Abstract

The color optical switching device by polymer network liquid crystal (PNLC) with color filter on a specular reflector shows excellent performance; white reflectance of 22%, color gamut of 32%, and contrast ratio up to 50:1 in reflective mode measurement. The view-angle dependence of the reflectance can be adjusted by changing the PNLC thickness. The color chromaticity shown by the device is close to the limit value of color filters, and its value nearly remains with respect to the operating voltage. These optical properties of the device can be explained from the prediction based on multiple interactions between the light and the droplets of liquid crystal. The high reflectance, vivid color image, and moderate responds time allow the PNLC device to drive good color moving image. It can widely extend the applications of the reflective device.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2011

2010

2009

A. Y.-G. Fuh, C.-C. Chen, C.-K. Liu, and K.-T. Cheng, “Polarizer-free, electrically switchable and optically rewritable displays based on dye-doped polymer-dispersed liquid crystals,” Opt. Express 17(9), 7088–7094 (2009).
[CrossRef] [PubMed]

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

2008

2006

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

2004

Y.-H. Fan, Y.-H. Lin, H. Ren, S. Gauza, and S.-T. Wu, “Fast-response and scattering-free polymer network liquid crystals for infrared light modulators,” Appl. Phys. Lett. 84(8), 1233 (2004).
[CrossRef]

2003

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425(6956), 383–385 (2003).
[CrossRef] [PubMed]

2002

J. M. Crowley, N. K. Sheridon, and L. Romano, “Dipole moments of gyricon balls,” J. Electrost. 55, 247 (2002).
[CrossRef]

1999

M. J. Escuti, C. C. Bowley, G. P. Crawford, and S. Zumer, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75(21), 3264 (1999).
[CrossRef]

1998

B. Comiskey, J. D. Albert, H. Yoshizawa, and J. Jacobson, “An electrophoretic ink for all-printed reflective electronic displays,” Nature 394(6690), 253–255 (1998).
[CrossRef]

1997

S. A. Carter, J. D. LeGrange, W. White, J. Boo, and P. Wiltzius, “Dependence of the morphology of polymer dispersed liquid crystals on the UV polymerization process,” J. Appl. Phys. 81(9), 5992 (1997).
[CrossRef]

1996

1993

J. R. Kelly and W. Wu, “Multiple scattering effects in polymer dispersed liquid crystals,” Liq. Cryst. 14(6), 1683–1694 (1993).
[CrossRef]

1988

S. Zǔmer, “Light scattering from nematic droplets: Anomalous-diffraction approach,” Phys. Rev. A 37(10), 4006–4015 (1988).
[CrossRef] [PubMed]

1987

N. A. Vaz, G. W. Smith, and G. P. Montgomery., “A light control film composed of liquid crystal droplets dispersed in a uv-curable polymer,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 146(1), 1–15 (1987).
[CrossRef]

Albert, J. D.

B. Comiskey, J. D. Albert, H. Yoshizawa, and J. Jacobson, “An electrophoretic ink for all-printed reflective electronic displays,” Nature 394(6690), 253–255 (1998).
[CrossRef]

Batey, J.

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

Bloisi, F.

Boo, J.

S. A. Carter, J. D. LeGrange, W. White, J. Boo, and P. Wiltzius, “Dependence of the morphology of polymer dispersed liquid crystals on the UV polymerization process,” J. Appl. Phys. 81(9), 5992 (1997).
[CrossRef]

Bowley, C. C.

M. J. Escuti, C. C. Bowley, G. P. Crawford, and S. Zumer, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75(21), 3264 (1999).
[CrossRef]

Bricker, R. L.

Bunning, T. J.

Carter, S. A.

S. A. Carter, J. D. LeGrange, W. White, J. Boo, and P. Wiltzius, “Dependence of the morphology of polymer dispersed liquid crystals on the UV polymerization process,” J. Appl. Phys. 81(9), 5992 (1997).
[CrossRef]

Chen, C.-C.

Cheng, K.-T.

Chui, C.

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

Clapp, L.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Comiskey, B.

B. Comiskey, J. D. Albert, H. Yoshizawa, and J. Jacobson, “An electrophoretic ink for all-printed reflective electronic displays,” Nature 394(6690), 253–255 (1998).
[CrossRef]

Crawford, G. P.

M. J. Escuti, C. C. Bowley, G. P. Crawford, and S. Zumer, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75(21), 3264 (1999).
[CrossRef]

Crowley, J. M.

J. M. Crowley, N. K. Sheridon, and L. Romano, “Dipole moments of gyricon balls,” J. Electrost. 55, 247 (2002).
[CrossRef]

Escuti, M. J.

M. J. Escuti, C. C. Bowley, G. P. Crawford, and S. Zumer, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75(21), 3264 (1999).
[CrossRef]

Fan, Y.-H.

Y.-H. Fan, Y.-H. Lin, H. Ren, S. Gauza, and S.-T. Wu, “Fast-response and scattering-free polymer network liquid crystals for infrared light modulators,” Appl. Phys. Lett. 84(8), 1233 (2004).
[CrossRef]

Feenstra, B. J.

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425(6956), 383–385 (2003).
[CrossRef] [PubMed]

Fuh, A. Y.-G.

Gally, B.

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

Gauza, S.

Y.-H. Fan, Y.-H. Lin, H. Ren, S. Gauza, and S.-T. Wu, “Fast-response and scattering-free polymer network liquid crystals for infrared light modulators,” Appl. Phys. Lett. 84(8), 1233 (2004).
[CrossRef]

Green, L.

Hayes, R. A.

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425(6956), 383–385 (2003).
[CrossRef] [PubMed]

Heikenfeld, J.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Hollfelder, N.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Hwang, K. Y.

Jacobson, J.

B. Comiskey, J. D. Albert, H. Yoshizawa, and J. Jacobson, “An electrophoretic ink for all-printed reflective electronic displays,” Nature 394(6690), 253–255 (1998).
[CrossRef]

Jang, J. E.

Jeng, S.-C.

Jin, Y. W.

Jung, J. E.

Kelly, J. R.

J. R. Kelly and W. Wu, “Multiple scattering effects in polymer dispersed liquid crystals,” Liq. Cryst. 14(6), 1683–1694 (1993).
[CrossRef]

Kilickiran, P.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Kothari, M.

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

Kreit, E.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Larson, E.

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

Lee, G. H.

Lee, S. Y.

LeGrange, J. D.

S. A. Carter, J. D. LeGrange, W. White, J. Boo, and P. Wiltzius, “Dependence of the morphology of polymer dispersed liquid crystals on the UV polymerization process,” J. Appl. Phys. 81(9), 5992 (1997).
[CrossRef]

Li, Q.

Liao, C.-C.

Lin, Y.-H.

Y.-H. Lin, J.-M. Yang, Y.-R. Lin, S.-C. Jeng, and C.-C. Liao, “A polarizer-free flexible and reflective electrooptical switch using dye-doped liquid crystal gels,” Opt. Express 16(3), 1777–1785 (2008).
[CrossRef] [PubMed]

Y.-H. Fan, Y.-H. Lin, H. Ren, S. Gauza, and S.-T. Wu, “Fast-response and scattering-free polymer network liquid crystals for infrared light modulators,” Appl. Phys. Lett. 84(8), 1233 (2004).
[CrossRef]

Lin, Y.-R.

Liu, C.-K.

Masutani, A.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Milarcik, A.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Miles, M.

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

Montgomery, G. P.

N. A. Vaz, G. W. Smith, and G. P. Montgomery., “A light control film composed of liquid crystal droplets dispersed in a uv-curable polymer,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 146(1), 1–15 (1987).
[CrossRef]

Natarajan, L. V.

Nelles, G.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Raj, B.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Ren, H.

Y.-H. Fan, Y.-H. Lin, H. Ren, S. Gauza, and S.-T. Wu, “Fast-response and scattering-free polymer network liquid crystals for infrared light modulators,” Appl. Phys. Lett. 84(8), 1233 (2004).
[CrossRef]

Roberts, T.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Romano, L.

J. M. Crowley, N. K. Sheridon, and L. Romano, “Dipole moments of gyricon balls,” J. Electrost. 55, 247 (2002).
[CrossRef]

Ruocchio, C.

Sakaigawa, A.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Schüller, B.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Schwartz, R.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Sheridon, N. K.

J. M. Crowley, N. K. Sheridon, and L. Romano, “Dipole moments of gyricon balls,” J. Electrost. 55, 247 (2002).
[CrossRef]

Smith, G. W.

N. A. Vaz, G. W. Smith, and G. P. Montgomery., “A light control film composed of liquid crystal droplets dispersed in a uv-curable polymer,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 146(1), 1–15 (1987).
[CrossRef]

Steckl, A. J.

H. You and A. J. Steckl, “Three-color electrowetting display device for electronic paper,” Appl. Phys. Lett. 97(2), 023514 (2010).
[CrossRef]

Sun, B.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Terrecuso, P.

Tondiglia, V. P.

Vaz, N. A.

N. A. Vaz, G. W. Smith, and G. P. Montgomery., “A light control film composed of liquid crystal droplets dispersed in a uv-curable polymer,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 146(1), 1–15 (1987).
[CrossRef]

Vicari, L.

White, T. J.

White, W.

S. A. Carter, J. D. LeGrange, W. White, J. Boo, and P. Wiltzius, “Dependence of the morphology of polymer dispersed liquid crystals on the UV polymerization process,” J. Appl. Phys. 81(9), 5992 (1997).
[CrossRef]

Wiltzius, P.

S. A. Carter, J. D. LeGrange, W. White, J. Boo, and P. Wiltzius, “Dependence of the morphology of polymer dispersed liquid crystals on the UV polymerization process,” J. Appl. Phys. 81(9), 5992 (1997).
[CrossRef]

Wu, S.-T.

Y.-H. Fan, Y.-H. Lin, H. Ren, S. Gauza, and S.-T. Wu, “Fast-response and scattering-free polymer network liquid crystals for infrared light modulators,” Appl. Phys. Lett. 84(8), 1233 (2004).
[CrossRef]

Wu, W.

J. R. Kelly and W. Wu, “Multiple scattering effects in polymer dispersed liquid crystals,” Liq. Cryst. 14(6), 1683–1694 (1993).
[CrossRef]

Yang, J.-M.

Yang, S.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Yasuda, A.

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

Yoshizawa, H.

B. Comiskey, J. D. Albert, H. Yoshizawa, and J. Jacobson, “An electrophoretic ink for all-printed reflective electronic displays,” Nature 394(6690), 253–255 (1998).
[CrossRef]

You, H.

H. You and A. J. Steckl, “Three-color electrowetting display device for electronic paper,” Appl. Phys. Lett. 97(2), 023514 (2010).
[CrossRef]

Zhou, K.

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Zumer, S.

M. J. Escuti, C. C. Bowley, G. P. Crawford, and S. Zumer, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75(21), 3264 (1999).
[CrossRef]

S. Zǔmer, “Light scattering from nematic droplets: Anomalous-diffraction approach,” Phys. Rev. A 37(10), 4006–4015 (1988).
[CrossRef] [PubMed]

Appl. Phys. Lett.

H. You and A. J. Steckl, “Three-color electrowetting display device for electronic paper,” Appl. Phys. Lett. 97(2), 023514 (2010).
[CrossRef]

M. J. Escuti, C. C. Bowley, G. P. Crawford, and S. Zumer, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75(21), 3264 (1999).
[CrossRef]

Y.-H. Fan, Y.-H. Lin, H. Ren, S. Gauza, and S.-T. Wu, “Fast-response and scattering-free polymer network liquid crystals for infrared light modulators,” Appl. Phys. Lett. 84(8), 1233 (2004).
[CrossRef]

A. Masutani, T. Roberts, B. Schüller, N. Hollfelder, P. Kilickiran, G. Nelles, A. Yasuda, and A. Sakaigawa, “Improvement of dichroic polymer dispersed liquid crystal performance using lift-off technique,” Appl. Phys. Lett. 89(18), 183514 (2006).
[CrossRef]

J. Appl. Phys.

S. A. Carter, J. D. LeGrange, W. White, J. Boo, and P. Wiltzius, “Dependence of the morphology of polymer dispersed liquid crystals on the UV polymerization process,” J. Appl. Phys. 81(9), 5992 (1997).
[CrossRef]

J. Electrost.

J. M. Crowley, N. K. Sheridon, and L. Romano, “Dipole moments of gyricon balls,” J. Electrost. 55, 247 (2002).
[CrossRef]

Liq. Cryst.

J. R. Kelly and W. Wu, “Multiple scattering effects in polymer dispersed liquid crystals,” Liq. Cryst. 14(6), 1683–1694 (1993).
[CrossRef]

Mol. Cryst. Liq. Cryst. (Phila. Pa.)

N. A. Vaz, G. W. Smith, and G. P. Montgomery., “A light control film composed of liquid crystal droplets dispersed in a uv-curable polymer,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 146(1), 1–15 (1987).
[CrossRef]

Nat. Photonics

J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz, “Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions,” Nat. Photonics 3(5), 292–296 (2009).
[CrossRef]

Nature

B. Comiskey, J. D. Albert, H. Yoshizawa, and J. Jacobson, “An electrophoretic ink for all-printed reflective electronic displays,” Nature 394(6690), 253–255 (1998).
[CrossRef]

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425(6956), 383–385 (2003).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. Zǔmer, “Light scattering from nematic droplets: Anomalous-diffraction approach,” Phys. Rev. A 37(10), 4006–4015 (1988).
[CrossRef] [PubMed]

Proc. Soc. Inf. Display

M. Miles, E. Larson, C. Chui, M. Kothari, B. Gally, and J. Batey, “Digital Paper for Reflective Displays,” Proc. Soc. Inf. Display 209 (2003).
[CrossRef]

Other

P. Nolan, M. Tillin, D. Coates, E. Ginter, E. Lueder, and T. Kallfass, “Reflective mode PDLC displays-paper white display,” Eurodisplay ’93, Proceedings of the International Display Research Conference (Society for Information Display, 1993), pp. 397–400.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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Figures (5)

Fig. 1
Fig. 1

Operation principle of color optical device switching by PNLC with color filter on a specular reflector.

Fig. 2
Fig. 2

(a) Measured reflectance of PNLC device with a specular reflector, an outside-absorption layer, and an inside-absorption layer. The reflectance with an inside-absorption layer was again shown in inset. (b) Schematics of light reflection in three types of cell.

Fig. 3
Fig. 3

Experimental setup for reflectance measurement with (a) specular collimated light source and (b) integrating sphere light source. Measured reflectance (c) in the specular geometry and (d) in the integrating sphere geometry. Measured contrast ratio (e) in the specular geometry and (f) in the integrating sphere geometry.

Fig. 4
Fig. 4

(a) Comparison between the normalized reflectance of PNLC device without color filter and with color filter. (b) Color chromaticity of color PNLC device in CIE 1931 color space with respect to the voltages. (c) Spectrum of color PNLC devices. (d) Images of color PNLC device. The PNLC with 20 μm thickness was used in (a) and (b), but with 5 μm thickness in (c) and (d).

Fig. 5
Fig. 5

Angle dependence of effective differential scattering cross section with respect to the number of scattering times (a) for forward direction and (b) for backward direction.

Equations (3)

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d σ d Ω = R 4 k 2 4 [ | H ( i v e , k R sin θ ) | 2 cos 2 α o + | H ( i v o , k R sin θ ) | 2 sin 2 α o ] .
σ 0 ( θ ) = d σ d Ω α o , ϕ = 1 4 π 0 2 π 0 π d σ d Ω sin ϕ d ϕ d α 0 .
σ i + 1 ( θ ) = ( n m c σ i ( θ n , φ m ) σ 0 ( θ , φ ) ) sin φ d φ .

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