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Fast electric field switched 2D-photonic liquid crystals S. P. Palto,* M. I. Barnik, A. R. Geivandov, I. V. Kasyanova, V. S. Palto, and N. M. Shtykov Institute of Crystallography of Russian Academy of Sciences, Leninsky pr. 59, 119333 Moscow, Russia *Corresponding author: [email protected] Received January 2, 2015; accepted February 10, 2015; posted February 17, 2015 (Doc. ID 231618); published March 19, 2015 We demonstrate field-induced 2D-photonic liquid crystals (LC). The 2D spatially periodic modulation of the LC director field is achieved using a geometry with two crossed interdigitated systems of electrodes located at opposite sides of the LC layer. With a special method of dual-field driving, a very fast switching between different spatially periodic LC director distributions is achieved. The director field distribution and potential use of these photonic crystals for fast switched multidirectional lasing is discussed. © 2015 Optical Society of America OCIS codes: (230.0250) Optoelectronics; (220.0220) Optical design and fabrication; (230.3720) Liquid-crystal devices; (230.5298) Photonic crystals; (250.6715) Switching. http://dx.doi.org/10.1364/OL.40.001254
Recently, photonic crystals [1] have been attracting high attention because of the perspectives of their use in many optical applications. Their periodic structure provides photonic band gaps (PBG) in a spectral range of Bragg reflection. In the region of PBG, because of strong interference effects, the propagation of light waves is forbidden, while at the edges of the band gaps the group velocity of light becomes very low, and high density of photonic states takes place. More than 40 years ago, it was shown by Kogelnik and Shank [2] that just at the edges of the band gaps the lowest threshold for the lasing effect can be achieved. PBG effects are also known in liquid crystals (LC) [3–9]. The most attractive property of PBG LC materials is that the band-gap effects can be controlled by an electric field. Perspective candidates for PBG microlasers are, for example, cholesteric LCs [3–6]. However, electric field lasing control was found to be rather difficult in these materials because of topological problems that prevent continuous unwinding of cholesteric helix [10]. Another approach is to utilize nematic LCs, either as intermediate layers [6] or as layers where the refractive index is modulated as a result of interference of pumping beams [7], by a spatially periodic electric field [9] or by covering a nano-patterned titanium dioxide photonic crystal within a well-oriented film of dye-doped LC [11]. This Letter is focused on fast field-induced LC PBG structures and consists of three closely associated parts. In the first part we describe a new and very fast method of electric field induction of PBG structures in nematic LCs. The method (referred to as “dual-field driving”) is based on fast switching of the direction of the electric field vector, which allows excluding the slow viscous-elastic relaxation characteristic of nematic LCs. Second, results of numerical simulations and experimental study of the LC director (local optical axis) distributions, which define field-induced refractive index patterns for the two switched electric field configurations, are discussed. Finally, using found field-induced director patterns, we demonstrate numerically the possibility of a fast switched low-threshold lasing effect in the visible spectral range, for which a huge assortment of lasing dyes for LC doping is available. The liquid crystal material used is a nematic LC (Merck ZLI-1957/5) with positive dielectric anisotropy Δε 4.5, 0146-9592/15/071254-04$15.00/0
optical anisotropy Δn 0.1213, and rotational viscosity γ ≅ 105 mPa · s (at a temperature of 20°C). The LC homeotropic alignment is provided by thin films of chromium distearyl chloride (chromolane) deposited onto glass substrate surfaces with electrodes. The electric field is created with the help of two crossed interdigitated systems of chromium electrodes located on the internal surfaces of the cell substrates, as shown in Fig. 1. It should be mentioned that interdigitated electrodes are widely used to electrically address LCs [12]. In addition, double-sided structures with striped electrodes patterned on both sides of LC were used for LC-based high resolution switchable gratings [13]. We use interdigitated electrodes of a width (w) of 2 μm and with in-plane gap (l) between the electrodes equal to four microns. They have been produced by a photolithography technique. The thickness of the electrodes is about 40 nm, while the thickness d of the LC layer is 3.2 μm. For special microscope studies, a particular LC cell is made with an increased thickness of LC layer (d 9 μm) and in-plane gap between the electrodes (l 40 μm, w 10 μm), which allows increasing a unit pixel area in between crossed pairs of electrodes (for example, bottom [A1, A2] and top [B1, B2]) to enable photo and video recording of detailed optical patterns formed in the pixel. Numerical calculations shown in this Letter are for the geometry l 4 μm, w 2 μm, and d 3.2 μm. The dual-field driving is achieved by using a homemade electronic circuit that provides fast TTL-driven switching of the preferable direction of the electric field in accordance with the states of electric potential waveforms φt shown in Table 1. At a high TTL pulse level
Fig. 1. Scheme of the experimental LC cell with two systems of interdigitated electrodes A1 ; A2 … and B1 ; B2 …. © 2015 Optical Society of America
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Table 1. Electric Potential Waveforms at Electrodes A1 ;A2 ;… and B1 ;B2 ;… at Two Levels of TTL Pulse and Input Voltage Waveform φt TTL Pulse Level (Type of Field) Low (normal) High (twistedplanar)
Waveform at Electrodes A1 ; A3 ; …A2k−1
Waveform at Electrodes A2 ; A4 ; …A2k
Waveform at Electrodes B1 ; B3 ; …B2k−1
Waveform at Electrodes B2 ; B4 ; …B2k
φt φt
φt −φt
−φt φt
−φt −φt
(2.5–5 V), the distribution of the electric potentials at the electrodes creates a so-called “twisted-planar” electric field. In this case, the electric field vector has planar orientation in the center of a pixel and is twisted across the LC layer thickness. At a low TTL pulse level (0–0.2 V), the normal (perpendicular to the LC layer) electric field is generated. In both cases, the electric field is inhomogeneous across the pixel volume and is referred to as “planar” and “normal” only for shortness. The strength of the electric field is defined by an amplitude U 0 of the waveform function φt that is in the simplest case has a sinusoidal shape (ϕt U 0 sin2πf t, where f ∼ 10 − 20 kHz). Numerical 3D simulations are based on the Ericksen– Leslie theory, and some details of the used approach can be found in [14]. The optical and lasing effect calculations are performed using an implementation described in [15], which is based on the well-known Berreman 4 × 4 matrix approach. The electrooptical response measured in the dualdriving mode at normal light incidence for the sample between the crossed polarizers is shown in Fig. 2 (curve 1). The achieved response time is indeed very small and belongs to the sub-millisecond range. The switching time to a black state driven by the normal field is about 0.2 ms at U 0 10 V, which is more than 10 times faster compared to free relaxation (Fig. 2, curve 2). The switching time to
Fig. 2. Electrooptical response in the visible spectral range (450–700 nm): (1) dual-field driving (U 0 10 V) when both twisted-planar (at 10 < t < 22 ms) and normal (t > 22 ms) electric field are applied; (2) only twisted-planar field is applied, and the normal field is zero at t > 22 ms; (3) standard twisted cell of the same LC material and thickness is subjected to ordinary driving (voltage pulse during time interval 10–22 ms is applied across the twisted LC layer between the two solid ITO electrodes). Inset: switching time versus voltage amplitude U 0 : (1) at normal field (t > 22 ms), (2) at twisted-planar field (10 ms < t < 22 ms).
each of the both states is defined by the voltage amplitude, as is shown in the inset of Fig. 2. However, we should emphasize that, even in the case of free relaxation (U 0 0) to the initial homeotropic state, the characteristic time is rather short (∼2.5 ms) compared to the time in a classical twist cell [16] (∼6 ms, Fig. 2, curve 3) driven in the standard way. Thus field-induced 2D-patterning resulting in localization of the deformation is also favorable for faster switching. In the case of LC materials, the PBG properties are characterized by the spatial distribution of the LC director field nx; y; z. The director n nx ; ny ; nz is a unit vector defining the orientation of the LC local optical axis. The local dielectric tensor components are defined by the director components as follows: εij ε⊥ δij ε∥ − ε⊥ ni nj ;
(1)
where i; j ∈ fx; y; zg, δij is Kronecker delta, ε∥;⊥ are the principal dielectric tensor components along and perpendicular to the director, respectively. Thus, if the LC director field is defined, then the spatial distribution of the dielectric tensor components is also known and can be used for further optical calculations. The numerical simulations show that, at the twistedplanar field, a specific twisted distribution of the LC director is induced, as shown in Fig. 3(a). Two separate regions of the twisted deformation appear in a single LC pixel located just in between two pairs of electrodes. These regions are distinguished by a sign of the director tilt angle and separated by domain walls with the homeotropic LC alignment (nz 1). The homeotropic walls are located along the diagonal planes between the topbottom electrode intersections, which is well visible on the maps of nx; y distribution shown in Fig. 3(a). These walls decrease the period of the director space modulation by a factor of two. They are also responsible for pronounced high-order harmonics in the spatial distribution of the dielectric tensor components, which is important for the lasing effect at higher orders of the Bragg reflections discussed below. Our theoretical calculations and, in particular, predictions of the domain walls are confirmed by experimental observations of optical patterns with polarized microscope. If the twisted-planar field is applied, then the diagonal domain walls are well visible in crossed polarizers as diagonal dark stripes between the electrodes intersections, Fig. 4(a). At the normal electric field, the LC director field still has deformation at the corners [see simulations in Fig. 3(b)] near the crossing of the top and bottom electrodes, which also can be seen in optics on the experimental photo in Fig. 4(b). However, in the last case, the modulation of the director field is weak (nz
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Fig. 3. Calculated LC director field distribution nx; y nx x; y; ny x; y; nz x; ynx; y nx x; y; ny x; y; nz x; y in the middle of the LC layer at the twisted planar field [top row, (a)] and normal field [bottom row, (b)].
component of the director is close to unit). This means that higher-order Fourier components in the dielectric tensor distribution are much less pronounced than in the case of the twisted-planar field driving. Pronounced 2D-modulation of the LC director field in the twisted-planar electric field results in deep modulation of refractive index n ≡ nrxy ; U 0 , where rxy is a unit in-plane radius vector pointing the chosen xy-direction of light propagation. If waveguide geometry is used so that light propagates in the plane of the LC layer, then such 2D modulation provides the photonic stop-bands at different orders m of the Bragg reflection. In our case, the free space wavelengths for the centers of the stop-bands can be expressed in a general form as λm rxy ; U 0 ; w l 2nrxy ; U 0 Prxy ; w l∕m ≡ 2nP∕m;
(2)
The estimated value of the lasing gain is low indeed. In practice the achieved [17] gain values for lasing dyes in nematic LCs are up to 0.05 μm−1 ; that is more than two orders higher than the estimated magnitude. However, as found from Eq. (3), the low value of the lasing gain is for the first-order Bragg reflection band, which in our case corresponds to the infrared spectral range. In practice, lasing dyes for doping LC materials are available only for the UV and visible range. Thus the question on the possibility of the lasing at higher orders of the Bragg reflections arises. In [9] it was experimentally demonstrated that the lasing effect can be observed, even for very high orders (m ∼ 75–79) of the in-plane Bragg reflection. The use of very high orders of Bragg reflections is impractical because the spectral interval between the two adjacent modes becomes too short, which is unfavorable for both single mode lasing and fine electric field tuning. In our
where P ≡ Prxy ; w l is a period of the spatial modulation depending on a period of the interdigitated grid (w l) of the electrodes and chosen propagation direction. According to Kogelnik and Shank [2], the lowest lasing threshold gain at a light wavelength λ1 corresponding to the edges of the first-order Bragg reflection stop-band is defined as α
λ21 : δ2 L 3
(3)
For a magnitude of the refractive index variation δ ≈ 0.01 (evaluated with Eq. (1) for the calculated director distribution) at xy-size L of the electrodes grating of 1 mm, one can estimate α 2.5 × 10−4 μm−1 .
Fig. 4. Experimental photos of a LC cell image between crossed polarizers (polarizers’ axes are along electrodes) at (a) the twisted-planar field and (b) the normal field. The experimental photos are done for the special LC sample with a gap between the electrodes l 40 μm and LC layer thickness d 9 μm.
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scope of this short article, shows that changing the strength of the twisted-planar electric field allows fine tuning of the lasing wavelength for the given diffraction order (the effect is caused by changing the effective value of the refractive index n). To conclude, we have demonstrated fast-switched field induced 2D photonic liquid crystals. The spatial distribution of the LC director field responsible for the dielectric tensor components distribution was studied. Based on the found spatial distribution of the dielectric tensor, the possibility of the low-threshold lasing effect controlled by fast-switched electric field is demonstrated as well. Fig. 5. Calculated transmittance and lasing spectra along x23line (see Fig. 3) at U 0 10 V: (a) transmittance in the twistedplanar field; (b) transmittance in the normal field; and (c) lasing in the twisted-planar field at a gain of 2 · 10−3 μm−1 . The length of the grating L 960 μm. The calculations are done for virtual LC with principal refractive indices n⊥ 1.52 and n∥ 1.77.
case, the spatial period of the interdigitated electrode system is 6 μm in x- and y-direction. However, because of the diagonal domain walls discussed above, the spatial period of the first Fourier harmonic of modulated refractive index is P 3 μm. Thus, according to Eq. (2), in the case of n 1.6 for wavelengths in a range of 500–600 nm, the Bragg reflection order m is defined by values of set of integers from 16 to 19, which are significantly lower than in [9], and the spectral interval between the neighboring modes is sufficiently wide (∼10 nm) to get both single mode lasing and electric field tuning. The numerical simulations for the director field distribution along the line marked as “x23” in Fig. 3(a) are shown in Fig. 5. By symmetry, for equivalent orthogonal directions, the results are the same. The transmission spectrum has quite a reach set of Bragg lines in a spectral range of 500–550 nm in the case of the twisted-planar field applied [Fig. 5(a)]. On the contrary, in the case of the normal field, few very weak Bragg lines are visible in the same spectral range [Fig. 5(b)]. As a result, the lasing effect is simulated at λ 531.6 nm and α 2 × 10−3 μm−1 only when the twisted-planar field is applied [Fig. 5(c)]. It is worth mentioning that a wavelength of 531 nm is quite convenient in the case of doping LC by Coumarin-6 lasing dye that has a maximum luminescence at this wavelength. At the normal field, the lasing is switched off. More detailed analysis, which is out of
The work has been supported by the Russian Scientific Foundation (project N 14-12-00553). References 1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). 2. H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972). 3. H. Takezoe, Liquid Crystals Beyond Displays: Chemistry, Physics, and Applications, Q. Li, ed. (Wiley, 2012), Chap. 1. 4. H. Coles and S. Morris, Nat. Photonics 4, 676 (2010). 5. V. A. Belyakov, Mol. Cryst. Liq. Cryst. 453, 43 (2006). 6. M. I. Barnik, L. M. Blinov, V. V. Lazarev, S. P. Palto, B. A. Umanskii, and N. M. Shtykov, J. Appl. Phys. 103, 123113 (2008). 7. T. Matsui, M. Ozaki, and K. Yoshino, J. Opt. Soc. Am. B 21, 1651 (2004). 8. M. Humar and I. Musevic, Opt. Express 18, 26995 (2010). 9. L. M. Blinov, G. Cipparrone, A. Mazzulla, P. Pagliusi, V. V. Lazarev, and S. P. Palto, Appl. Phys. Lett. 90, 131103 (2007). 10. L. M. Blinov and S. P. Palto, Liq. Cryst. 36, 1037 (2009). 11. D. H. Ko, S. M. Morris, A. Lorenz, F. Castles, H. Butt, D. J. Gardiner, M. M. Qasim, B. Wallikewitz, P. J. W. Hands, T. D. Wilkinson, G. Amaratunga, H. J. Coles, and R. H. Friend, Appl. Phys. Lett. 103, 051101 (2013). 12. A. Lorenz, D. J. Gardiner, S. M. Morris, F. Castles, M. M. Qasim, S. S. Choi, W.-S. Kim, H. J. Coles, and T. D. Wilkinson, Appl. Phys. Lett. 104, 071102 (2014). 13. L. Gu, “Micro- and nano-periodic-structure-based,” Ph.D. dissertation (University of Texas, 2007), paper 3278169. 14. S. P. Palto, N. J. Mottram, and M. A. Osipov, Phys. Rev. E 75, 061707 (2007). 15. S. P. Palto, Liquid Crystal Microlasers, L. M. Blinov and R. Bartolino, eds. (Transworld Research Network, 2010), pp. 141–165. 16. M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971). 17. N. M. Shtykov, M. I. Barnik, L. M. Blinov, B. A. Umanskii, and S. P. Palto, J. Exp. Theor. Phys. Lett. 85, 602 (2007).
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Fast electric field switched 2D-photonic liquid crystals S. P. Palto,* M. I. Barnik, A. R. Geivandov, I. V. Kasyanova, V. S. Palto, and N. M. Shtykov Institute of Crystallography of Russian Academy of Sciences, Leninsky pr. 59, 119333 Moscow, Russia *Corresponding author: [email protected] Received January 2, 2015; accepted February 10, 2015; posted February 17, 2015 (Doc. ID 231618); published March 19, 2015 We demonstrate field-induced 2D-photonic liquid crystals (LC). The 2D spatially periodic modulation of the LC director field is achieved using a geometry with two crossed interdigitated systems of electrodes located at opposite sides of the LC layer. With a special method of dual-field driving, a very fast switching between different spatially periodic LC director distributions is achieved. The director field distribution and potential use of these photonic crystals for fast switched multidirectional lasing is discussed. © 2015 Optical Society of America OCIS codes: (230.0250) Optoelectronics; (220.0220) Optical design and fabrication; (230.3720) Liquid-crystal devices; (230.5298) Photonic crystals; (250.6715) Switching. http://dx.doi.org/10.1364/OL.40.001254
Recently, photonic crystals [1] have been attracting high attention because of the perspectives of their use in many optical applications. Their periodic structure provides photonic band gaps (PBG) in a spectral range of Bragg reflection. In the region of PBG, because of strong interference effects, the propagation of light waves is forbidden, while at the edges of the band gaps the group velocity of light becomes very low, and high density of photonic states takes place. More than 40 years ago, it was shown by Kogelnik and Shank [2] that just at the edges of the band gaps the lowest threshold for the lasing effect can be achieved. PBG effects are also known in liquid crystals (LC) [3–9]. The most attractive property of PBG LC materials is that the band-gap effects can be controlled by an electric field. Perspective candidates for PBG microlasers are, for example, cholesteric LCs [3–6]. However, electric field lasing control was found to be rather difficult in these materials because of topological problems that prevent continuous unwinding of cholesteric helix [10]. Another approach is to utilize nematic LCs, either as intermediate layers [6] or as layers where the refractive index is modulated as a result of interference of pumping beams [7], by a spatially periodic electric field [9] or by covering a nano-patterned titanium dioxide photonic crystal within a well-oriented film of dye-doped LC [11]. This Letter is focused on fast field-induced LC PBG structures and consists of three closely associated parts. In the first part we describe a new and very fast method of electric field induction of PBG structures in nematic LCs. The method (referred to as “dual-field driving”) is based on fast switching of the direction of the electric field vector, which allows excluding the slow viscous-elastic relaxation characteristic of nematic LCs. Second, results of numerical simulations and experimental study of the LC director (local optical axis) distributions, which define field-induced refractive index patterns for the two switched electric field configurations, are discussed. Finally, using found field-induced director patterns, we demonstrate numerically the possibility of a fast switched low-threshold lasing effect in the visible spectral range, for which a huge assortment of lasing dyes for LC doping is available. The liquid crystal material used is a nematic LC (Merck ZLI-1957/5) with positive dielectric anisotropy Δε 4.5, 0146-9592/15/071254-04$15.00/0
optical anisotropy Δn 0.1213, and rotational viscosity γ ≅ 105 mPa · s (at a temperature of 20°C). The LC homeotropic alignment is provided by thin films of chromium distearyl chloride (chromolane) deposited onto glass substrate surfaces with electrodes. The electric field is created with the help of two crossed interdigitated systems of chromium electrodes located on the internal surfaces of the cell substrates, as shown in Fig. 1. It should be mentioned that interdigitated electrodes are widely used to electrically address LCs [12]. In addition, double-sided structures with striped electrodes patterned on both sides of LC were used for LC-based high resolution switchable gratings [13]. We use interdigitated electrodes of a width (w) of 2 μm and with in-plane gap (l) between the electrodes equal to four microns. They have been produced by a photolithography technique. The thickness of the electrodes is about 40 nm, while the thickness d of the LC layer is 3.2 μm. For special microscope studies, a particular LC cell is made with an increased thickness of LC layer (d 9 μm) and in-plane gap between the electrodes (l 40 μm, w 10 μm), which allows increasing a unit pixel area in between crossed pairs of electrodes (for example, bottom [A1, A2] and top [B1, B2]) to enable photo and video recording of detailed optical patterns formed in the pixel. Numerical calculations shown in this Letter are for the geometry l 4 μm, w 2 μm, and d 3.2 μm. The dual-field driving is achieved by using a homemade electronic circuit that provides fast TTL-driven switching of the preferable direction of the electric field in accordance with the states of electric potential waveforms φt shown in Table 1. At a high TTL pulse level
Fig. 1. Scheme of the experimental LC cell with two systems of interdigitated electrodes A1 ; A2 … and B1 ; B2 …. © 2015 Optical Society of America
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Table 1. Electric Potential Waveforms at Electrodes A1 ;A2 ;… and B1 ;B2 ;… at Two Levels of TTL Pulse and Input Voltage Waveform φt TTL Pulse Level (Type of Field) Low (normal) High (twistedplanar)
Waveform at Electrodes A1 ; A3 ; …A2k−1
Waveform at Electrodes A2 ; A4 ; …A2k
Waveform at Electrodes B1 ; B3 ; …B2k−1
Waveform at Electrodes B2 ; B4 ; …B2k
φt φt
φt −φt
−φt φt
−φt −φt
(2.5–5 V), the distribution of the electric potentials at the electrodes creates a so-called “twisted-planar” electric field. In this case, the electric field vector has planar orientation in the center of a pixel and is twisted across the LC layer thickness. At a low TTL pulse level (0–0.2 V), the normal (perpendicular to the LC layer) electric field is generated. In both cases, the electric field is inhomogeneous across the pixel volume and is referred to as “planar” and “normal” only for shortness. The strength of the electric field is defined by an amplitude U 0 of the waveform function φt that is in the simplest case has a sinusoidal shape (ϕt U 0 sin2πf t, where f ∼ 10 − 20 kHz). Numerical 3D simulations are based on the Ericksen– Leslie theory, and some details of the used approach can be found in [14]. The optical and lasing effect calculations are performed using an implementation described in [15], which is based on the well-known Berreman 4 × 4 matrix approach. The electrooptical response measured in the dualdriving mode at normal light incidence for the sample between the crossed polarizers is shown in Fig. 2 (curve 1). The achieved response time is indeed very small and belongs to the sub-millisecond range. The switching time to a black state driven by the normal field is about 0.2 ms at U 0 10 V, which is more than 10 times faster compared to free relaxation (Fig. 2, curve 2). The switching time to
Fig. 2. Electrooptical response in the visible spectral range (450–700 nm): (1) dual-field driving (U 0 10 V) when both twisted-planar (at 10 < t < 22 ms) and normal (t > 22 ms) electric field are applied; (2) only twisted-planar field is applied, and the normal field is zero at t > 22 ms; (3) standard twisted cell of the same LC material and thickness is subjected to ordinary driving (voltage pulse during time interval 10–22 ms is applied across the twisted LC layer between the two solid ITO electrodes). Inset: switching time versus voltage amplitude U 0 : (1) at normal field (t > 22 ms), (2) at twisted-planar field (10 ms < t < 22 ms).
each of the both states is defined by the voltage amplitude, as is shown in the inset of Fig. 2. However, we should emphasize that, even in the case of free relaxation (U 0 0) to the initial homeotropic state, the characteristic time is rather short (∼2.5 ms) compared to the time in a classical twist cell [16] (∼6 ms, Fig. 2, curve 3) driven in the standard way. Thus field-induced 2D-patterning resulting in localization of the deformation is also favorable for faster switching. In the case of LC materials, the PBG properties are characterized by the spatial distribution of the LC director field nx; y; z. The director n nx ; ny ; nz is a unit vector defining the orientation of the LC local optical axis. The local dielectric tensor components are defined by the director components as follows: εij ε⊥ δij ε∥ − ε⊥ ni nj ;
(1)
where i; j ∈ fx; y; zg, δij is Kronecker delta, ε∥;⊥ are the principal dielectric tensor components along and perpendicular to the director, respectively. Thus, if the LC director field is defined, then the spatial distribution of the dielectric tensor components is also known and can be used for further optical calculations. The numerical simulations show that, at the twistedplanar field, a specific twisted distribution of the LC director is induced, as shown in Fig. 3(a). Two separate regions of the twisted deformation appear in a single LC pixel located just in between two pairs of electrodes. These regions are distinguished by a sign of the director tilt angle and separated by domain walls with the homeotropic LC alignment (nz 1). The homeotropic walls are located along the diagonal planes between the topbottom electrode intersections, which is well visible on the maps of nx; y distribution shown in Fig. 3(a). These walls decrease the period of the director space modulation by a factor of two. They are also responsible for pronounced high-order harmonics in the spatial distribution of the dielectric tensor components, which is important for the lasing effect at higher orders of the Bragg reflections discussed below. Our theoretical calculations and, in particular, predictions of the domain walls are confirmed by experimental observations of optical patterns with polarized microscope. If the twisted-planar field is applied, then the diagonal domain walls are well visible in crossed polarizers as diagonal dark stripes between the electrodes intersections, Fig. 4(a). At the normal electric field, the LC director field still has deformation at the corners [see simulations in Fig. 3(b)] near the crossing of the top and bottom electrodes, which also can be seen in optics on the experimental photo in Fig. 4(b). However, in the last case, the modulation of the director field is weak (nz
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Fig. 3. Calculated LC director field distribution nx; y nx x; y; ny x; y; nz x; ynx; y nx x; y; ny x; y; nz x; y in the middle of the LC layer at the twisted planar field [top row, (a)] and normal field [bottom row, (b)].
component of the director is close to unit). This means that higher-order Fourier components in the dielectric tensor distribution are much less pronounced than in the case of the twisted-planar field driving. Pronounced 2D-modulation of the LC director field in the twisted-planar electric field results in deep modulation of refractive index n ≡ nrxy ; U 0 , where rxy is a unit in-plane radius vector pointing the chosen xy-direction of light propagation. If waveguide geometry is used so that light propagates in the plane of the LC layer, then such 2D modulation provides the photonic stop-bands at different orders m of the Bragg reflection. In our case, the free space wavelengths for the centers of the stop-bands can be expressed in a general form as λm rxy ; U 0 ; w l 2nrxy ; U 0 Prxy ; w l∕m ≡ 2nP∕m;
(2)
The estimated value of the lasing gain is low indeed. In practice the achieved [17] gain values for lasing dyes in nematic LCs are up to 0.05 μm−1 ; that is more than two orders higher than the estimated magnitude. However, as found from Eq. (3), the low value of the lasing gain is for the first-order Bragg reflection band, which in our case corresponds to the infrared spectral range. In practice, lasing dyes for doping LC materials are available only for the UV and visible range. Thus the question on the possibility of the lasing at higher orders of the Bragg reflections arises. In [9] it was experimentally demonstrated that the lasing effect can be observed, even for very high orders (m ∼ 75–79) of the in-plane Bragg reflection. The use of very high orders of Bragg reflections is impractical because the spectral interval between the two adjacent modes becomes too short, which is unfavorable for both single mode lasing and fine electric field tuning. In our
where P ≡ Prxy ; w l is a period of the spatial modulation depending on a period of the interdigitated grid (w l) of the electrodes and chosen propagation direction. According to Kogelnik and Shank [2], the lowest lasing threshold gain at a light wavelength λ1 corresponding to the edges of the first-order Bragg reflection stop-band is defined as α
λ21 : δ2 L 3
(3)
For a magnitude of the refractive index variation δ ≈ 0.01 (evaluated with Eq. (1) for the calculated director distribution) at xy-size L of the electrodes grating of 1 mm, one can estimate α 2.5 × 10−4 μm−1 .
Fig. 4. Experimental photos of a LC cell image between crossed polarizers (polarizers’ axes are along electrodes) at (a) the twisted-planar field and (b) the normal field. The experimental photos are done for the special LC sample with a gap between the electrodes l 40 μm and LC layer thickness d 9 μm.
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scope of this short article, shows that changing the strength of the twisted-planar electric field allows fine tuning of the lasing wavelength for the given diffraction order (the effect is caused by changing the effective value of the refractive index n). To conclude, we have demonstrated fast-switched field induced 2D photonic liquid crystals. The spatial distribution of the LC director field responsible for the dielectric tensor components distribution was studied. Based on the found spatial distribution of the dielectric tensor, the possibility of the low-threshold lasing effect controlled by fast-switched electric field is demonstrated as well. Fig. 5. Calculated transmittance and lasing spectra along x23line (see Fig. 3) at U 0 10 V: (a) transmittance in the twistedplanar field; (b) transmittance in the normal field; and (c) lasing in the twisted-planar field at a gain of 2 · 10−3 μm−1 . The length of the grating L 960 μm. The calculations are done for virtual LC with principal refractive indices n⊥ 1.52 and n∥ 1.77.
case, the spatial period of the interdigitated electrode system is 6 μm in x- and y-direction. However, because of the diagonal domain walls discussed above, the spatial period of the first Fourier harmonic of modulated refractive index is P 3 μm. Thus, according to Eq. (2), in the case of n 1.6 for wavelengths in a range of 500–600 nm, the Bragg reflection order m is defined by values of set of integers from 16 to 19, which are significantly lower than in [9], and the spectral interval between the neighboring modes is sufficiently wide (∼10 nm) to get both single mode lasing and electric field tuning. The numerical simulations for the director field distribution along the line marked as “x23” in Fig. 3(a) are shown in Fig. 5. By symmetry, for equivalent orthogonal directions, the results are the same. The transmission spectrum has quite a reach set of Bragg lines in a spectral range of 500–550 nm in the case of the twisted-planar field applied [Fig. 5(a)]. On the contrary, in the case of the normal field, few very weak Bragg lines are visible in the same spectral range [Fig. 5(b)]. As a result, the lasing effect is simulated at λ 531.6 nm and α 2 × 10−3 μm−1 only when the twisted-planar field is applied [Fig. 5(c)]. It is worth mentioning that a wavelength of 531 nm is quite convenient in the case of doping LC by Coumarin-6 lasing dye that has a maximum luminescence at this wavelength. At the normal field, the lasing is switched off. More detailed analysis, which is out of
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