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Dependence of second-harmonic generation in ferroelectric liquid crystals with a defect on the optical gap
Physics Letters A 372 (2008) 4754–4755
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Dependence of second-harmonic generation in ferroelectric liquid crystals with a defect on the optical gap Hajime Hoshi ∗ Department of Materials Science and Environmental Engineering, Tokyo University of Science, Yamaguchi, 1-1-1 Daigaku-dori, Sanyo-Onoda, Yamaguchi 756-0844, Japan
a r t i c l e
i n f o
Article history: Received 3 March 2008 Received in revised form 26 April 2008 Accepted 29 April 2008 Available online 15 May 2008 Communicated by R. Wu
a b s t r a c t Second-harmonic generation (SHG) in ferroelectric liquid crystal with a twist defect is studied by de Vries’ solution. It is shown that the width of the photonic band gap for the SH wave is an important factor to determine the saturation length. SHG is effectively enhanced by using narrow photonic band gaps. © 2008 Elsevier B.V. All rights reserved.
PACS: 42.70.Qs 42.65.Ky 42.70.Df Keywords: Photonic band gap Defect Ferroelectric liquid crystal Second-harmonic generation
The photonic effect in condensed media has attracted considerable attention from both scientific and application viewpoints. Chiral liquid crystals (LCs) form one-dimensional helical structures and photonic band gaps are produced in the visible-near-infrared region spontaneously. Interesting photonic effects are observed when light waves are influenced by the photonic band gap. One well-known phenomenon is a selective reflection due to the helical structures in cholesteric liquid crystals (CLCs) [1]. In nonlinear optics, phase matchings have been realized by the use of photonic effects in LCs. Shelton and Shen showed many phase matchings for the third-harmonic generation in CLC [2]. Belyakov and Shipov pointed out that the harmonic generation is enhanced when the harmonic wave is located near the photonic band edge [3,4]. Kajikawa et al. observed that the second-harmonic generation (SHG) in ferroelectric LCs (FLCs) is enhanced when the harmonic wave is near the photonic band edge [5]. Experimental and theoretical studies have been carried out to understand the mechanism of the enhanced SHG in FLCs [6–9]. Recently, photonic defect modes in helical LCs received much attention [10–17]. For example, an introduction of an isotropic layer or a twist defect in the middle of a CLC produced defect
*
Tel.: +81 836 88 3500; fax: +81 836 88 3400. E-mail address: [email protected].
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.04.062
modes in the photonic band gap [10,11]. Nonlinear optical properties of FLCs with a twist defect have been studied by numerical simulations [16,17]. In the first study, FLC cells were sliced into small pieces and the SH spectra in FLC with a twist defect were calculated under the assumption that the power depletion of the fundamental wave is neglected [16]. It was shown that the SHG is enhanced when the SH wave is located at the defect mode in the photonic band gap. In the preceding study, thickness dependence of SH intensity was studied based on de Vries’ solution [17]. The twist angle was assumed to be proportional to the thickness except for the location of the defect. The waves propagating parallel to the helical axis in FLC were described by de Vries’ solution. This approach enabled us to study the thickness dependent properties efficiently and showed that the SH intensity enhanced by the defect is restricted by a crossover thickness (L co ). L co is the cell thickness at which the linear spectrum for a circularly polarized incident wave shows 50% reflectance at the defect mode [11]. It was shown that the SH intensity increases with the thickness below L co and saturation of SH intensity occurs beyond L co . For applications, saturation of SH intensity is the problematic point. In the present Letter, dependence of the SHG on the optical parameters is studied based on de Vries’ solution. In particular, the effect of the dielectric constants at SH frequency (ε˜ 1 , ε˜ 2 , ε˜ 3 ) and the tilt angle θ is studied. The optical geometry and other optical
H. Hoshi / Physics Letters A 372 (2008) 4754–4755
Fig. 1. Thickness dependence of the transmission SH peak intensity. M = −0.03687 and L co = 200p. Three tilt angles were used.
Fig. 2. Thickness dependence of the transmission SH peak intensity. M = −0.13687 and L co = 40–42p. Three tilt angles were used.
parameters are identical to those in the previous studies [16,17]. Namely, in the middle of an FLC, the twist angle is jumped by 90◦ . The results are analyzed based on Ref. [9] that describes optical band gap properties in FLCs. In this study, L was fixed at L = 2.13687, where L = {˜ε1 + ε˜ 2 + (˜ε3 − ε˜ 1 )˜ε1 sin2 θ/(˜ε1 sin2 θ + ε˜ 3 cos2 θ)}/2 [9]. This value of L was also used in the previous studies [16,17]. By assuming ε˜ 1 = ε˜ 2 , L is the function of ε˜ 1 , ε˜ 3 , and θ . L determines the location of the gap for the SH wave [9]. 90◦ twist defect in the middle of a cell produces the defect mode in the middle of the gap. Moreover, SHG is enhanced when the SH wave matches with the defect mode. Thus, the enhancement of SHG is expected to occur at the same wavelength by fixing the value of L. This expectation was actually verified except for thin cells. On the other hand, the width of optical gap is determined by M, M = − L + ε˜ 2 [9]. It was found that the SHG is significantly affected by M, the width of the gap. Fig. 1 shows the SH peak intensity dependence on the thickness of the cell. The parameters used are M = −0.03687 and θ = 28◦ , 35◦ , and 40◦ , while M = −0.03687 and θ = 23◦ were used in the previous studies [16, 17]. L co was calculated based on the defect mode in the linear
4755
Fig. 3. Thickness dependence of the transmission SH peak intensity. M = −0.00687 and L co = 1410p. Two tilt angles were used.
selective reflection spectra [11] and L co = 200p was obtained for three θ values in Fig. 1, where p denotes the helical pitch. Thus, it is suggested that M is an important factor to determine L co and L co determines the saturation thickness for the enhanced SHG. Figs. 2 and 3 show the results for different M values. In Fig. 2, M = −0.13687 was used for three different θ values. The optical gap in Fig. 2 is wider than that in Fig. 1. The linear selective reflection spectra showed L co = 40–42p for the condition in Fig. 2. The value of L co for Fig. 2 is smaller than that for Fig. 1, which leads to the weaker saturation intensity. On the other hand, Fig. 3 shows the results for M = −0.00687, which is narrower band gap than that in Fig. 1. L co = 1410p was obtained based on the linear selective reflection. Saturation SH intensity in Fig. 3 is stronger than that in Fig. 1, which is enabled by large L co value due to the narrow optical gap. In summary, the numerical simulations based on de Vries’ solution were carried out to study the dependence of the SHG enhanced by the twist defect on the optical parameters. It was revealed that the optical band gap affects the crossover length and the saturated SH intensity significantly. It is possible to obtain strong SH intensity by using narrow optical band gaps. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
S. Chandrasekhar, Liquid Crystals, Cambridge Univ. Press, London, 1992. J.W. Shelton, Y.R. Shen, Phys. Rev. A 5 (1972) 1867. V.A. Belyakov, N.V. Shipov, Phys. Lett. A 86 (1981) 94. V.A. Belyakov, N.V. Shipov, Sov. Phys. JETP 55 (1982) 674. K. Kajikawa, T. Isozaki, H. Takezoe, A. Fukuda, Jpn. J. Appl. Phys. Part 2 31 (1992) 679. M. Copic, I. Drevensek-Olenik, Liq. Cryst. 21 (1996) 233. I. Drevensek-Olenik, M. Copic, Phys. Rev. E 56 (1997) 581. J.-G. Yoo, S.-W. Choi, H. Hoshi, K. Ishikawa, H. Takezoe, M. Schadt, Jpn. J. Appl. Phys. Part 2 36 (1997) 1168. H. Hoshi, D.-H. Chung, K. Ishikawa, H. Takezoe, Phys. Rev. E 63 (2001) 056610. Y.-C. Yang, C.-S. Kee, J.-E. Kim, H.-Y. Park, J.-C. Lee, Y.-J. Jeon, Phys. Rev. E 60 (1999) 6852. V.I. Kopp, A.Z. Genack, Phys. Rev. Lett. 89 (2002) 033901. V.I. Kopp, R. Bose, A.Z. Genack, Opt. Lett. 28 (2003) 349. J. Schmidtke, W. Stille, H. Finkelmann, Phys. Rev. Lett. 90 (2003) 083902. M.H. Song, N.Y. Ha, K. Amemiya, B. Park, Y. Takanishi, K. Ishikawa, J.W. Wu, S. Nishimura, T. Toyooka, H. Takezoe, Adv. Mater. 18 (2006) 193. R. Ozaki, T. Sanda, H. Yoshida, Y. Matsuhisa, M. Ozaki, K. Yoshino, Jpn. J. Appl. Phys. Part 1 45 (2006) 493. H. Hoshi, K. Ishikawa, H. Takezoe, Phys. Rev. E 68 (2003) 020701. H. Hoshi, H. Takezoe, Phys. Lett. A 358 (2006) 242.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Dependence of second-harmonic generation in ferroelectric liquid crystals with a defect on the optical gap Hajime Hoshi ∗ Department of Materials Science and Environmental Engineering, Tokyo University of Science, Yamaguchi, 1-1-1 Daigaku-dori, Sanyo-Onoda, Yamaguchi 756-0844, Japan
a r t i c l e
i n f o
Article history: Received 3 March 2008 Received in revised form 26 April 2008 Accepted 29 April 2008 Available online 15 May 2008 Communicated by R. Wu
a b s t r a c t Second-harmonic generation (SHG) in ferroelectric liquid crystal with a twist defect is studied by de Vries’ solution. It is shown that the width of the photonic band gap for the SH wave is an important factor to determine the saturation length. SHG is effectively enhanced by using narrow photonic band gaps. © 2008 Elsevier B.V. All rights reserved.
PACS: 42.70.Qs 42.65.Ky 42.70.Df Keywords: Photonic band gap Defect Ferroelectric liquid crystal Second-harmonic generation
The photonic effect in condensed media has attracted considerable attention from both scientific and application viewpoints. Chiral liquid crystals (LCs) form one-dimensional helical structures and photonic band gaps are produced in the visible-near-infrared region spontaneously. Interesting photonic effects are observed when light waves are influenced by the photonic band gap. One well-known phenomenon is a selective reflection due to the helical structures in cholesteric liquid crystals (CLCs) [1]. In nonlinear optics, phase matchings have been realized by the use of photonic effects in LCs. Shelton and Shen showed many phase matchings for the third-harmonic generation in CLC [2]. Belyakov and Shipov pointed out that the harmonic generation is enhanced when the harmonic wave is located near the photonic band edge [3,4]. Kajikawa et al. observed that the second-harmonic generation (SHG) in ferroelectric LCs (FLCs) is enhanced when the harmonic wave is near the photonic band edge [5]. Experimental and theoretical studies have been carried out to understand the mechanism of the enhanced SHG in FLCs [6–9]. Recently, photonic defect modes in helical LCs received much attention [10–17]. For example, an introduction of an isotropic layer or a twist defect in the middle of a CLC produced defect
*
Tel.: +81 836 88 3500; fax: +81 836 88 3400. E-mail address: [email protected].
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.04.062
modes in the photonic band gap [10,11]. Nonlinear optical properties of FLCs with a twist defect have been studied by numerical simulations [16,17]. In the first study, FLC cells were sliced into small pieces and the SH spectra in FLC with a twist defect were calculated under the assumption that the power depletion of the fundamental wave is neglected [16]. It was shown that the SHG is enhanced when the SH wave is located at the defect mode in the photonic band gap. In the preceding study, thickness dependence of SH intensity was studied based on de Vries’ solution [17]. The twist angle was assumed to be proportional to the thickness except for the location of the defect. The waves propagating parallel to the helical axis in FLC were described by de Vries’ solution. This approach enabled us to study the thickness dependent properties efficiently and showed that the SH intensity enhanced by the defect is restricted by a crossover thickness (L co ). L co is the cell thickness at which the linear spectrum for a circularly polarized incident wave shows 50% reflectance at the defect mode [11]. It was shown that the SH intensity increases with the thickness below L co and saturation of SH intensity occurs beyond L co . For applications, saturation of SH intensity is the problematic point. In the present Letter, dependence of the SHG on the optical parameters is studied based on de Vries’ solution. In particular, the effect of the dielectric constants at SH frequency (ε˜ 1 , ε˜ 2 , ε˜ 3 ) and the tilt angle θ is studied. The optical geometry and other optical
H. Hoshi / Physics Letters A 372 (2008) 4754–4755
Fig. 1. Thickness dependence of the transmission SH peak intensity. M = −0.03687 and L co = 200p. Three tilt angles were used.
Fig. 2. Thickness dependence of the transmission SH peak intensity. M = −0.13687 and L co = 40–42p. Three tilt angles were used.
parameters are identical to those in the previous studies [16,17]. Namely, in the middle of an FLC, the twist angle is jumped by 90◦ . The results are analyzed based on Ref. [9] that describes optical band gap properties in FLCs. In this study, L was fixed at L = 2.13687, where L = {˜ε1 + ε˜ 2 + (˜ε3 − ε˜ 1 )˜ε1 sin2 θ/(˜ε1 sin2 θ + ε˜ 3 cos2 θ)}/2 [9]. This value of L was also used in the previous studies [16,17]. By assuming ε˜ 1 = ε˜ 2 , L is the function of ε˜ 1 , ε˜ 3 , and θ . L determines the location of the gap for the SH wave [9]. 90◦ twist defect in the middle of a cell produces the defect mode in the middle of the gap. Moreover, SHG is enhanced when the SH wave matches with the defect mode. Thus, the enhancement of SHG is expected to occur at the same wavelength by fixing the value of L. This expectation was actually verified except for thin cells. On the other hand, the width of optical gap is determined by M, M = − L + ε˜ 2 [9]. It was found that the SHG is significantly affected by M, the width of the gap. Fig. 1 shows the SH peak intensity dependence on the thickness of the cell. The parameters used are M = −0.03687 and θ = 28◦ , 35◦ , and 40◦ , while M = −0.03687 and θ = 23◦ were used in the previous studies [16, 17]. L co was calculated based on the defect mode in the linear
4755
Fig. 3. Thickness dependence of the transmission SH peak intensity. M = −0.00687 and L co = 1410p. Two tilt angles were used.
selective reflection spectra [11] and L co = 200p was obtained for three θ values in Fig. 1, where p denotes the helical pitch. Thus, it is suggested that M is an important factor to determine L co and L co determines the saturation thickness for the enhanced SHG. Figs. 2 and 3 show the results for different M values. In Fig. 2, M = −0.13687 was used for three different θ values. The optical gap in Fig. 2 is wider than that in Fig. 1. The linear selective reflection spectra showed L co = 40–42p for the condition in Fig. 2. The value of L co for Fig. 2 is smaller than that for Fig. 1, which leads to the weaker saturation intensity. On the other hand, Fig. 3 shows the results for M = −0.00687, which is narrower band gap than that in Fig. 1. L co = 1410p was obtained based on the linear selective reflection. Saturation SH intensity in Fig. 3 is stronger than that in Fig. 1, which is enabled by large L co value due to the narrow optical gap. In summary, the numerical simulations based on de Vries’ solution were carried out to study the dependence of the SHG enhanced by the twist defect on the optical parameters. It was revealed that the optical band gap affects the crossover length and the saturated SH intensity significantly. It is possible to obtain strong SH intensity by using narrow optical band gaps. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
S. Chandrasekhar, Liquid Crystals, Cambridge Univ. Press, London, 1992. J.W. Shelton, Y.R. Shen, Phys. Rev. A 5 (1972) 1867. V.A. Belyakov, N.V. Shipov, Phys. Lett. A 86 (1981) 94. V.A. Belyakov, N.V. Shipov, Sov. Phys. JETP 55 (1982) 674. K. Kajikawa, T. Isozaki, H. Takezoe, A. Fukuda, Jpn. J. Appl. Phys. Part 2 31 (1992) 679. M. Copic, I. Drevensek-Olenik, Liq. Cryst. 21 (1996) 233. I. Drevensek-Olenik, M. Copic, Phys. Rev. E 56 (1997) 581. J.-G. Yoo, S.-W. Choi, H. Hoshi, K. Ishikawa, H. Takezoe, M. Schadt, Jpn. J. Appl. Phys. Part 2 36 (1997) 1168. H. Hoshi, D.-H. Chung, K. Ishikawa, H. Takezoe, Phys. Rev. E 63 (2001) 056610. Y.-C. Yang, C.-S. Kee, J.-E. Kim, H.-Y. Park, J.-C. Lee, Y.-J. Jeon, Phys. Rev. E 60 (1999) 6852. V.I. Kopp, A.Z. Genack, Phys. Rev. Lett. 89 (2002) 033901. V.I. Kopp, R. Bose, A.Z. Genack, Opt. Lett. 28 (2003) 349. J. Schmidtke, W. Stille, H. Finkelmann, Phys. Rev. Lett. 90 (2003) 083902. M.H. Song, N.Y. Ha, K. Amemiya, B. Park, Y. Takanishi, K. Ishikawa, J.W. Wu, S. Nishimura, T. Toyooka, H. Takezoe, Adv. Mater. 18 (2006) 193. R. Ozaki, T. Sanda, H. Yoshida, Y. Matsuhisa, M. Ozaki, K. Yoshino, Jpn. J. Appl. Phys. Part 1 45 (2006) 493. H. Hoshi, K. Ishikawa, H. Takezoe, Phys. Rev. E 68 (2003) 020701. H. Hoshi, H. Takezoe, Phys. Lett. A 358 (2006) 242.