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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 2 — Jan. 23, 2006
  • pp: 914–925
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Photonic crystal-liquid crystal fibers for single-polarization or high-birefringence guidance

D. C. Zografopoulos, E. E. Kriezis, and T. D. Tsiboukis  »View Author Affiliations


Optics Express, Vol. 14, Issue 2, pp. 914-925 (2006)
http://dx.doi.org/10.1364/OPEX.14.000914


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Abstract

The dispersive characteristics of a photonic crystal fiber enhanced with a liquid crystal core are studied using a planewave expansion method. Numerical results demonstrate that by appropriate design such fibers can function in a single-mode/single-polarization operation, exhibit high- or low- birefringence behavior, or switch between an on-state and an off-state (no guided modes supported). All of the above can be controlled by the application of an external electric field, the specific liquid crystal anchoring conditions and the fiber structural parameters.

© 2006 Optical Society of America

1. Introduction

Photonic crystal fibers (PCFs), or holey fibers, have been under intense study during the last years since they offer a promising alternative to conventional fibers for a wide range of applications, such as in telecommunication, sensor, or quantum cryptography technology [1–3

01. M. D. Nielsen, C. Jacobsen, N.A. Mortensen, J.R. Folkenberg, and H.R. Simonsen, “Low-loss photonic crystal fibers for transmission systems and their dispersion properties,” Opt. Express 12, 1372–1376 (2004),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1372. [CrossRef] [PubMed]

]. Holey fibers are normally composed of a transverse periodical arrangement of air holes opened in some dielectric material - silica being the rule - which is broken by a defect core. Light can be guided through the fiber by two distinct mechanisms, namely the total internal reflection (TIR) and the bandgap effect [4

04. J. Broeng, D. Mogilevtsev, S. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Techn. 5, 305–330 (1999). [CrossRef]

]. Fiber guiding by the TIR effect relies on index-guiding by a core, whose refractive index is higher than the cladding’s effective index, while the bandgap effect guiding holey fibers take advantage of the full 2-D bandgaps exhibited by the cladding’s periodical structure, which confine light in a hollow core along the fiber axis.

Holey fibers operating under the TIR principle show some significant advantages compared to their conventional counterparts. For instance, they can be exclusively fabricated out of a single dielectric, thus avoiding interfaces between different materials that might undermine the fiber’s performance. The dependence of the cladding’s effective index on wavelength permits the manifestation of only a finite number of modes at relatively high frequencies, due to the reduction of the difference between the refractive indices of the core and the cladding for increasing frequencies [5

05. T.A. Birks, J.C. Knight, and P. St. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]. Furthermore, it has been theoretically shown and experimentally verified that holey fibers with a triangular lattice of air-holes can exhibit endlessly single-mode operation provided the diameter of the cladding’s air holes and the centre-to-centre spacing of two adjacent holes (lattice constant) are properly chosen [5

05. T.A. Birks, J.C. Knight, and P. St. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

].

The variation of the cladding’s effective index can be optimally turned to advantage, as far as the fiber’s dispersion properties are concerned. The wide tailoring of the fiber’s design in terms of the air hole sizes, shapes and arrangements provides several degrees of freedom whose proper combination can tune the fiber’s dispersion curve. The influence of the design parameters on the dispersion properties of holey fibers has been extensively studied, concerning both the classical triangular air hole-silica core lattice and other configurations that include additional features, e.g. air hole rings of variable diameter, air holes with elliptical cross-sections or doped-silica cylindrical cores [6–9

06. T.-L. Wu and C.-H. Chao, “A novel ultraflattened dispersion photonic crystal fiber,” IEEE Phot. Tech. Let. 17, 67–69 (2005). [CrossRef]

]. Several types of holey fibers were designed that exhibit favorable dispersion properties, such as zero-dispersion at the 1.55 μm operation wavelength, and ultra-flattened group-velocity dispersion curves of zero, positive or negative dispersion over a significant wavelength regime [10–11

10. A. Ferrando, E. Silvestre, and P. Andrés, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express 9, 687–697 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687. [CrossRef] [PubMed]

].

Nematic liquid crystals are anisotropic materials consisting of rod-like molecules whose axis coincides with the anisotropy’s optical axis. When confined in closed cylindrical cavities in the absence of external stimuli, the liquid crystal’s director distribution is determined by the physics of elastic theory and the anchoring conditions at the cavity’s surface [12–13

12. G.P. Crawford, D.W. Allender, and J.W. Doane, “Surface elastic and molecular-anchoring properties of nematic liquid crystals confined to cylindrical cavities,” Phys. Rev. A 45, 8693–8710 (1992). [CrossRef] [PubMed]

]. Under the application of a static electric field the director’s orientation can be controlled, since the liquid crystal molecules tend to align their axis according to the applied field. In an alternative approach, the properties of nematic liquid crystals can be tuned thermally owing to the dependence of the refractive index values on temperature. The above features have favored their utilization in a number of recently proposed photonic crystal based optoelectronic devices [14–18

14. F. Du, Y.-Q. Lu, and S.-T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183 (2004). [CrossRef]

].

2. Photonic crystal-liquid crystal fiber analysis

2.1. Structural parameters and fiber layout

The fiber under study is shown in Fig. 1 and consists of three rings of a triangular lattice of air holes of radius r opened in a lossless optical fiber glass, whose refractive index ng is considered constant. The transverse periodicity is broken by a defect core with diameter d c. The core is filled with a typical nematic liquid crystal material characterized by ordinary and extraordinary refractive indices of no and ne, respectively. The selection of the fiber materials is only limited by the requirement that the refractive index of the fiber glass matches the extraordinary index of the nematic liquid crystal, as will be evidenced by the analysis hereafter. Extraordinary indices within the range 1.5 < ne < 1.8 are typical for nematic materials [21

21. B. BahadurLiquid crystals: applications and uses, vol. 1 (World Scientific Publishing, 1990). [CrossRef]

], and their values can be sufficiently tuned by factors such as molecular design, chemical synthesis conditions, and operating temperature. Additionally, nonsilica fiber glasses whose refractive index obtains values as high as ng = 2.2 are commercially available and they have already been proposed and studied as candidates for the fabrication of holey fibers [22

22. X. Feng, A.K. Mairaj, D.W. Hewak, and T.M. Monro, “Nonsilica glasses for holey fibers,” IEEE J. Lightwave Tech. 23, 2046–2054 (2005). [CrossRef]

]. Among the various types of available fiber glasses for which 1.5 < ng < 1.8 are the high-lead silicate, barium crown, lanthanum crown, or barium flint fiberglasses [23

23. M.J. Weber, Handbook of optical materials (CRC Press, 2003).

]. The refractive index of these glasses can be tuned, for instance, by adjusting the doping percentage of metal oxides such as lead (PbO) or lanthanum (La2O3) oxides. Particular examples of high index glasses in the above range were reported in [24–25

24. X. Feng, T.M. Monro, P. Petropoulos, V. Finazzi, and D. Hewak, “Solid microstructured optical fiber,” Opt. Express 11, 2225–2230 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2225. [CrossRef] [PubMed]

]; in the first reference a lead oxide borosilicate glass of ng = 1.76 was utilized in making microstructured optical fibers, whereas in the second a barium crown glass of n g = 1.61 was adopted in a conventional wavelength insensitive coupler. For the purposes of the present analysis we selected the common nematic material E7 (ne = 1.68, no = 1.5) and a fiberglass with ng = 1.68. The fiber is considered to be uniform along the z-axis, which coincides with the axis of the cylindrical defect core.

Fig. 1. Cross-sectional view of the proposed type of PC-LC fiber: hole-to-hole spacing Λ, hole radius r, and central defect core diameter dc. The figure corresponds to parameters r = 0.2Λ, and dc = Λ.

Before proceeding to the analysis of the LC-PC fiber, we provide a reference diagram (Fig. 2) where the dispersion curve for the fundamental degenerate HE x and HEy modes, along with the dispersion curve of the first higher order mode and the cladding’s fundamental space-filling mode (FSM) - namely the radiation line - are plotted for a conventional (solid core) photonic crystal fiber with ng = 1.68 and r = 0.2Λ. Noticeable in the figure, the fiber is endlessly-single mode; the first higher-order mode dispersion curve always remains below the radiation line (see inset in Fig. 2).

We study the case where the nematic director in the fiber core is uniform and parallel to the x- or y-axis of the fiber’s lattice. Such alignment can be theoretically exhibited under the influence of the appropriate homeotropic anchoring conditions. The planar-polar (PP) profile, a commonly observed director pattern for nematic liquid crystals confined in cylindrical capillaries comprises an example, where almost uniform alignment of the director can be achieved. In general, the PP texture can be analytically calculated [12

12. G.P. Crawford, D.W. Allender, and J.W. Doane, “Surface elastic and molecular-anchoring properties of nematic liquid crystals confined to cylindrical cavities,” Phys. Rev. A 45, 8693–8710 (1992). [CrossRef] [PubMed]

] under the one elastic constant approximation K 11 = K 33 = K for the nematic material. The Euler-Lagrange equation for minimizing the elastic free energy takes the form of the two-dimensional Laplace equation (1), expressed in polar coordinates,

2ψrϕr2+1rψrϕr+1r22ψrϕϕ2=0,
(1)
Fig. 2. Dispersion curve for the degenerate HEx and HEy modes of a triangular lattice PCF with r = 0.2Λ and ng = 1.68. The notations nFSMeff and n(eff) (1) refer to the effective indices of the FSM mode and the first higher order mode, respectively.

Φ being the angle between the director and the radial unit vector. The solution of (1) yields the following relation

ψrϕ=π2tan1[(R2+γr2)(R2γr2)tanϕ],
(2)

where γ= (ξ 2 +1)1/2 -ξ,ξ= 2K/RW 0 and RW 0/K is the effective anchoring strength, which determines the orientation of the molecules at the cavity wall. At the strong anchoring limit the director is locally perpendicular to the surface (Fig. 3(a)), while as the anchoring strength relaxes, the molecules tend to adopt a more uniform orientation towards the uniform profile acquired at the weak anchoring limit (Fig. 3(b)).

Another way to generate the uniform profile is by controlling the nematic director through the application of an external electric (or magnetic) static field. Figure 4 shows how the application of a sufficiently strong voltage [26

26. Y. Jeong, B. Yang, B. Lee, H.S. Seo, S. Choi, and K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Phot. Tech. Let. 12, 519–521 (2000). [CrossRef]

] between two pairs of electrodes can align the nematic molecules along the x- or y-axis and lead to an almost uniform director profile.

2.2. Single-mode/single-polarization guidance

The uniform alignment of the nematic director has a direct impact on the fiber’s waveguiding behaviour. Let us assume, for instance, the profile of Fig. 3(b), were the optical axis coincides with the y-axis. Owing to the birefringence of the defect core’s nematic material, the effective refractive index of the core shall differ for light of different polarizations. In the case of y-polarized light, the core is practically homogenous - since we have assumed that for the fiberglass and the chosen nematic material the condition ng = ne is met - and, therefore, the fiber’s operation in this case shall not differ significantly from a common solid core PCF. Thus, the fiber can exhibit properties such as the endlessly-single mode operation provided that the cladding’s hole radius is smaller than a threshold value. On the contrary, a x-polarized field senses a more complex core region where a low-refractive index cylinder is embedded in the fiberglass background. A rough estimation of the core’s effective refractive index ncoreeff,x at the low-frequency regime is obtained by considering the core as a region occupied by two circular discs with radii rdef and rcore = 0.5Λ [5

05. T.A. Birks, J.C. Knight, and P. St. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]; the effective indexcoreeff, x referring to the x- polarization, is then given by

neff,xcore=ne[1(2rdefΛ)2]+no(2rdefΛ)2,λ>>Λ.
(3)
Fig. 3. The planar-polar director profile at the strong (a) and the weak (b) anchoring limit.
Fig. 4. Layout for arbitrarily controlling the orientation of the nematic director profile in the xOy plane: for instance, uniform x-parallel alignment (Vx = V 0, Vy = 0), and uniform y-parallel alignment (Vy =V 0, Vx = 0).

The total internal reflection waveguidance criterion at a given wavelength λ demands that the core’s effective index is higher than that of the cladding. In the case studied, this criterion is interpreted as ncladeff(λ) < ncoreeff,x (λ), where ncladeff(λ) = nFSMeff(λ) is the index of the cladding’s fundamental space-filling mode. Thus, the satisfaction of the opposite condition (n cladeff(λ) > ncoreeff,x(λ)) within a wavelength range of interest is expected to permit the exhibition of single-polarization (HEy) operation. Single-polarization PCFs have also been proposed based on a special design of the cladding [27–30

27. A. Ferrando and J.J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystal fibers,” Appl. Phys. Lett. 78, 3184–3186 (2001). [CrossRef]

], lacking, though, the versatility of selective operation between the two polarizations (HEx either HEy) of the proposed fiber.

Fig. 5. Dispersion curves for the fundamental HEx and HEy modes and radiation line for the PC-LC fiber of Fig. 4 in the planar polar weak anchoring limit case for rdef - 0.5Λ and r = 0.2Λ. As noticed, the fiber exhibits both endlessly single-mode and single-polarization behavior.

2.3. Controllable birefringence guidance

Fig. 6. Modal intensity profiles at Λ/λ = 1.5 for the fundamental y- and x-polarized modes for the dispersion curves of Fig. 5: a) HEy mode for rdef = 0.5Λ, b) HEx mode for rdef = 0.5Λ. The air hole radius is r = 0.2Λ. The HEy mode, which senses a homogenous core of ne = 1.68, shows a regular hexagonal profile. On the contrary, the HEX mode radiates into the cladding.

Fig. 7. Dispersion curves for the fundamental HEx and HEy modes and radiation line for the planar polar weak anchoring limit case. HEx mode curves correspond to different rdef values.
Fig. 8. Modal intensity profiles at Λ/λ = 1.5 for the fundamental x-polarized mode for different radii of the defect core: a) rdef = 0.1 Λ, b) rdef = 0.15Λ, c) rdef = 0.2Λ, and d) rdef - 0.25Λ. The radius of the air holes is kept constant at r = 0.2Λ.

Another point of interest is the investigation of the fiber’s properties when the condition ng = ne is relaxed. As the equality refers to an ideal case, it is possible that this condition cannot be met exactly in practice, at least for broad wavelength ranges, such as those of the present analysis. We will assume that the refractive indices of the nematic material are kept fixed, while that of the glass can obtain values around ne. In case ng < ne, the core obtains a more intense guiding behavior, while the radiation line, which depends solely on the fiberglass cladding, is expected to drop; these modifications shall eventually lead to the emergence of more propagating modes, turning the fiber from single-mode to multimode. On the contrary, if ng > ne the radiation line rises and, thus, no more propagating modes emerge. In addition, the fundamental single-polarized mode’s dispersion curve is bound to drop below the radiation line at a certain frequency threshold, above which the fiber shall support no guided modes whatsoever. Figure 10 elucidates the reasoning of the above remarks; for ng = 1.67 the fiber supports a second guided mode for Λ/λ > 3 (Fig. 10(a)), while for ng = 1.69 the fiber ceases to guide any modes for Λ/λ > 4. In both cases, though, the fiber is shown to operate under the single-mode single-polarization property within a large wavelength regime, despite the deviation from the ideal condition ng = ne.

Fig. 9. Modal birefringence of the fundamental HEx and HEy modes for rdef = 0.1Λ, 0.15Λ, 0.2Λ, and 0.25λ.
Fig. 10. Modal dispersion curves for (a) ng = 1.67 and (b) ng = 1.69. The parameters used in both cases are r = 0.2Λ, rdef = 0.4Λ, no = 1.5, and ne = 1.68.

2.4. Switching between on-off states

Finally, we shall examine the case of a uniform director profile where the optical axis is now parallel to the z-axis (axial configuration). This profile can be exhibited under homogenous anchoring conditions and can be at large approached under homeotropic anchoring condition at the weak anchoring limit of the escaped - radial (ER) configuration. The director for the ER pattern remains radial on the transverse plane very close to the cavity wall, yet escapes in the third dimension while moving towards the cylinder axis. The director profile in polar coordinates is given by

ψrϕ=0,
(4)
θ(r)=π22tan1(rRtan(a2)),
(5)

where Φ, θ are the angles between the director and the radial unit vector and the z-axis, respectively, a = π/2 - θ(r = R) = cos-1(1/σ), and σ = RW 0/K+K 24/K- 1 is the parameter that expresses the anchoring strength, considered to be larger than unity. In the case of strong anchoring conditions (σ →∞) the molecules are perpendicularly aligned to the cavity wall (Fig. 11(a)). While moving towards the weak anchoring limit (σ = 1) the molecules obtain an alignment parallel to the z-axis (Fig. 11(b)).

Fig. 11. The escaped - radial profile at the strong (a) and the weak (b) anchoring limit.

Thus, a fiber exhibiting the above texture (ER) under the application of an external electric field - with electrodes placed as in Fig. 4 - can operate alternatively in a single-polarization single-mode operation for Vx,y > Vth (director parallel to the x- or y- axis) and an off-mode state for Vx,y = 0 (director parallel to the z-axis). This property could be exploited for instance in switching protection applications of optical streams, with the switching speed solely limited by the response time of the LC molecules. Response times in the order of a few ms are typical for nematic liquid crystals, providing a switching performance compliant with telecommunication standards, such as those of the Synchronous Digital Hierarchy (SDH), which require a protection switchover in less than 50 ms.

A final issue regarding the potential use of such fibers in optical systems is that of the involved overall losses. The principal loss mechanisms in this case are absorption and scattering losses associated with the nematic material. It has been shown that bulk scattering losses obtain values two orders of magnitude higher than absorption, ranging between 15~40 dB/cm [33

33. C. Hu and J.R. Whinnery, “Losses of a nematic liquid-crystal optical waveguide,” J. Opt. Soc. Am. 64, 1424–1432 (1974). [CrossRef]

]. Fortunately, scattering loss is significantly suppressed (1~3 dB/cm) in case the nematic material is confined in capillaries of small core diameter (2~8 μm) [34

34. M. Green and S.J. Madden, “Low loss nematic liquid crystal cored fiber waveguides,” Appl. Opt. 28, 5202–5203 (1989). [CrossRef] [PubMed]

]. Leakage losses due to the finite dimensions of the fiber can be minimized by adding more ring holes in the periodic cladding. Finally, numerous studies have shown that splicing issues between conventional and microstructured fibers can also be addressed; splicing losses of 0.2~0.9 dB have been reported for various techniques such as arc-fusion splicing [35

35. O. Frazão, J.P. Carvalho, and H.M. Salgado, “Low-loss splice in a microstructured fibre using a conventional fusion splicer,” Microw. Opt. Tech. Let. 46, 172–174 (2005). [CrossRef]

] or laser splicing [36

36. J. H. Chong and M.K. Rao, “Development of a system for laser splicing photonic crystal fiber,” Opt. Express 11, 1365–1370 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1365. [CrossRef] [PubMed]

], and a method for the formation of a splice-free low-loss interface between the two fiber types has been also successfully implemented [37

37. S.G. Leon-Saval, T.A. Birks, N.Y. Joly, A.K. George, W.J. Wadsworth, G. Karakantzas, and P.St.J. Russell, “Splice-free interfacing of photonic crystal fibers,” Opt. Lett. 30, 1629–1631 (2005). [CrossRef] [PubMed]

].

Fig. 12. Dispersion curves for the axial director profile case for rdef = 0.5Λ and r = 0.2Λ. The fiber operates in an off-state since all supported modes are evanescent.

3. Conclusions

A novel liquid-crystal core photonic-crystal fiber design has been proposed for single-polarization and high-birefringence fibers. It was found that the proposed type of nematic core fibers can support an endlessly single-mode/single-polarization operation under the appropriate anchoring conditions. It is noteworthy that the fiber’s lattice symmetry is not disturbed; thus, no orientation issues arise concerning the coupling and splicing problems that appear for single-polarization fibers with stress induced or geometrical birefringence. Moreover, under the control of an external electric field, the fiber proved capable of functioning with preferential polarization characteristics, a feature that might be directly exploited in polarization control elements or other similar devices. In addition, the fiber design allows for the fabrication of fibers with controllable birefringence values, from remarkably high values (more than 4×10-2) down to zero birefringence. More realistic patterns of the nematic director as well as other nematic materials are currently under investigation, in order to predict the fiber’s behaviour under different conditions; these will be reported in the context of a future publication.

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A. Argyros, N. Issa, I. Bassett, and M.A. van Eijkelenborg, “Microstructured optical fiber for single-polarization air guidance,” Opt. Lett. 29, 20–22 (2004). [CrossRef] [PubMed]

31.

S. Gauza, J. Li, S.-T. Wu, A. Spadlo, R. Da̧browski, Y.-N. Tzeng, and K.-L. Cheng, “High birefringence and high resistivity isothiocyanate-based nematic liquid crystal mixtures,” Liq. Cryst. 32, 1077–1085 (2005). [CrossRef]

32.

J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, Art. 073501 (2005). [CrossRef]

33.

C. Hu and J.R. Whinnery, “Losses of a nematic liquid-crystal optical waveguide,” J. Opt. Soc. Am. 64, 1424–1432 (1974). [CrossRef]

34.

M. Green and S.J. Madden, “Low loss nematic liquid crystal cored fiber waveguides,” Appl. Opt. 28, 5202–5203 (1989). [CrossRef] [PubMed]

35.

O. Frazão, J.P. Carvalho, and H.M. Salgado, “Low-loss splice in a microstructured fibre using a conventional fusion splicer,” Microw. Opt. Tech. Let. 46, 172–174 (2005). [CrossRef]

36.

J. H. Chong and M.K. Rao, “Development of a system for laser splicing photonic crystal fiber,” Opt. Express 11, 1365–1370 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1365. [CrossRef] [PubMed]

37.

S.G. Leon-Saval, T.A. Birks, N.Y. Joly, A.K. George, W.J. Wadsworth, G. Karakantzas, and P.St.J. Russell, “Splice-free interfacing of photonic crystal fibers,” Opt. Lett. 30, 1629–1631 (2005). [CrossRef] [PubMed]

OCIS Codes
(060.2400) Fiber optics and optical communications : Fiber properties
(060.2420) Fiber optics and optical communications : Fibers, polarization-maintaining
(060.2430) Fiber optics and optical communications : Fibers, single-mode
(230.3720) Optical devices : Liquid-crystal devices

ToC Category:
Photonic Crystal Fibers

Citation
D. C. Zografopoulos, E. E. Kriezis, and T. D. Tsiboukis, "Photonic crystal-liquid crystal fibers for single-polarization or high-birefringence guidance," Opt. Express 14, 914-925 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-914


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