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Measurement of group velocity dispersion in a solid-core photonic crystal fiber filled with a nematic liquid crystal Markus Wahle and Heinz Kitzerow* Department of Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany *Corresponding author: heinz.kitzerow@uni‑paderborn.de Received June 13, 2014; revised July 11, 2014; accepted July 11, 2014; posted July 16, 2014 (Doc. ID 214091); published August 11, 2014 Liquid crystal-filled photonic crystal fibers (PCFs) are promising candidates for electrically tunable integrated photonic devices. In this Letter, we present group velocity measurements on such fibers. A large mode area PCF, LMA8, was infiltrated with the liquid crystal mixture, E7. The measurements were performed with an interferometric setup. The fiber exhibits several spectral transmission windows in the visible wavelength regime that originate from the bandgap guiding mechanism. The dispersion of these windows is very unusual compared to typical fibers. Our measurements show that it can change from −2500 ps km−1 nm−1 to 2500 ps km−1 nm−1 within a spectral range of only 15 nm. This leads to multiple zero dispersion wavelengths in the visible wavelength range. © 2014 Optical Society of America OCIS codes: (060.5295) Photonic crystal fibers; (060.2270) Fiber characterization; (060.5530) Pulse propagation and temporal solitons; (230.3720) Liquid-crystal devices. http://dx.doi.org/10.1364/OL.39.004816
The invention of photonic crystal fibers (PCFs) has led to a multitude of new designs for optical fibers [1]. While conventional fibers guide light due to total internal reflection (TIR) between a high index core and a low index cladding, the guiding in PCFs is different due to the periodic cladding that surrounds the core [1]. In the case of a 2D array of air inclusions embedded in a silica background material (Fig. 1), where the core is formed by a missing inclusion, the guiding mechanism is very similar to TIR [2] and thus is called modified total internal reflection (mTIR) [1]. The use of high index rods instead of air inclusions leads to the formation of photonic bandgaps that confine the light inside the core [3–8]. This mechanism is very different from the conventional TIR because the light is now confined in a region of low refractive index, which is incompatible with a mTIR approach. Due to the unusual guiding mechanism, the latter fibers are called photonic bandgap fibers (PBGFs) [9]. Liquid crystals (LCs) can serve as an anisotropic high index material if filled in the inclusions [10–13]. This case is of special interest because LCs can be tuned by heat and electric or magnetic fields [13–15]. Based on this idea, many different devices have been designed, for example tunable polarisers [16–18], attenuators [19] and fiber couplers [20]. A review on various liquid crystal photonic crystal fiber (LCPCF) devices can be found in Ref. [21]. So far, very little attention has been paid on the influence of LCPCFs on the propagation of optical pulses. The formation of photonic bandgaps usually leads to interesting effects with regard to group velocity and the group velocity dispersion (GVD) [3,4,22]. These quantities determine how fast the pulse envelope moves through the fiber and how its shape changes during propagation. The addressability of the LC by external fields can be expected to offer the possibility of tuning the group velocity and its dispersion. In this Letter, we explore the GVD of a commercially available large mode area (LMA) PCF filled with the nematic liquid crystal, E7 (Merck), in the visible wavelength range. 0146-9592/14/164816-04$15.00/0
The bandgap guiding mechanism leads to broad spectral windows of high transmission (each corresponding to the wavelength range of a photonic bandgap), that are separated by the transmission dips. The position of the dips in transmission can be predicted with the help of the antiresonant reflecting optical waveguiding (ARROW) model. The model ascribes the dips to waveguide modes originating from the high index rods to which the core PCF mode couples [23,24]. This is equivalent to a loss of confinement and increases gradually if one approaches the band edges. The coupling tends to bend the effective refractive index, neff , of the PCF mode [25] and also the group velocity and its dispersion. The group velocity, vg , and neff are related by −1 ∂β −1 ∂n c neff − λ eff ; ∂λ ∂ω
vg
(1)
where ω is the angular frequency, λ is the wavelength, β is the propagation constant, and c is the speed of light. The propagation constant and the effective refractive index are related by β neff k0 , where k0 is the wavenumber in a vacuum. The GVD describes the change of the inverse group velocity when changing the angular frequency,
Fig. 1. (a) Schematic of a solid core PCF: air holes (white) in a silica background material (gray). (b) Enlargement of the periodic cladding with inclusion diameter, d, and lattice pitch, Λ. © 2014 Optical Society of America
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
GVD
∂v−1 ∂2 β g 2: ∂ω ∂ω
4817
(2)
More commonly, instead of the GVD, the dispersion, D, is given to characterize optical waveguides. It is related to GVD by [26], D−
λ2 GVD: 2πc
(3)
The dispersion, D, is a well measurable quantity because it describes the change in the delay, τ, between two pulses traveling through a fiber of length L which are centered around different wavelengths, λ and λ dλ, D
∂v−1 1 ∂τ g : ∂λ L ∂λ
(4)
It is convenient to fit the experimentally acquired values to a Sellmeier equation, n2eff − 1
X ai λ2 ; λ2 − λ2i i
(5)
where ai and λi are the parameters to be fitted. The group delay can then be found by using Eq. (1), L L ∂neff : τ neff − λ ∂λ vg c
(6)
A least square fit of the group delay, τ, based on Eqs. (5) and (6), to the experimental values of τλ is performed for each spectral transmission window. As starting values for λi , the approximate positions of the short and far wavelength edges of the spectral transmission windows are used. Usually, a third term is needed for an adequate fit, with its λi located in the infrared. The investigated fibers are prepared in a two step process. First, an alignment layer is created on the inclusion walls; second, the nematic liquid crystal is filled into these inclusions. A high quality alignment is very important because inhomogeneities change the guiding properties locally and also the dispersion of the fiber. To this end, a linear photopolymerization technique was used to ensure a uniaxial orientation of the LC. This technique was initially applied to liquid crystal cells [27] and has proven to be applicable to capillaries as well [28]. The fiber (LMA8: d 2.27 μm, Λ 5.35 μm, made from silica, n 1.4584 at 589 nm, NKT Photonics) was cut to a length of 10–20 cm. In the next step, the fiber was filled with a 2% solution of poly-vinylmethylcinnamate in cyclopentanone (ROLIC JP265). Subsequently, the solution was extracted by a nitrogen flow at 3 bars, which took approximately 1 h. This procedure leaves a thin film of the solution on the inner inclusion walls. The residual solvent was removed by heating the sample at 135°C for 1 h. The photopolymerization was performed with a UV source (Hönle bluepoint 4) equipped with a linear polariser and a short pass filter which transmits light below 380 nm. The polarization of the UV light is perpendicular to the fiber axis [27] to obtain alignment along the fiber
Fig. 2. Mach–Zehnder interferometer for measuring the pulse delay. M, mirror; BS, beamsplitter; Col, collimator; Si PD, silicon photodiode. The delay of the upper path is adjustable by a moving stage.
axis. Due to the limited spot size, the fiber was continuously rotated under the UV source, exposing only about 1 cm of the fiber to the light at the same time. The sample was exposed to 5 J of radiation at an intensity of approximately 20 mW∕cm2 . Finally, the liquid crystal mixture, E7 (no 1.5222, ne 1.7390 at 589 nm), was filled into inclusions by applying a vacuum from the opposite side. The delay in group velocity is measured with an interferometric setup [29]. This setup (Fig. 2) is necessary because the fiber length is limited to a few ten centimeters due to scattering losses inside of the liquid crystal [30]. Thus, one has to be able to detect small changes in the group velocity. Light from a supercontinuum source (NKT SuperK compact) is passed through a monochromator onto a beamsplitter. where it is split into a delay path and a fiber path. The length of the delay path can be adjusted by a motorized stage. The collimated beam of the fiber path is coupled into the test fiber and collimated again at the other end with a microscope objective. Both collimated beams are recombined at a second beamsplitter. One of the outgoing beams is directed to a CCD camera to observe the pattern and the overlap of the two beams. The second outgoing beam is coupled into a single mode fiber and then detected with a silicon photodiode. For every wavelength, an interferogram is recorded. The change in group delay, Δτ, between two different wavelengths can then be found by evaluating the change of the peak position, 2Δ, of the interference pattern. They are related by Δτ 2Δ∕c. The accuracy of the method depends on finding the maximum value of the pulse envelope and was evaluated by measuring the interference patterns multiple times at a specific wavelength. The standard deviation was determined to be less than 0.01 ps. Although the absolute value of the group delay cannot be measured directly, it is possible to deduce this quantity by calibrating the setup using a fiber with known group delay. Figure 3 shows the results of the group delay measurement in the visible wavelength range of a 12.5 cm long piece of LMA8 fiber filled with the nematic liquid crystal mixture, E7. This is probably a result of the Gaussian input pulse and the higher attenuation of these modes due to lower confinement and higher scattering losses. The sample shows spectral transmission windows [Fig. 3(a), gray line], which are expected in the case of photonic bandgap guiding. For each of these transmission windows, the group delays (red circles) have been measured
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OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
liquid crystal leads to a change in guiding mechanism from mTIR to photonic bandgap guiding, which results in the appearance of spectral transmission windows. The group delay is significantly increased close to the edges of the transmission windows and, therefore, the dispersion is strongly bent. It reaches the point of zero dispersion and proceeds into the regime of anomalous dispersion for every transmission window. Previous experiments [15] have shown thermally and electrically induced continuous tuning of the position of the spectral transmission windows. From this observation, together with the work presented here and the theoretical analysis presented in [30], the opportunity of tuning the group velocity and the dispersion along with the transmission windows is expected.
Fig. 3. (a) Measured group delay, τ, (circles) and fitted Sellmeier curve (solid black line). The gray line represents the measured normalized intensity. (b) Dispersion, D, calculated from the Sellmeier fit for LMA8 filled with E7 (solid). For comparison, the dispersion of an unfilled (air filled) LMA8 (dashed) is plotted.
and a Sellmeier-based fit (solid black line) was used to obtain a differentiable function for the dispersion, which is displayed in Fig. 3(b). The accuracy of the fit is assessed by the average deviation, which is 0.132 ps in this case. In principle, higher-order modes are possible for these fiber specifications, but were not observed during the measurements. In general, the group delay tends to shift to lower values with increasing wavelengths, which can be attributed to the decreasing group index of silica, i.e., a material property. However, focusing on the individual transmission windows close to the edges of the windows, a clear increase in group delay can be observed. This is a result of the increasing amount of modal power guided inside the LC inclusions instead of the silica core. The Sellmeier-based fit appears to be a suitable approximation for the measured group delay. The dispersion [Fig. 3(b), solid line] was obtained from the fitted group delay by use of Eq. (4). As for the group delay, there is a material dominated shift of the infliction points with increasing wavelength. The waveguiderelated dispersion results in a strong bending at the edges that leads the dispersion coming from the regime of normal dispersion to cross through zero into the regime of anomalous dispersion for every window except the first. In the latter case, the intensity at the large wavelength edge was insufficient for reliable measurements. For comparison, the dispersion of a 38.5 cm piece of an unfilled LMA8 fiber was measured [Fig. 3(b), dashed line]. In summary, we have measured the group delay of the commercially available PCF LMA8 with a uniaxial alignment layer infiltrated with the liquid crystal mixture E7. From this, we derived the dispersion by using a Sellmeier equation-based fit. The infiltration with a high index
Financial support by the Deutsche Forschungsgemeinschaft (DFG Research Training Group 1464, Micro- and Nanostructures in Optoelectronics and Photonics) is gratefully acknowledged. The authors would like to thank Christine Silberhorn and Benjamin Brecht for illuminating discussions. References 1. P. S. J. Russell, J. Lightwave Technol. 24, 4729 (2006). 2. T. A. Birks, J. C. Knight, and P. S. Russell, Opt. Lett. 22, 961 (1997). 3. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. S. J. Russell, Opt. Lett. 29, 2369 (2004). 4. A. Fuerbach, P. Steinvurzel, J. A. Bolger, A. Nulsen, and B. J. Eggleton, Opt. Lett. 30, 830 (2005). 5. A. Argyros, T. A. Birks, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, Opt. Express 13, 309 (2005). 6. O. N. Egorova, S. L. Semjonov, A. F. Kosolapov, A. N. Denisov, A. D. Pryamikov, D. A. Gaponov, A. S. Biriukov, E. M. Dianov, M. Y. Salganskii, V. F. Khopin, M. V. Yashkov, A. N. Gurianov, and D. V. Kuksenkov, Opt. Express 16, 11735 (2008). 7. B. Kuhlmey, B. Eggleton, and D. Wu, J. Lightwave Technol. 27, 1617 (2009). 8. N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. S. J. Russell, J. Non-Cryst. Solids 356, 1829 (2010). 9. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, Science 282, 1476 (1998). 10. T. Larsen, A. Bjarklev, D. Hermann, and J. Broeng, Opt. Express 11, 2589 (2003). 11. T. Alkeskjold, J. Laegsgaard, A. Bjarklev, D. Hermann, A. Anawati, J. Broeng, J. Li, and S.-T. Wu, Opt. Express 12, 5857 (2004). 12. T. R. Wolinski, K. Szaniawska, K. Bondarczuk, P. Lesiak, and A. W. Domañski, Opto-Electron. Rev. 13, 177 (2005). 13. A. Lorenz, H.-S. Kitzerow, A. Schwuchow, J. Kobelke, and H. Bartelt, Opt. Express 16, 19375 (2008). 14. T. R. Wolinski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, Meas. Sci. Technol. 17, 985 (2006). 15. L. Wei, L. Eskildsen, J. Weirich, L. Scolari, T. T. Alkeskjold, and A. Bjarklev, Appl. Opt. 48, 497 (2009). 16. D. Zografopoulos, E. Kriezis, and T. Tsiboukis, Opt. Express 14, 914 (2006). 17. A. K. Pitilakis, D. C. Zografopoulos, and E. E. Kriezis, J. Lightwave Technol. 29, 2560 (2011). 18. A. Lorenz and H.-S. Kitzerow, Appl. Phys. Lett. 98, 241106 (2011). 19. M. Tefelska, M. Chychlowski, T. R. Wolinski, R. Dkabrowski, and J. Wójcik, Photon. Lett. Poland 1, 97 (2009).
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS 20. D. J. J. Hu, P. Shum, C. Lu, X. Sun, G. Ren, X. Yu, and G. Wang, Opt. Commun. 282, 2343 (2009). 21. H.-S. Kitzerow, in Handbook of Liquid Crystals, J. W. Goodby, P. J. Collings, H. Gleeson, T. Kato, P. Raynes, and C. Tschierske, eds., 2nd ed. (Wiley, 2014), Vol. 8, Chap. 13, pp. 373–426. 22. P. D. Rasmussen, J. Laegsgaard, and O. Bang, Opt. Express 16, 4059 (2008). 23. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, Opt. Express 11, 1243 (2003). 24. N. Litchinitser, S. Dunn, P. Steinvurzel, B. Eggleton, T. White, R. McPhedran, and C. de Sterke, Opt. Express 12, 1540 (2004).
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25. G. Renversez, P. Boyer, and A. Sagrini, Opt. Express 14, 5682 (2006). 26. J. A. Buck, Fundamentals of Optical Fibers (Wiley, 1995). 27. M. Schadt, K. Schmitt, V. Kozinkov, and V. Chigrinov, Jpn. J. Appl. Phys. 31, 2155 (1992). 28. H. Kitzerow, B. Liu, F. Xu, and P. Crooker, Phys. Rev. E 54, 568 (1996). 29. T. G. Euser, M. A. Schmidt, N. Y. Joly, C. Gabriel, C. Marquardt, L. Y. Zang, M. Förtsch, P. Banzer, A. Brenn, D. Elser, M. Scharrer, G. Leuchs, and P. S. J. Russell, J. Opt. Soc. Am. B 28, 193 (2011). 30. A. Lorenz, R. Schuhmann, and H.-S. Kitzerow, Opt. Express 18, 3519 (2010).
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
Measurement of group velocity dispersion in a solid-core photonic crystal fiber filled with a nematic liquid crystal Markus Wahle and Heinz Kitzerow* Department of Chemistry, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany *Corresponding author: heinz.kitzerow@uni‑paderborn.de Received June 13, 2014; revised July 11, 2014; accepted July 11, 2014; posted July 16, 2014 (Doc. ID 214091); published August 11, 2014 Liquid crystal-filled photonic crystal fibers (PCFs) are promising candidates for electrically tunable integrated photonic devices. In this Letter, we present group velocity measurements on such fibers. A large mode area PCF, LMA8, was infiltrated with the liquid crystal mixture, E7. The measurements were performed with an interferometric setup. The fiber exhibits several spectral transmission windows in the visible wavelength regime that originate from the bandgap guiding mechanism. The dispersion of these windows is very unusual compared to typical fibers. Our measurements show that it can change from −2500 ps km−1 nm−1 to 2500 ps km−1 nm−1 within a spectral range of only 15 nm. This leads to multiple zero dispersion wavelengths in the visible wavelength range. © 2014 Optical Society of America OCIS codes: (060.5295) Photonic crystal fibers; (060.2270) Fiber characterization; (060.5530) Pulse propagation and temporal solitons; (230.3720) Liquid-crystal devices. http://dx.doi.org/10.1364/OL.39.004816
The invention of photonic crystal fibers (PCFs) has led to a multitude of new designs for optical fibers [1]. While conventional fibers guide light due to total internal reflection (TIR) between a high index core and a low index cladding, the guiding in PCFs is different due to the periodic cladding that surrounds the core [1]. In the case of a 2D array of air inclusions embedded in a silica background material (Fig. 1), where the core is formed by a missing inclusion, the guiding mechanism is very similar to TIR [2] and thus is called modified total internal reflection (mTIR) [1]. The use of high index rods instead of air inclusions leads to the formation of photonic bandgaps that confine the light inside the core [3–8]. This mechanism is very different from the conventional TIR because the light is now confined in a region of low refractive index, which is incompatible with a mTIR approach. Due to the unusual guiding mechanism, the latter fibers are called photonic bandgap fibers (PBGFs) [9]. Liquid crystals (LCs) can serve as an anisotropic high index material if filled in the inclusions [10–13]. This case is of special interest because LCs can be tuned by heat and electric or magnetic fields [13–15]. Based on this idea, many different devices have been designed, for example tunable polarisers [16–18], attenuators [19] and fiber couplers [20]. A review on various liquid crystal photonic crystal fiber (LCPCF) devices can be found in Ref. [21]. So far, very little attention has been paid on the influence of LCPCFs on the propagation of optical pulses. The formation of photonic bandgaps usually leads to interesting effects with regard to group velocity and the group velocity dispersion (GVD) [3,4,22]. These quantities determine how fast the pulse envelope moves through the fiber and how its shape changes during propagation. The addressability of the LC by external fields can be expected to offer the possibility of tuning the group velocity and its dispersion. In this Letter, we explore the GVD of a commercially available large mode area (LMA) PCF filled with the nematic liquid crystal, E7 (Merck), in the visible wavelength range. 0146-9592/14/164816-04$15.00/0
The bandgap guiding mechanism leads to broad spectral windows of high transmission (each corresponding to the wavelength range of a photonic bandgap), that are separated by the transmission dips. The position of the dips in transmission can be predicted with the help of the antiresonant reflecting optical waveguiding (ARROW) model. The model ascribes the dips to waveguide modes originating from the high index rods to which the core PCF mode couples [23,24]. This is equivalent to a loss of confinement and increases gradually if one approaches the band edges. The coupling tends to bend the effective refractive index, neff , of the PCF mode [25] and also the group velocity and its dispersion. The group velocity, vg , and neff are related by −1 ∂β −1 ∂n c neff − λ eff ; ∂λ ∂ω
vg
(1)
where ω is the angular frequency, λ is the wavelength, β is the propagation constant, and c is the speed of light. The propagation constant and the effective refractive index are related by β neff k0 , where k0 is the wavenumber in a vacuum. The GVD describes the change of the inverse group velocity when changing the angular frequency,
Fig. 1. (a) Schematic of a solid core PCF: air holes (white) in a silica background material (gray). (b) Enlargement of the periodic cladding with inclusion diameter, d, and lattice pitch, Λ. © 2014 Optical Society of America
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
GVD
∂v−1 ∂2 β g 2: ∂ω ∂ω
4817
(2)
More commonly, instead of the GVD, the dispersion, D, is given to characterize optical waveguides. It is related to GVD by [26], D−
λ2 GVD: 2πc
(3)
The dispersion, D, is a well measurable quantity because it describes the change in the delay, τ, between two pulses traveling through a fiber of length L which are centered around different wavelengths, λ and λ dλ, D
∂v−1 1 ∂τ g : ∂λ L ∂λ
(4)
It is convenient to fit the experimentally acquired values to a Sellmeier equation, n2eff − 1
X ai λ2 ; λ2 − λ2i i
(5)
where ai and λi are the parameters to be fitted. The group delay can then be found by using Eq. (1), L L ∂neff : τ neff − λ ∂λ vg c
(6)
A least square fit of the group delay, τ, based on Eqs. (5) and (6), to the experimental values of τλ is performed for each spectral transmission window. As starting values for λi , the approximate positions of the short and far wavelength edges of the spectral transmission windows are used. Usually, a third term is needed for an adequate fit, with its λi located in the infrared. The investigated fibers are prepared in a two step process. First, an alignment layer is created on the inclusion walls; second, the nematic liquid crystal is filled into these inclusions. A high quality alignment is very important because inhomogeneities change the guiding properties locally and also the dispersion of the fiber. To this end, a linear photopolymerization technique was used to ensure a uniaxial orientation of the LC. This technique was initially applied to liquid crystal cells [27] and has proven to be applicable to capillaries as well [28]. The fiber (LMA8: d 2.27 μm, Λ 5.35 μm, made from silica, n 1.4584 at 589 nm, NKT Photonics) was cut to a length of 10–20 cm. In the next step, the fiber was filled with a 2% solution of poly-vinylmethylcinnamate in cyclopentanone (ROLIC JP265). Subsequently, the solution was extracted by a nitrogen flow at 3 bars, which took approximately 1 h. This procedure leaves a thin film of the solution on the inner inclusion walls. The residual solvent was removed by heating the sample at 135°C for 1 h. The photopolymerization was performed with a UV source (Hönle bluepoint 4) equipped with a linear polariser and a short pass filter which transmits light below 380 nm. The polarization of the UV light is perpendicular to the fiber axis [27] to obtain alignment along the fiber
Fig. 2. Mach–Zehnder interferometer for measuring the pulse delay. M, mirror; BS, beamsplitter; Col, collimator; Si PD, silicon photodiode. The delay of the upper path is adjustable by a moving stage.
axis. Due to the limited spot size, the fiber was continuously rotated under the UV source, exposing only about 1 cm of the fiber to the light at the same time. The sample was exposed to 5 J of radiation at an intensity of approximately 20 mW∕cm2 . Finally, the liquid crystal mixture, E7 (no 1.5222, ne 1.7390 at 589 nm), was filled into inclusions by applying a vacuum from the opposite side. The delay in group velocity is measured with an interferometric setup [29]. This setup (Fig. 2) is necessary because the fiber length is limited to a few ten centimeters due to scattering losses inside of the liquid crystal [30]. Thus, one has to be able to detect small changes in the group velocity. Light from a supercontinuum source (NKT SuperK compact) is passed through a monochromator onto a beamsplitter. where it is split into a delay path and a fiber path. The length of the delay path can be adjusted by a motorized stage. The collimated beam of the fiber path is coupled into the test fiber and collimated again at the other end with a microscope objective. Both collimated beams are recombined at a second beamsplitter. One of the outgoing beams is directed to a CCD camera to observe the pattern and the overlap of the two beams. The second outgoing beam is coupled into a single mode fiber and then detected with a silicon photodiode. For every wavelength, an interferogram is recorded. The change in group delay, Δτ, between two different wavelengths can then be found by evaluating the change of the peak position, 2Δ, of the interference pattern. They are related by Δτ 2Δ∕c. The accuracy of the method depends on finding the maximum value of the pulse envelope and was evaluated by measuring the interference patterns multiple times at a specific wavelength. The standard deviation was determined to be less than 0.01 ps. Although the absolute value of the group delay cannot be measured directly, it is possible to deduce this quantity by calibrating the setup using a fiber with known group delay. Figure 3 shows the results of the group delay measurement in the visible wavelength range of a 12.5 cm long piece of LMA8 fiber filled with the nematic liquid crystal mixture, E7. This is probably a result of the Gaussian input pulse and the higher attenuation of these modes due to lower confinement and higher scattering losses. The sample shows spectral transmission windows [Fig. 3(a), gray line], which are expected in the case of photonic bandgap guiding. For each of these transmission windows, the group delays (red circles) have been measured
4818
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
liquid crystal leads to a change in guiding mechanism from mTIR to photonic bandgap guiding, which results in the appearance of spectral transmission windows. The group delay is significantly increased close to the edges of the transmission windows and, therefore, the dispersion is strongly bent. It reaches the point of zero dispersion and proceeds into the regime of anomalous dispersion for every transmission window. Previous experiments [15] have shown thermally and electrically induced continuous tuning of the position of the spectral transmission windows. From this observation, together with the work presented here and the theoretical analysis presented in [30], the opportunity of tuning the group velocity and the dispersion along with the transmission windows is expected.
Fig. 3. (a) Measured group delay, τ, (circles) and fitted Sellmeier curve (solid black line). The gray line represents the measured normalized intensity. (b) Dispersion, D, calculated from the Sellmeier fit for LMA8 filled with E7 (solid). For comparison, the dispersion of an unfilled (air filled) LMA8 (dashed) is plotted.
and a Sellmeier-based fit (solid black line) was used to obtain a differentiable function for the dispersion, which is displayed in Fig. 3(b). The accuracy of the fit is assessed by the average deviation, which is 0.132 ps in this case. In principle, higher-order modes are possible for these fiber specifications, but were not observed during the measurements. In general, the group delay tends to shift to lower values with increasing wavelengths, which can be attributed to the decreasing group index of silica, i.e., a material property. However, focusing on the individual transmission windows close to the edges of the windows, a clear increase in group delay can be observed. This is a result of the increasing amount of modal power guided inside the LC inclusions instead of the silica core. The Sellmeier-based fit appears to be a suitable approximation for the measured group delay. The dispersion [Fig. 3(b), solid line] was obtained from the fitted group delay by use of Eq. (4). As for the group delay, there is a material dominated shift of the infliction points with increasing wavelength. The waveguiderelated dispersion results in a strong bending at the edges that leads the dispersion coming from the regime of normal dispersion to cross through zero into the regime of anomalous dispersion for every window except the first. In the latter case, the intensity at the large wavelength edge was insufficient for reliable measurements. For comparison, the dispersion of a 38.5 cm piece of an unfilled LMA8 fiber was measured [Fig. 3(b), dashed line]. In summary, we have measured the group delay of the commercially available PCF LMA8 with a uniaxial alignment layer infiltrated with the liquid crystal mixture E7. From this, we derived the dispersion by using a Sellmeier equation-based fit. The infiltration with a high index
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