Abstract

We demonstrate a hysteresis-free polymer-stabilized blue phase liquid crystals (PSBP-LCs) using thermal recycles by means of enlarging the domain size of PSBP-LC and improving the uniformity of the crystal orientations of PSBP-LC. The crystal growth of PSBP-LCs bases on a competition between the heterogeneous nucleation and the homogeneous nucleation is also demonstrated and discussed. This study opens a new window for understanding the hysteresis of the PSBP-LC and assisting in designing many PSBP-LC-based photonic devices, such as displays, electro-optical switches, and electrically tunable focusing LC lenses.

© 2012 OSA

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  1. H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
    [CrossRef] [PubMed]
  2. Y. H. Lin, H. S. Chen, T. H. Chiang, C. H. Wu, and H. K. Hsu, “A reflective polarizer-free electro-optical switch using dye-doped polymer-stabilized blue phase liquid crystals,” Opt. Express19(3), 2556–2561 (2011).
    [CrossRef] [PubMed]
  3. Y. H. Lin, H. S. Chen, and T. H. Chiang, “A reflective polarizer-free display using dye-doped polymer-stabilized blue-phase liquid crystals,” J. Soc. Inf. Disp.20(6), 333–336 (2012).
    [CrossRef]
  4. Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
    [CrossRef]
  5. K. M. Chen, S. Gauza, H. Xianyu, and S. T. Wu, “Hysteresis Effects in Blue-Phase Liquid Crystals,” J. Disp. Technol.6(8), 318–322 (2010).
    [CrossRef]
  6. L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
    [CrossRef]
  7. C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
    [CrossRef]
  8. W. Y. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002).
    [CrossRef] [PubMed]
  9. Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
    [CrossRef]
  10. P. Pieranski, R. Barbet-Massin, and P. E. Cladis, “Steps on surfaces of liquid-crystal blue phase I,” Phys. Rev. A31(6), 3912–3923 (1985).
    [CrossRef] [PubMed]
  11. H. J. Coles and M. N. Pivnenko, “Liquid crystal ‘blue phases’ with a wide temperature range,” Nature436(7053), 997–1000 (2005).
    [CrossRef] [PubMed]
  12. A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
    [CrossRef]
  13. H. S. Kitzerow, “Blue phases come of age: a review,” Proc. SPIE7232, 723205 (2009).
    [CrossRef]
  14. S. K. Hong, G. H. Lim, and H. Kikuchi, “Thickness dependence of blue phase transition behavior of chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst.511(1), 248/[1718]–254/[1724] (2009).
    [CrossRef]
  15. M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
    [CrossRef]
  16. Y. H. Lin, H. S. Chen, C. H. Wu, and H. K. Hsu, “Measuring electric-field-induced birefringence in polymer stabilized blue-phase liquid crystals based on phase shift measurements,” J. Appl. Phys.109(10), 104503 (2011).
    [CrossRef]
  17. P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
    [CrossRef]
  18. P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
    [CrossRef]
  19. A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
    [CrossRef]

2012 (4)

Y. H. Lin, H. S. Chen, and T. H. Chiang, “A reflective polarizer-free display using dye-doped polymer-stabilized blue-phase liquid crystals,” J. Soc. Inf. Disp.20(6), 333–336 (2012).
[CrossRef]

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

2011 (4)

Y. H. Lin, H. S. Chen, C. H. Wu, and H. K. Hsu, “Measuring electric-field-induced birefringence in polymer stabilized blue-phase liquid crystals based on phase shift measurements,” J. Appl. Phys.109(10), 104503 (2011).
[CrossRef]

Y. H. Lin, H. S. Chen, T. H. Chiang, C. H. Wu, and H. K. Hsu, “A reflective polarizer-free electro-optical switch using dye-doped polymer-stabilized blue phase liquid crystals,” Opt. Express19(3), 2556–2561 (2011).
[CrossRef] [PubMed]

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

2010 (2)

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

K. M. Chen, S. Gauza, H. Xianyu, and S. T. Wu, “Hysteresis Effects in Blue-Phase Liquid Crystals,” J. Disp. Technol.6(8), 318–322 (2010).
[CrossRef]

2009 (4)

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
[CrossRef]

H. S. Kitzerow, “Blue phases come of age: a review,” Proc. SPIE7232, 723205 (2009).
[CrossRef]

S. K. Hong, G. H. Lim, and H. Kikuchi, “Thickness dependence of blue phase transition behavior of chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst.511(1), 248/[1718]–254/[1724] (2009).
[CrossRef]

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

2005 (1)

H. J. Coles and M. N. Pivnenko, “Liquid crystal ‘blue phases’ with a wide temperature range,” Nature436(7053), 997–1000 (2005).
[CrossRef] [PubMed]

2002 (2)

W. Y. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002).
[CrossRef] [PubMed]

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
[CrossRef] [PubMed]

1997 (1)

A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
[CrossRef]

1985 (1)

P. Pieranski, R. Barbet-Massin, and P. E. Cladis, “Steps on surfaces of liquid-crystal blue phase I,” Phys. Rev. A31(6), 3912–3923 (1985).
[CrossRef] [PubMed]

Asagi, H.

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

Barbet-Massin, R.

P. Pieranski, R. Barbet-Massin, and P. E. Cladis, “Steps on surfaces of liquid-crystal blue phase I,” Phys. Rev. A31(6), 3912–3923 (1985).
[CrossRef] [PubMed]

Cao, W. Y.

W. Y. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002).
[CrossRef] [PubMed]

Chen, H. S.

Y. H. Lin, H. S. Chen, and T. H. Chiang, “A reflective polarizer-free display using dye-doped polymer-stabilized blue-phase liquid crystals,” J. Soc. Inf. Disp.20(6), 333–336 (2012).
[CrossRef]

Y. H. Lin, H. S. Chen, T. H. Chiang, C. H. Wu, and H. K. Hsu, “A reflective polarizer-free electro-optical switch using dye-doped polymer-stabilized blue phase liquid crystals,” Opt. Express19(3), 2556–2561 (2011).
[CrossRef] [PubMed]

Y. H. Lin, H. S. Chen, C. H. Wu, and H. K. Hsu, “Measuring electric-field-induced birefringence in polymer stabilized blue-phase liquid crystals based on phase shift measurements,” J. Appl. Phys.109(10), 104503 (2011).
[CrossRef]

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

Chen, H. Y.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Chen, K. M.

K. M. Chen, S. Gauza, H. Xianyu, and S. T. Wu, “Hysteresis Effects in Blue-Phase Liquid Crystals,” J. Disp. Technol.6(8), 318–322 (2010).
[CrossRef]

Cheng, K. L.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Chiang, T. H.

Y. H. Lin, H. S. Chen, and T. H. Chiang, “A reflective polarizer-free display using dye-doped polymer-stabilized blue-phase liquid crystals,” J. Soc. Inf. Disp.20(6), 333–336 (2012).
[CrossRef]

Y. H. Lin, H. S. Chen, T. H. Chiang, C. H. Wu, and H. K. Hsu, “A reflective polarizer-free electro-optical switch using dye-doped polymer-stabilized blue phase liquid crystals,” Opt. Express19(3), 2556–2561 (2011).
[CrossRef] [PubMed]

Chiu, Y. H.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Cladis, P. E.

P. Pieranski, R. Barbet-Massin, and P. E. Cladis, “Steps on surfaces of liquid-crystal blue phase I,” Phys. Rev. A31(6), 3912–3923 (1985).
[CrossRef] [PubMed]

Coles, H. J.

H. J. Coles and M. N. Pivnenko, “Liquid crystal ‘blue phases’ with a wide temperature range,” Nature436(7053), 997–1000 (2005).
[CrossRef] [PubMed]

Fan, C.-Y.

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

Fujii, A.

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

Gauza, S.

K. M. Chen, S. Gauza, H. Xianyu, and S. T. Wu, “Hysteresis Effects in Blue-Phase Liquid Crystals,” J. Disp. Technol.6(8), 318–322 (2010).
[CrossRef]

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
[CrossRef]

Ge, Z.

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
[CrossRef]

Hauser, A.

A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
[CrossRef]

Heppke, G.

A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
[CrossRef]

Hisakado, Y.

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
[CrossRef] [PubMed]

Hong, S. H.

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Hong, S. K.

S. K. Hong, G. H. Lim, and H. Kikuchi, “Thickness dependence of blue phase transition behavior of chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst.511(1), 248/[1718]–254/[1724] (2009).
[CrossRef]

Hsieh, P. J.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Hsu, H. K.

Y. H. Lin, H. S. Chen, T. H. Chiang, C. H. Wu, and H. K. Hsu, “A reflective polarizer-free electro-optical switch using dye-doped polymer-stabilized blue phase liquid crystals,” Opt. Express19(3), 2556–2561 (2011).
[CrossRef] [PubMed]

Y. H. Lin, H. S. Chen, C. H. Wu, and H. K. Hsu, “Measuring electric-field-induced birefringence in polymer stabilized blue-phase liquid crystals based on phase shift measurements,” J. Appl. Phys.109(10), 104503 (2011).
[CrossRef]

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

Huang, T.-H.

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

Jeong, H.

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Jiao, M.

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
[CrossRef]

Kajiyama, T.

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
[CrossRef] [PubMed]

Kang, S.-W.

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

Kikuchi, H.

S. K. Hong, G. H. Lim, and H. Kikuchi, “Thickness dependence of blue phase transition behavior of chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst.511(1), 248/[1718]–254/[1724] (2009).
[CrossRef]

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
[CrossRef] [PubMed]

Kim, H. S.

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

Kitzerow, H. S.

H. S. Kitzerow, “Blue phases come of age: a review,” Proc. SPIE7232, 723205 (2009).
[CrossRef]

Kruerke, D.

A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
[CrossRef]

Lai, Y. C.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Lee, G.-D.

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

Lee, H. J.

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Lee, S. H.

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

Li, W. Y.

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

Liang, C. C.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Lim, G. H.

S. K. Hong, G. H. Lim, and H. Kikuchi, “Thickness dependence of blue phase transition behavior of chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst.511(1), 248/[1718]–254/[1724] (2009).
[CrossRef]

Lin, H. C.

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

Lin, T.-H.

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

Lin, Y. H.

Y. H. Lin, H. S. Chen, and T. H. Chiang, “A reflective polarizer-free display using dye-doped polymer-stabilized blue-phase liquid crystals,” J. Soc. Inf. Disp.20(6), 333–336 (2012).
[CrossRef]

Y. H. Lin, H. S. Chen, T. H. Chiang, C. H. Wu, and H. K. Hsu, “A reflective polarizer-free electro-optical switch using dye-doped polymer-stabilized blue phase liquid crystals,” Opt. Express19(3), 2556–2561 (2011).
[CrossRef] [PubMed]

Y. H. Lin, H. S. Chen, C. H. Wu, and H. K. Hsu, “Measuring electric-field-induced birefringence in polymer stabilized blue-phase liquid crystals based on phase shift measurements,” J. Appl. Phys.109(10), 104503 (2011).
[CrossRef]

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

Liu, C.-Y.

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

Liu, S. H.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Mukherjee, A.

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Muñoz, A.

W. Y. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002).
[CrossRef] [PubMed]

Nayek, P.

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Noma, T.

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

Ojima, M.

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

Ozaki, M.

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

Palffy-Muhoray, P.

W. Y. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002).
[CrossRef] [PubMed]

Park, H. R.

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

Park, H. S.

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

Park, H.-S.

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

Pieranski, P.

P. Pieranski, R. Barbet-Massin, and P. E. Cladis, “Steps on surfaces of liquid-crystal blue phase I,” Phys. Rev. A31(6), 3912–3923 (1985).
[CrossRef] [PubMed]

Pivnenko, M. N.

H. J. Coles and M. N. Pivnenko, “Liquid crystal ‘blue phases’ with a wide temperature range,” Nature436(7053), 997–1000 (2005).
[CrossRef] [PubMed]

Rao, L.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Saupe, A.

A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
[CrossRef]

Shin, S.-T.

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Sugiura, N.

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

Taheri, B.

W. Y. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002).
[CrossRef] [PubMed]

Thieme, M.

A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
[CrossRef]

Tsou, Y. S.

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

Wang, C.-T.

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

Wu, C. H.

Y. H. Lin, H. S. Chen, T. H. Chiang, C. H. Wu, and H. K. Hsu, “A reflective polarizer-free electro-optical switch using dye-doped polymer-stabilized blue phase liquid crystals,” Opt. Express19(3), 2556–2561 (2011).
[CrossRef] [PubMed]

Y. H. Lin, H. S. Chen, C. H. Wu, and H. K. Hsu, “Measuring electric-field-induced birefringence in polymer stabilized blue-phase liquid crystals based on phase shift measurements,” J. Appl. Phys.109(10), 104503 (2011).
[CrossRef]

Wu, C. M.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Wu, S. T.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

K. M. Chen, S. Gauza, H. Xianyu, and S. T. Wu, “Hysteresis Effects in Blue-Phase Liquid Crystals,” J. Disp. Technol.6(8), 318–322 (2010).
[CrossRef]

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
[CrossRef]

Xianyu, H.

K. M. Chen, S. Gauza, H. Xianyu, and S. T. Wu, “Hysteresis Effects in Blue-Phase Liquid Crystals,” J. Disp. Technol.6(8), 318–322 (2010).
[CrossRef]

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
[CrossRef]

Yan, J.

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

Yang, G. H.

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Yang, H.

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
[CrossRef] [PubMed]

Yokota, M.

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
[CrossRef] [PubMed]

Yu, F.-C.

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

Appl. Phys. Express (1)

P. Nayek, H. Jeong, H. R. Park, S.-W. Kang, S. H. Lee, H. S. Park, H. J. Lee, and H. S. Kim, “Tailoring Monodomain in Blue Phase Liquid Crystal by Surface Pinning Effect,” Appl. Phys. Express5(5), 051701 (2012).
[CrossRef]

Appl. Phys. Lett. (2)

Y. H. Lin, H. S. Chen, H. C. Lin, Y. S. Tsou, H. K. Hsu, and W. Y. Li, “Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystals,” Appl. Phys. Lett.96(11), 113505 (2010).
[CrossRef]

Z. Ge, S. Gauza, M. Jiao, H. Xianyu, and S. T. Wu, “Electro-optics of polymer-stabilized blue phase liquid crystal displays,” Appl. Phys. Lett.94(10), 101104 (2009).
[CrossRef]

J. Appl. Phys. (1)

Y. H. Lin, H. S. Chen, C. H. Wu, and H. K. Hsu, “Measuring electric-field-induced birefringence in polymer stabilized blue-phase liquid crystals based on phase shift measurements,” J. Appl. Phys.109(10), 104503 (2011).
[CrossRef]

J. Disp. Technol. (2)

K. M. Chen, S. Gauza, H. Xianyu, and S. T. Wu, “Hysteresis Effects in Blue-Phase Liquid Crystals,” J. Disp. Technol.6(8), 318–322 (2010).
[CrossRef]

L. Rao, J. Yan, S. T. Wu, Y. C. Lai, Y. H. Chiu, H. Y. Chen, C. C. Liang, C. M. Wu, P. J. Hsieh, S. H. Liu, and K. L. Cheng, “Critical Field for a Hysteresis-Free BPLC Device,” J. Disp. Technol.7(12), 627–629 (2011).
[CrossRef]

J. Mater. Chem. (1)

A. Hauser, M. Thieme, A. Saupe, G. Heppke, and D. Kruerke, “Surface-imaging of frozen blue phases in a discotic liquid crystal with atomic force microscopy,” J. Mater. Chem.7(11), 2223–2229 (1997).
[CrossRef]

J. Soc. Inf. Disp. (2)

Y. H. Lin, H. S. Chen, and T. H. Chiang, “A reflective polarizer-free display using dye-doped polymer-stabilized blue-phase liquid crystals,” J. Soc. Inf. Disp.20(6), 333–336 (2012).
[CrossRef]

P. Nayek, H. Jeong, S.-W. Kang, S. H. Lee, H.-S. Park, H. J. Lee, H. S. Kim, and G.-D. Lee, “Effect of the grain size on hysteresis of liquid-crystalline Blue Phase I,” J. Soc. Inf. Disp.20(6), 318–325 (2012).
[CrossRef]

Liq. Cryst. (1)

A. Mukherjee, G. H. Yang, H. Jeong, P. Nayek, S.-W. Kang, S. H. Lee, S. H. Hong, H. J. Lee, and S.-T. Shin, “Emergence of a novel optically isotropic transient state with low frequency in a blue phase liquid crystal mixture,” Liq. Cryst.39(2), 231–237 (2012).
[CrossRef]

Mol. Cryst. Liq. Cryst. (2)

S. K. Hong, G. H. Lim, and H. Kikuchi, “Thickness dependence of blue phase transition behavior of chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst.511(1), 248/[1718]–254/[1724] (2009).
[CrossRef]

M. Ojima, T. Noma, H. Asagi, A. Fujii, M. Ozaki, and H. Kikuchi, “Effect of mixed cellulose ester membrane structure on appearance of cholesteric blue phases,” Mol. Cryst. Liq. Cryst.512(1), 136/[1982]–[142/1988] (2009).
[CrossRef]

Nat. Mater. (2)

H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nat. Mater.1(1), 64–68 (2002).
[CrossRef] [PubMed]

W. Y. Cao, A. Muñoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nat. Mater.1(2), 111–113 (2002).
[CrossRef] [PubMed]

Nature (1)

H. J. Coles and M. N. Pivnenko, “Liquid crystal ‘blue phases’ with a wide temperature range,” Nature436(7053), 997–1000 (2005).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. A (1)

P. Pieranski, R. Barbet-Massin, and P. E. Cladis, “Steps on surfaces of liquid-crystal blue phase I,” Phys. Rev. A31(6), 3912–3923 (1985).
[CrossRef] [PubMed]

Proc. SPIE (1)

H. S. Kitzerow, “Blue phases come of age: a review,” Proc. SPIE7232, 723205 (2009).
[CrossRef]

SID Int. Symp. Digest Tech. Papers (1)

C.-Y. Fan, C.-T. Wang, T.-H. Lin, F.-C. Yu, T.-H. Huang, C.-Y. Liu, and N. Sugiura, “Hysteresis and Residual Birefringence Free Polymer-stabilized Blue Phase Liquid Crystal,” SID Int. Symp. Digest Tech. Papers42(1), 213–215 (2011).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

The observed morphologies of PSBP-LC samples at 28 °C when the samples were photo-polymerized at the (a) 0th, (b) 5th and (c) 10th thermal recycle. The morphologies of the PSBP-LC samples (d) right before and (e) right after the 5th thermal recycling processes, respectively. (f) The morphologies of the PSBP-LC sample when the sample was photo-polymerized after the crystal re-growth was completed. The thermal rates were 0.5 °C min−1.

Fig. 2
Fig. 2

, Kossel diagram of the blue area with monochromatic light λ = (a) 427 nm (b) 488 nm, and λ = (c) 510 nm. T = 28°C.

Fig. 3
Fig. 3

The area ratio of the PSBP-LCs as a function of the number of thermal recycles. Blue dots line represents the crystal plan of (2, 2, 0) and black squares represent the other crystal planes. The thermal rate was 0.5 °C min−1.

Fig. 4
Fig. 4

The domain size as a function of the number of thermal recycles at thermal rate of 0.5 °C min−1.

Fig. 5
Fig. 5

The illustrated mechanism of PSBP-LC under the thermal recycles. (a) The unit cell of PSBP-LC. (b) The PSBP-LC cell has blue phase (the crystal planes of (2, 2, 0), (2, 1, 1) and the others) and isotropic phase (gray region). (c) When the temperature decreases, the number of all color cubes grows. (d) When the temperature increases, the number of all color cubes decreases. (e) When the temperature decreases again, the number of all color cubes re-grow again. (f) After several thermal recycles, the area of blue cubes is enlarged and the area of the other color cubes is reduced.

Fig. 6
Fig. 6

The voltage-dependent transmittance of PSBP-LCs with an increasing voltage (red and gray lines) and a decreasing voltage (pink and black line). “With TR” stands for “with the thermal recycles” and “W/O TR” stands for “without the thermal recycles”.

Equations (1)

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λ= 2nacosθ h 2 + k 2 + l 2 ,

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