica microstructured optical fiber gratings,” in OFC conference 2000, 3, pp. 134–136.
12.	C. Kerbage, P. Steinvurzel, A. Hale, R. S. Windeler, and B. J. Eggleton, “Birefringent tunable hybrid microstructured optical fiber,” in CLEO conference 2002.
13.	T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Tech. 155, 1277–1294, (1997). [CrossRef]
14.	C. Kerbage, P. Steinvurzel, P. Reyes, P. S. Westbrook, R. S. Windeler, A. Hale, and B.J. Eggleton, “Highly tunable birefringent microstructured photonic crystal optical fiber,” Opt. Lett, in press , (2002).
15.	C. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–123,(2000), http://www.opticsexpress.org/oearchive/source/22997.htm [CrossRef] [PubMed]
16.	B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Tech.. 18, 1084–1100, (2000). [CrossRef]
17.	M. D. Feit and J. A. Fleck, ”Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” App. Opt. 19, 2240–2246, (1980). [CrossRef]
18.	H. G. Jerrard, “Transmission of light through birefringent and optically active media: the Poincaré sphere,” J. Opt. Soc. Am 44, 634–640, (1954). [CrossRef]
19.	B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices,” IEEE Phot. Tech. Lett. 4, 451–454, (1992). [CrossRef]

OCIS Codes
(060.2310) Fiber optics and optical communications : Fiber optics
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Research Papers

History
Original Manuscript: February 13, 2002
Revised Manuscript: February 27, 2002
Published: March 11, 2002

Citation
C. Kerbage and B. Eggleton, "Numerical analysis and experimental design of tunable birefringence in microstructured optical fiber," Opt. Express 10, 246-255 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-5-246

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References

P. V. Kaiser and H. W. Astle, "Low-loss single-material fibers made from pure fused silica," The Bell System Technical Journal 53, 1021-1039, (1974).
T. A. Birks, D. Mogilevstev, J. C. Knight, and P. S.J . Russell, "Dispersion Compensation Using Single-Material Fibers," IEEE Phot. Tech. Lett. 11, 674-676, (1999). [CrossRef]
J. K. Ranka, R. S. Windeler, and A. J. Stentz, "Optical properties of high-delta air-silica microstructure optical fibers," Opt. Lett. 25, 796-798, (2000). [CrossRef]
T. P. Hansen, J. Broeng, E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, "Highly birefringent index-guiding photonic crystal fibers," IEEE Phot. Tech. Lett. 13, 588-590, (2001). [CrossRef]
A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. S. J. Russell, "Highly birefringent photonic crystal fibers," Opt. Lett. 25, 1325-1327, (2000). [CrossRef]
M. Steel and J. R. M. Osgood, "Elliptical-hole photonic crystal fibers," Opt. Lett. 26, 229-231, (2001). [CrossRef]
M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L. C. Botten, "Symmetry and degeneracy in microstructured optical fibers," Opt. Lett. 26, 488-490, (2001). [CrossRef]
B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, "Microstructured optical fiber devices," Opt. Express 9, 698-713, (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698</a>. [CrossRef]
J. K. Chandalia, B. J. Eggleton, R. S. Windeler, S. G. Kosinski, X. Liu, and C. Xu, "Adiabatic Coupling in Tapered Air-Silica Microstructured Optical Fiber," IEEE Phot. Tech. Lett. 13, 52-54, (2001). [CrossRef]
C. Kerbage, A. Hale, A. Yablon, R. S. Windeler, and B. J. Eggleton, "Integrated all-fiber variable attenuator based on hybrid microstructure fiber," Appl. Phys. Lett. 79, 3191-3193, (2001). [CrossRef]
P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. Burdge, "Control of waveguide properties in hybrid polymer-silica microstructured optical fiber gratings," in OFC conference 2000, 3, pp. 134-136.
C. Kerbage, P. Steinvurzel, A. Hale, R. S. Windeler, and B. J. Eggleton, "Birefringent tunable hybrid microstructured optical fiber," in CLEO conference (Optical Society of America, Washington, D.C., 2002).
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