Infiltrated photonic crystal fiber: experiments and ... - OSA Publishing

Infiltrated photonic crystal fiber: experiments and liquid crystal scattering model Alexander Lorenz,1 Rolf Schuhmann,2 and Heinz-Siegfried Kitzerow1 1

Department of Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany Fachgebiet Theoretische Elektrotechnik, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

2

Abstract: Experimental results obtained by means of a cut-back technique indicate low attenuations (< 1 dB·cm−1) for a solid core photonic crystal fiber filled with the nematic liquid crystal E7. These results observed in the visible wavelength range are compared with electromagnetic field simulations. The latter are carried out with a full vectorial finite element algorithm. Based on the modal properties under the condition of perpendicular anchoring of the liquid crystal molecules, the wavelength dependent attenuation is estimated using a power loss model considering the turbidity of the nematic liquid crystal. The results indicate that the scattering properties of this type of materials make them extremely interesting for fiber optical filters in the visible wavelength range and that filling materials with a relatively high turbidity are in general potentially useful as filling materials for solid core photonic crystal fibers. ©2009 Optical Society of America OCIS codes: (060.5295) Photonic crystal fibers; (260.1440) Birefringence; (230.3990) Microstructure devices; (230.7370) Waveguides; (160.3710) Liquid crystals; (060.2270) Fiber characterization; (230.3720) Liquid-crystal devices; (230.7408) Wavelength filtering devices

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

P. S. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13(1), 309–314 (2005). G. B. Ren, P. Shum, L. R. Zhang, X. Yu, W. J. Tong, and J. Luo, “Low-loss all-solid photonic bandgap fiber,” Opt. Lett. 32(9), 1023–1025 (2007). M. A. Schmidt, N. Granzow, N. Da, M. Peng, L. Wondraczek, and P. S. J. Russell, “All-solid bandgap guiding in tellurite-filled silica photonic crystal fibers,” Opt. Lett. 34(13), 1946–1948 (2009). C. Hu, and J. R. Whinnery, “Losses Of a Nematic Liquid-Crystal Optical-Waveguide,” J. Opt. Soc. Am. 64(11), 1424–1432 (1974). A. Lorenz, H.-S. Kitzerow, A. Schwuchow, J. Kobelke, and H. Bartelt, “Photonic crystal fiber with a dualfrequency addressable liquid crystal: behavior in the visible wavelength range,” Opt. Express 16(23), 19375– 19381 (2008). H.-S. Kitzerow, A. Lorenz, and H. Matthias, “Tuneable photonic crystals obtained by liquid crystal infiltration,” Phys. Status Solidi 204(11), 3754–3767 (2007) (a). G. D. Ziogos, and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40(10), 733–748 (2008). LMA-10, NKT Photonics A/S, Denmark (formerly Crystal Fiber A/S). J. Jasapara, T. H. Her, R. Bise, R. Windeler, and D. J. DiGiovanni, “Group-velocity dispersion measurements in a photonic bandgap fiber,” J. Opt. Soc. Am. B 20(8), 1611–1615 (2003). M. A. Duguay, Y. Kokubun, T. L. Koch, L. Pfeiffer, “Antiresonant Reflecting Optical Wave-Guides in SiO2-Si Multilayer Structures,” Appl. Phys. Lett. 49, 13–15 (1986). N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27(18), 1592–1594 (2002). L. Scolari, S. Gauza, H. Q. Xianyu, L. Zhai, L. Eskildsen, T. T. Alkeskjold, S. T. Wu, and A. Bjarklev, “Frequency tunability of solid-core photonic crystal fibers filled with nanoparticle-doped liquid crystals,” Opt. Express 17(5), 3754–3764 (2009). G. Tartarini, T. Alkeskjold, L. Scolari, A. Bjarklev, and P. Bassi, “Spectral properties of liquid crystal photonic bandgap fibres with splay-aligned mesogens,” Opt. Quantum Electron. 39(10-11), 913–925 (2007). S. V. Burylov, “Equilibrium configuration of a nematic liquid crystal confined to a cylindrical cavity,” Sov. Phys. JETP 85(5), 873–886 (1997). M. Green, and S. J. Madden, “Low loss nematic liquid crystal cored fiber waveguides,” Appl. Opt. 28(24), 5202– 5203 (1989).

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Received 28 Sep 2009; revised 5 Nov 2009; accepted 9 Nov 2009; published 3 Feb 2010

15 February 2010 / Vol. 18, No. 4 / OPTICS EXPRESS 3519

17. R. D. Polak, G. P. Crawford, B. C. Kostival, J. W. Doane, and S. Zumer, “Optical determination of the saddlesplay elastic constant K24 in nematic liquid crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(2), R978–R981 (1994). 18. J. Sun, C. C. Chan, and N. Ni, “Analysis of photonic crystal fibers infiltrated with nematic liquid crystal,” Opt. Commun. 278(1), 66–70 (2007). 19. S. M. Hsu, and H. C. Chang, “Characteristic investigation of 2D photonic crystals with full material anisotropy under out-of-plane propagation and liquid-crystal-filled photonic-band-gap-fiber applications using finite element methods,” Opt. Express 16(26), 21355–21368 (2008). 20. J. Weirich, J. Laegsgaard, L. Scolari, L. Wei, T. T. Alkeskjold, and A. Bjarklev, “Biased liquid crystal infiltrated photonic bandgap fiber,” Opt. Express 17(6), 4442–4453 (2009). 21. COMSOL 3.5a, Comsol Multiphysics®, http://www.comsol.com. 22. T. T. Alkeskjold, J. Lægsgaard, A. Bjarklev, D. Hermann, A. Anawati, J. Broeng, J. Li, and S. T. Wu, “Alloptical modulation in dye-doped nematic liquid crystal photonic bandgap fibers,” Opt. Express 12(24), 5857– 5871 (2004). 23. G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101(7), 73105 (2007). 24. P. G. de Gennes, “Long Range Order and Thermal Fluctuations in Liquid Crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 7(1), 325–345 (1969). 25. D. Langevin, M.A. Bouchiat, “Anisotropy of the turbidity of an oriented nematic liquid crystal,” J. Physique Colloques, C197 (1975). 26. K. Simonyi, Foundations of Electrical Engineering (Elsevier 1964). 27. M. A. Khashan, and A. Y. Nassif, “Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2-3 µm,” Opt. Commun. 188(1-4), 129–139 (2001).

1. Introduction Photonic crystal fibers open new possibilities to design waveguides with outstanding properties [1]. For example, hollow core photonic band gap fibers guiding light in air have gained great interest over the past years. Surprisingly, all solid photonic band gap fibers have become candidates to achieve low attenuations by applying the photonic band gap effect and maintaining an all solid structure [2, 3]. Commonly, in this type of fiber, cylindrical high index inclusions in a background material with lower refractive index are arranged in a trigonal array forming a 2-dimensional microstructure. In the center of this microstructure, one inclusion is missing and guided modes are confined in this central low index core. Principally, there are two ways to realize all solid micro structured fibers. Fibers longer than several m are drawn from a macroscopic preform already consisting of the intended materials in an adequate geometry. But also, shorter pieces are sufficing for optical modulators or filters: Schmidt et al. have shown recently [4], that such short pieces of all solid photonic band gap fibers can be fabricated by pressing molten high index tellurite glass into a micro structured silica glass fiber with a silica glass core surrounded by an array of air inclusions. Photonic crystal fibers with air inclusions can more easily be filled with liquids or liquid crystals. While bulk wave guiding in liquid crystals is limited by high attenuation in the range of 20 to 40 dB·cm−1[5], a solid core photonic crystal fiber with liquid crystal filled inclusions exhibits guided core modes which have just evanescent field components in the liquid crystal filled sections. Thereby, wave guiding becomes possible combining the intense, fast and reversible response of liquid crystals with reasonable transmission, even with the possibility of dual frequency addressing [6]. Consequently, solid core photonic crystal fibers with a high index liquid crystal filled microstructure show lower attenuation than micro structured fibers with nematic liquid crystals in the core [7, 8]. In this paper, the commercially available photonic crystal fiber LMA-10 [9] is filled with the liquid crystal E7 (Merck, Germany) and experimental data is reported as well as results of electromagnetic field simulations. In these simulations, the loss is assumed to be based mainly on scattering by the liquid crystal inclusions due to thermal fluctuations of the director. In contrast, previous simulations were based on the assumption that a guided mode in the core region of a photonic crystal fiber may penetrate the finite micro structured region and gradually leak into the outer cladding region consisting of solid silica. The experimental cutback technique applied to record transmission spectra in the present study does not allow distinguishing between different origins of loss. However, the experimental results are in

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reasonable agreement with the assumption of a scattering loss mechanism; the simulations show strongly attenuated modes once the guided intensity enters the first and second ring of inclusions at least partially. Characteristically for this type of waveguide, low attenuations are only possible in spectral regions with severe modal confinement to the core. As the simulations regard wavelength dependent refractive indices, even theoretical analysis of the chromatic dispersion is possible.

Fig. 1. Representative structure of a solid core photonic crystal fiber used in the simulation process. Light microscopic image of the LMA-10 fiber as inset.

2. Experiments The LMA-10 fiber consists of silica glass with a trigonal array of cylindrical air holes in the cladding region with a missing hole as defect in the center (Fig. 1). The central defect is the core of the fiber. Light coupled into the bare fiber is guided due to modified total internal reflection [1]. The core has a higher refractive index than the average refractive index of the unfilled microstructure surrounding it. However, due to its holey character, the microstructure acts as a modal sieve, allowing selectively the propagation of just the fundamental mode even in the visible wavelength range for quite large core diameters [1]. In the unfilled state, the fiber shows continuous transmission. By filling the air holes with liquid crystal, the situation is changed. The birefringent liquid crystal material has two refractive indices (ordinary refractive index no and extraordinary refractive index ne) both larger than the refractive index of silica glass: ne > no > nsilica. After filling, the average refractive index of the core material is smaller than the average refractive index of the microstructure surrounding it. Therefore, light cannot be guided by total internal reflection. Nevertheless, the filled fiber is an excellent waveguide because cylindrical high index inclusions are created by the filling. Photonic crystal fibers with cylindrical high index inclusions in a background material with lower refractive index have gained growing interest over the past years. The wave guiding mechanism of this type of fiber can be explained either by Bragg scattering at the high index inclusions [10] or by the ARROW-model [11, 12]. By light microscopic imaging of the fiber end face (inset, Fig. 1.) a pitch of p = (6.5±0.5) µm and a hole diameter of 2·Ri = (3.1±0.5) µm were measured. Differing data was reported for the LMA-10 fiber previously: p = 7.2 µm and 2·Ri = 3.1 µm [13], p = 7 µm and 2·Ri = 3 µm [14]. In the literature, a LMA-10 fiber with 7 rings of holes was investigated [13]. In contrast, the fiber analyzed here has only 4 rings of holes (inset, Fig. 1). #117869 - $15.00 USD

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The air holes of the fiber were filled with the nematic liquid crystal E7 (Merck). The alignment of the liquid crystal is essential for optical applications. It has been shown earlier [6] that by just varying the boundary conditions, the transmission properties of a liquid crystal filled photonic crystal fiber device can be changed dramatically. Nematic liquid crystals consist of rod like molecules and due to their shape, the orientation of neighboring molecules is almost uniform within a certain critical length. The orientation of the molecules is commonly described using a pseudo vector field, the so called director field v(r). In the nematic phase, the local director indicates the average orientation of the long axes of neighboring molecules. There are numerous well known anchoring agents that generate a preferred molecular alignment at the solid boundary faces of liquid crystal cells. Lecithin, for example, is a hydro-lipid which enforces a so called homeotropic or perpendicular molecular alignment on a glass surface. The hydrophilic parts of the lecithin molecules cover the glass surface, thereby aligning the organic nematic molecules perpendicular to the surface. The equilibrium configuration of the director field corresponds to a minimum of the free elastic energy. For cylindrical confinement, there are several well known possible configurations [15]. Defects of the director field can e.g. lead to higher attenuation in liquid crystal waveguides [16]. Therefore, to generate a defect free alignment, single capillaries as well as the micro structured LMA-10 fibers are coated with a dilute solution of lecithin in petroleum ether using a mild pressure gradient (< 1 bar). Subsequently the solvent is allowed to evaporate at 110 °C for two h. In a second step the samples are filled with E7 using a mild pressure gradient supporting the capillary forces again. The E7-filled samples are heated to 110 °C and slowly cooled to room temperature. By this process pieces with a length of several cm can be fabricated. By monochromatic polarizing optical microscopy of single capillaries with diameters from 2 to 40 µm, the typical stripes textures were observed [17]. Comparing the textures with simulated results, the escaped radial director field (Fig. 2) could clearly be identified for the single capillaries treated with lecithin and filled with E7. The stripes textures observed here indicate a defect free alignment of the director over the length of several cm. Because of the great number of inclusions, polarizing optical microscopy of the filled LMA10 fibers is not as sound as for the single capillaries. Nevertheless, it could be used to exclude defective samples from further experimental analysis.

Fig. 2. Escaped radial director field. a) y,z-cross section b) x,y-cross section.

A special splicing technique needed to be developed to make the fiber samples stable enough to record attenuation spectra with a cut-back technique. Because of a limited sample length, the recording of high quality spectra is a challenging task for liquid crystal filled fibers. By using cut-back techniques, the spectral characteristics of fiber transmission are most accurately reproduced: Light is coupled to a long piece of a fiber and the transmitted optical power Il is detected. Subsequently the length of the piece is reduced by cutting away a shorter piece with a length d. Now the optical output power Il-d is recorded. The attenuation a, for instance in dB·cm−1, can then be calculated:

I a = 10 ⋅ log  l − d  Il

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 −1 ⋅d . 

(1)



Equation (1) describes the transmission characteristics of the sample appropriately, since the emission spectrum of the light source and the insertion efficiency of the light insertion into the fiber core are eliminated by this calculation. For this purpose, a special technique for the spectral analysis of photonic crystal fibers filled with liquid crystals was developed. In contrast to other groups, an adjustable monochromatic light source was used consisting of a xenon-arc-lamp and a fiber coupled monochromator. The white light from the arc-lamp is transmitted through the computer controlled grating monochromator with a focal length of 300 mm and is then coupled to a conventional optical fiber with a core diameter of 9 µm. This fiber is then used to couple light into the core of the filled photonic crystal fiber. Both fibers are cut precisely with an optical fiber cleaver to provide plane end faces. Subsequently, the fiber ends are exactly adjusted face to face so that the light coming from the standard optical fiber is inserted into the core of the photonic crystal fiber. The free end of the photonic crystal fiber is meanwhile observed with a microscope lens optic (using a CCD-camera). This near field analysis helps to make sure that the transmitted light is guided in the fiber core and not in the cladding. The coupled end pieces of the fibers are then embedded in a droplet of photo curable optical adhesive. Observing a high transmission, the fibers are once more readjusted and then the optical adhesive is photo cured with UV-radiation. Thereby, a stable splice of the two fibers is generated. And this is the precondition for cutting away a piece at the free end of the filled photonic crystal fiber without modifying the coupling situation. The transmitted monochromatic light can now be observed with almost any kind of optical detector. In our case the light was collected with a microscope lens and analyzed by means of a photomultiplier. The optical output power is then recorded in the visible wavelength range adjusting the monochromator with a step width of 2 nm and observing the free end of the filled photonic crystal fiber. Cutting length in the range of cm already suffice to observe the full transmission characteristics (Fig. 3). As expected, the filled fiber shows a windowed transmission. The transmission windows, regions of rather high transmission, are separated by attenuation maxima (462, 508, 568, 638, 702, 760 and 836 nm). In the transmission windows attenuations as low as 0.84 dB·cm−1 are observed (λmin= 670 nm and 800 nm). Five of these six windows are quite well pronounced: The contrast of the minimum and maximum attenuation is ≈3 dB·cm−1. The transmission window located between 702 and 760 nm has a higher average attenuation than the neighboring windows.

Fig. 3. Measured attenuation of the filled fiber versus wavelength.

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3. Theory

Some groups have studied basic effects theoretically assuming a uniform alignment of the birefringent liquid crystals [18, 19]. Currently, there is a trend of using more realistic expressions for the anisotropic dielectric permittivity tensor ε(r) in the liquid crystal filled regions. To mention two examples, theoretical investigations considering the infrared spectral region were published for the LMA-10 fiber [14] with splay aligned nematic liquid crystals and for a similar system using a LMA-13 fiber with larger inclusion diameters [20]. The infrared transmission spectra were estimated by calculating the coupling loss of a filled and an unfilled photonic crystal fiber eventually taking into account the influence of external electric fields. In contrast to these previous considerations, the current paper presents simulations that are based on the assumption that the propagation losses are caused by scattering due to thermal fluctuations of the molecular alignment of the liquid crystal. This requires a focus on the transmission properties in the visible wavelength range, though here the transmission spectra are more complicated than in the infrared. In the visible spectral region, the optical filtering properties of the filled fibers are especially pronounced, because the turbidity of the liquid crystal rises with decreasing wavelength. This model ideally fits the experimental data. Due to the cut-back technique, it can be excluded that coupling effects have influence on the experimental spectra. 3.1 Model for a modal analysis An accurate full-vectorial finite element algorithm [21] was used to compute the modal properties of the filled fibers. The simulations show that the shape of the spectra is roughly maintained and shifts to higher wavelength with rising hole diameter. Therefore, different hole diameters were considered in the range of 3 to 3.2 µm and the best fitting results were obtained for a value of 3.1 µm. In order to minimize the area of calculation, just 3 rings of filled holes were considered (Fig. 1) in our simulations. The simulations were performed in a two dimensional geometry to provide a high spatial resolution especially inside the filled inclusions. In order to make use the computer performance efficiently, the area of calculation was reduced to a quarter of the fiber cross section by exploiting the symmetry. A variation technique of the boundary conditions for the electric and magnetic field at the boundaries of the calculation area was then used to find all guided modes: The curved boundary is constantly treated as perfect electric conductor while the x- and y-boundaries (Fig. 1) are varied between perfect electric and perfect magnetic conductor approximations. Consequently, 4 runs of the simulation are performed at each wavelength, one for each permutation of the x- and y-boundary conditions. During the modal analysis, Maxwells equations are solved requiring the dielectric permittivity tensor ε. Inside the liquid crystal inclusions, the tensor ε represents an escaped radial director field, which is described by an analytical approximation [15]. The elements permittivity tensor are given by Eq. (2):

ε αβ = no 2 ⋅ δ αβ + ( ne 2 − no 2 ) ⋅ vα ⋅ vβ ,

(2)

where δαβ is the Kronecker-symbol, ne and no are the extraordinary and ordinary refractive indices of the birefringent liquid crystal, and va, vb are components of the director. Equation (3) describes the z-component of the escaped radial director field v(ri) (Fig. 2) which is non zero if the norm ri of the radial coordinate ri:=(xi,yi)T of the inclusion i (Fig. 1) is small (ri < Ri):

vz =

Ri2 − ( xi 2 + yi 2 )

Ri2 + ( xi 2 + yi 2 )

.

(3)

However, for calculating the optical properties, it is sufficient to consider only the contributions of the dielectric anisotropy to the x- and y-components of the dielectric tensor. #117869 - $15.00 USD

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This is a very simplified assumption. However, using this approximation the experimental data is resembled well, although in earlier experimental works with uniaxial director fields [22] there were discrepancies (at least to a certain degree). According to the analytical expressions for the escaped radial director field given in Ref [15], the dielectric tensor is given by Eq. (4):

 no 2 0  x2 0  xi ⋅ yi 0  1 − vz 2 )  i (    2 2 2 ε =  0 no 0  + ( ne − no ) 2 0 . (4) x ⋅ yi yi 2 2  i xi + yi )  ( 2  0  0 0 0 0 n o     The wavelength dependences of the (real parts of the) refractive indices of the materials were considered by using Eq. (5), the three-parameter Cauchy formula: n(λ0 ) = A + B ⋅ λ0−2 + C ⋅ λ0−4 .

(5)

The Cauchy coefficients for E7 (Ao = 1.49669; Bo = 0.00785 µm2; Co = 0.00026 µm4; Ae = 1.67798; Be = 0.01696 µm2 Ce = 0.00127 µm4) reported by Abbate [23] were used. The Cauchy coefficients of fused silica were obtained from a fit to data for Suprasil® glass by Heraeus (nC = 1.45637 at 656.3 nm; nd = 1.45846 at 587.6 nm; nF = 1,46313 at 486,1 nm; ng = 1.46669 at 435.8 nm; n248 = 1.50855 at 248 nm; Asuprasil= 1.44855; Bsuprasil= 0.00334 µm2; Csuprasil= 2.14528·10−5 µm4). So far, just the real parts of the refractive indices for both the liquid crystal and the glass were considered. To add an estimation of the losses, the simulated electric and magnetic fields were used in a power loss model using the corresponding scattering and absorption coefficients. The modal analysis yields the propagation constants β and the effective mode indices neff for the guided modes (neff (λ0) is real because no complex parts of the refractive indices were considered up to this point). From the effective indices neff(λ0), the waveguide dispersion can already be calculated using Eq. (6) and Eq. (7) (ω = 2π·c/λ0, c: speed of light, vg: group velocity): D=

vg =

ω 2 dvg , 2π cvg d ω c

neff (ω ) + ω

dneff

(6)

.

(7)

dω

3.2 Power loss estimation The confinement loss is usually calculated and interpreted as the attenuation for the core modes of photonic crystal fibers with high index inclusions. When using isotropic materials for the filling, light is guided not only in the fiber core, but also by the modes of the single high index inclusions (total internal reflection) [2]. In order to separate the fraction of light guided by the core from the light transmitted through the cladding, a small aperture has to be properly adjusted to the fiber core allowing an analysis of the core mode exclusively. In our experiments on E7 filled fibers no light guidance by the single inclusions could be detected because of the high attenuation in these inclusions. As mentioned above, waveguides with a core consisting of nematic liquid crystals show attenuations in the range of 20 to 40 dB·cm−1 [5]. Therefore, the intensity decays rapidly in the E7-inclusions. These high losses inside the inclusions can be evaluated to estimate the power loss of the whole structure very well. For this purpose, the refractive indices n and the dielectric constants of the materials are treated as being imaginary in the Eq. (8)a) to Eq. (8)d): n = n '− in '', #117869 - $15.00 USD

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(8a)



ε r = n 2 ⇒ ε r = n '2 − 2in ' n ''− n ''2 ,

(8b)

Re(ε r ) = n '2 − n ''2 ,

(8c)

Im(ε r ) = −2n ' n ''. (8d) In the visible spectral region, typical nematic liquid crystals show no absorption bands. Thus, the major source of loss is scattering. In any medium, small local changes of the density or the temperature can cause local variations of the dielectric tensor ε. De Gennes [24] has shown that in nematic liquid crystals, fluctuations of ε are dominantly caused by fluctuations of the orientation of the director v. The extent of fluctuations depends on the elastic constants Kii (i = 1, 2, 3). Considering the free elastic energy of bulk material in a thermal equilibrium, de Gennes theoretically derived dσ/dΩ, the differential scattering cross section of a nematic liquid crystal per solid angle. The total scattering cross section or turbidity of oriented nematic liquid crystals was then calculated by Langevin and Bouchiat [25]. Using their model, they could successfully extract the three elastic constants K11, K22, K33 from experimental light scattering data of a nematic liquid crystal in three selected geometries. Although their experiments were performed with a laser at one wavelength only, their model had to consider the wavelength dependence of the turbidity. From the turbidity for different molecular alignments, they derived a formula for an average scattering coefficient α0,lc. In contrast to their dimensionless scattering parameter, we use here an average scattering coefficient (SI unit m−1) α0,lc given by Eq. (9).

α 0,lc =

π kT ∆ε . λ0 2 K33 no '2

(9)

Equation (10a) to Eq. (10d) are the basic equations of the power loss model [26] for dielectric waveguides and yield the absorption coefficient of a waveguide, e. g. αfiber, as function of the transmitted power N(z) and the dielectric power loss density pV: 1 ( E(z ) × H* (z) ) ⋅ dA, 2 A

N ( z ) = ∫ S( z ) ⋅ dA = ∫ A

(10a)

1 pV = ωε 0ε r '' E 2 , 2

(10b)

P '( z ) = ∫ pV dA,

(10c)

A

α fiber =

P '( z ) , N (z )

(10d)

where S(z) is the Poynting vector, A the area of the waveguide cross section, E the electric field vector, H the magnetic field vector, ε0 the dielectric permittivity of the vacuum, and εr” the imaginary part of the relative dielectric constant. To calculate the integrals in the Eqs. (10a) to (10c), the absorption coefficient of silica glass reported in Ref [27]. was used in the glass areas of the fiber. The fields are simulated by considering the real part of the dielectric tensor. In the liquid crystal filled sections, the real part of the dielectric tensor for the escaped radial director field is considered with respect to the anisotropy of the liquid crystal. The imaginary part of the refractive index is in the liquid crystal filled sections approximated by Eq. (9). I. e., the turbidity of the liquid crystal and hence the imaginary part of the refractive index is treated as quasi isotropic (n´´lc ≈ne´´ ≈no´´). The relation α = 2k0 n '' = 4π n '' λ0 helps to extract n´´lc from Eq. (9):

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4π n ''

π kT ∆ε λ0 λ0 2 K 33 no '2 1 kT ∆ε . ⇔ n ''lc = 4λ0 K33 no '2 =

(11)

In order to obtain from Eq. (11) the imaginary part of the dielectric permittivity Eq. (8d) for the liquid crystal filled sections, the average value of the real parts of the refractive indices Eq. (12) is used: n'=

2 1 2 2 ( no ') + ( ne ') , 3 3

(12)

1 kT ∆ε 2no '2 + ne '2 . (13) 2λ0 K 33 no '2 3 Finally, this yields an expression for P´i(z) for the individual inclusions which can easily be evaluated using the simulated electric fields in the inclusion areas Ai:

ε '' =

P 'i ( z ) =

1

∫ 2 ωε '' E

2

dA = π

Ai

π c kT ∆ε ⇔ P 'i ( z ) = 2λ0 2 K 33 no '2

c

λ0

ε '' ∫ E 2 dA Ai

2no ' + ne '2 3 2

(14)

∫E

2

dA.

Ai

3.3 Theoretical Results Selected simulated modal intensity profiles are seen in Fig. 4 and Fig. 5. Light coupled in the core is guided with a low attenuation except a good confinement cannot be guaranteed by the surrounding microstructure: For example at 614 nm (Fig. 4(a)), a good confinement and a low attenuation are observed. The effective refractive index of the fundamental core mode is lower than the refractive index of fused silica. As for the spectral regions with low attenuation, the relation neff < nsilica is true for the spectral regions with high attenuation of the fundamental core mode, too. But in the latter case, the core mode has a large fraction of the intensity located in the high index anisotropic inclusions (Fig. 4(b)). This causes a high attenuation of the fundamental core mode. The modes of the liquid crystal inclusions (Fig. 5) have a relatively high attenuation. Therefore, light coupled into the inclusions is attenuated strongly. Near 614 nm, where the core mode has a very low attenuation, there is an inclusion mode (Fig. 5(a)) with an attenuation of 18 dB·cm−1 but this inclusion mode is not in a resonant state and the core mode cannot couple to it. The inclusion mode shown in Fig. 5(b) is in a resonant state at 707 nm (neff,inclusion= nsilica). Although the inclusion shows resonance already at 705 nm, the local maximum of the attenuation of the fundamental core mode is reached at little higher wavelength (≈707 nm) where neff,inclusion = 1.4548 < nsilica. For isotropic high index inclusions or inclusions of nematic liquid crystals with vz || z, the intensity of the modes is regularly located near the center of an inclusion. Notice, that this is not the case for the mayor number of modes of escaped radial inclusions. The modal intensity profiles avoid the center of escaped radial inclusions.

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Fig. 4. Core mode of the liquid crystal filled photonic crystal fiber. The intensity is plotted as shade and the x- and y-components of the electric field vector are indicated by arrows. a) core mode at 614 nm, amode= 0.24 dB·cm−1, neff= 1.4569, nsilica= 1.4576, b) core mode at 707 nm, amode= 3.7 dB·cm−1, neff= 1.4543, nsilica= 1.4553.

Fig. 5. Selected inclusion modes of the liquid crystal filled photonic crystal fiber. The intensity is plotted as shade and the x- and y-components of the electric field vector are indicated by arrows. a) inclusion mode at 616 nm, amode= 18.2 dB·cm−1, neff= 1.4579, nsilica= 1.4575, b) inclusion mode at 705 nm, amode= 11.8 dB·cm−1, neff= 1.4554= nsilica.

To give a direct comparison, the measured and simulated spectra are plotted together in Fig. 6. The main transmission characteristics are found by the simulation. As expected, the spectra of the real system are noisier. As a consequence of the idealized geometry, the simulated attenuations of the fundamental core mode are lower than the observed attenuations. The six transmission bands in the experimentally investigated spectral region are reproduced by the simulation. In Fig. 6, these transmission windows are indicated by bars. The attenuation maxima at 508, 568, 638 and 836 nm are accurately reproduced by the simulation. Unfortunately, the measured spectrum is quite noisy at lower wavelengths. At wavelengths smaller than 638 nm, the simulation reveals the tendency that the transmission windows are getting narrower with decreasing wavelength. This tendency can also be seen in the measured spectra. The transmission window around 730 nm, which was already discussed as being unusually small, is seen in the measurement as well as in the simulation. However, the shape of the simulation spectrum does not resemble the experimental one exactly. The position of the attenuation maxima in the simulations is in good agreement with the experimental maxima for most of the transmission windows. This can be due to the fact, that leakage loss was not included in the simulations. On the other hand, this problem occurs also for simulations evaluating the leakage loss [4]. The fact that a simplified permittivity tensor was used can lead to a different position of the band gap edges, depending on which capillary modes have a strong Ez direction. This might be the reason that some of the attenuation maxima match very well the experimental results, while some do not.

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Fig. 6. Measured attenuation (dots) and simulated attenuation (stars) of the filled fiber versus wavelength.

The simulated attenuation spectra should additionally be interpreted with the help of the calculated chromatic dispersion (Fig. 7). In a photonic band gap, the chromatic dispersion typically crosses the zero line one time and additionally D(λ) should have an inflexion point [10]. D should be negative in the short wavelength regime of a band gap and grow with rising wavelength. Therefore, the dispersion curves can help to identify the single band gaps. The individual transmission windows seem to consist of several band gaps from this point of view. Notice, that the calculated chromatic dispersion curves again show irregularities at wavelength around 730 nm. The experimentally observed transmission windows exhibit different wave guiding properties: The simulation predicts the position of zero dispersion wavelengths, but not every transmission window seems to show them.

Fig. 7. Simulated attenuation (left scale, solid line) and simulated chromatic dispersion D (right scale, dotted line) of the filled fiber versus wavelength.

4. Conclusions

In summary, it was found that micro structured solid core fibers filled with E7 exhibiting perpendicular anchoring can lead to attenuations below 1 dB·cm−1 in certain spectral regions. Even fiber pieces with a length as short as ≈1 cm show a very pronounced spectral distribution of the transmitted intensity, so that spectral filters with a high contrast ratio are feasible. The observation of transmission windows with low attenuation is in promising contradiction to earlier observations on the wave guiding properties of liquid crystal filled

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slab waveguides [5], which indicated a very high attenuation in the entire visible wavelength range due to scattering losses. Obviously, the micro structured fibers investigated in this work show a core with very high transmission, while losses are caused by scattering due to the liquid crystal inclusions. In order to study this behavior experimentally in more detail, a method was developed in order to fabricate rather long infiltrated fibers, so that the well-known cut-back technique could be applied. The latter method can be used to measure attenuation spectra that show exclusively the propagation loss of fibers, independent of the properties of the light source and the fiber splice. Thanks to this new development, it is possible to compare the experimental results with theoretical calculations. Theoretical spectral mode analysis indicates that guided modes appear in the entire visible wavelength range, both for the core and the liquid crystal inclusions of the fiber. However, near field optical experiments indicate that high transmitted intensities appear exclusively in the fiber core. Comparison between the theoretical mode analysis and this experimental observation confirms that the resonant or anti resonant coupling between the core and the inclusions is essential for the exchange of energy, and thus for the final transmission of the fiber. For quantitative calculations, we introduced a scattering coefficient that describes the power loss due to the scattering caused by thermal fluctuations of the liquid crystal director. The simulations fit the experimental data well. In conclusion, the spectral distribution of the transmission of liquid crystal filled solid core photonic crystal fibers can be calculated very well, by a model, which takes into account the geometry and material parameters of the microstructure and a scattering loss mechanism based on thermal fluctuations of the liquid crystal. The practical advantage of this type of fibers is the low attenuation of exclusively the core in certain spectral regions, in spite of the high attenuation of the liquid crystal filled sections. This finding should also encourage trying other filling materials with high turbidity like colored liquids or colored or colloidal solutions. In addition, the unique sensitivity of the optical properties of liquid crystals to a static electric field and their giant optical nonlinearity can be used to achieve tunable devices [6,7]. Acknowledgments

Support by the German Research Foundation (GRK 1464) is gratefully acknowledged.

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