Abstract

We investigate the reflection at the interface between a homogeneous medium and a semi-infinite periodic layered medium. The result provides independent confirmation of the form birefringence of the periodic layered medium. In addition, our result shows a fundamental difference between the reflection at the interface between a homogeneous medium and a semi-infinite periodic layered medium and that of a Bragg reflector that consists of a large number of layers. We analyze the form birefringence induced by volume index gratings in a homogeneous medium. The extension of our analysis to multiple volume index gratings provides an explanation of the relationship between birefringence and crystal symmetry.

© 1995 Optical Society of America

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References

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  1. P. Yeh, W. J. Gunning, J. P. Eblen, Jr., and M. Khoshnevisan, "Compensator for liquid crystal displays having two types of layers with different refractive indices alternating," U.S. patent 5,196,953 (March 23, 1993).
  2. J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.
  3. J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
    [CrossRef]
  4. See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 118–138, and references therein.
  5. See, for example, A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–208, and references therein.
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 705–708.
  7. D. E. Aspnes, "Optical properties of thin films," Thin Solid Films 89, 249–262 (1982).
    [CrossRef]
  8. J. K. Moyle, W. J. Gunning, and W. H. Southwell, "Microstructure modeling: scattering and form birefringence in dielectric thin films," in Modeling of Optical Thin Films, M. R. Jacobson, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 821, 157 (1987).
    [CrossRef]
  9. I. J. Hodgekinson and P. W. Wilson, "Microstructuralinduced anisotropy in thin films for optical applications," CRC Crit. Rev. Solid State Mater. Sci. 15, 27 (1988).
    [CrossRef]
  10. Q. Wu and I. J. Hodgekinson, "Materials for birefringent coatings," Appl. Opt. 33, 8109–8110 (1994).
    [CrossRef] [PubMed]
  11. F. Horowitz, "Structure-induced optical anisotropy in thin films," Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1983).

1994 (1)

1988 (1)

I. J. Hodgekinson and P. W. Wilson, "Microstructuralinduced anisotropy in thin films for optical applications," CRC Crit. Rev. Solid State Mater. Sci. 15, 27 (1988).
[CrossRef]

1982 (1)

D. E. Aspnes, "Optical properties of thin films," Thin Solid Films 89, 249–262 (1982).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, "Optical properties of thin films," Thin Solid Films 89, 249–262 (1982).
[CrossRef]

Beedy, J.

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 705–708.

Eblen, J. P.

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

P. Yeh, W. J. Gunning, J. P. Eblen, Jr., and M. Khoshnevisan, "Compensator for liquid crystal displays having two types of layers with different refractive indices alternating," U.S. patent 5,196,953 (March 23, 1993).

Gunning, W. J.

P. Yeh, W. J. Gunning, J. P. Eblen, Jr., and M. Khoshnevisan, "Compensator for liquid crystal displays having two types of layers with different refractive indices alternating," U.S. patent 5,196,953 (March 23, 1993).

J. K. Moyle, W. J. Gunning, and W. H. Southwell, "Microstructure modeling: scattering and form birefringence in dielectric thin films," in Modeling of Optical Thin Films, M. R. Jacobson, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 821, 157 (1987).
[CrossRef]

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

Hale, L.

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

Hodgekinson, I. J.

Q. Wu and I. J. Hodgekinson, "Materials for birefringent coatings," Appl. Opt. 33, 8109–8110 (1994).
[CrossRef] [PubMed]

I. J. Hodgekinson and P. W. Wilson, "Microstructuralinduced anisotropy in thin films for optical applications," CRC Crit. Rev. Solid State Mater. Sci. 15, 27 (1988).
[CrossRef]

Horowitz, F.

F. Horowitz, "Structure-induced optical anisotropy in thin films," Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1983).

Khoshnevisan, M.

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

P. Yeh, W. J. Gunning, J. P. Eblen, Jr., and M. Khoshnevisan, "Compensator for liquid crystal displays having two types of layers with different refractive indices alternating," U.S. patent 5,196,953 (March 23, 1993).

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

Moyle, J. K.

J. K. Moyle, W. J. Gunning, and W. H. Southwell, "Microstructure modeling: scattering and form birefringence in dielectric thin films," in Modeling of Optical Thin Films, M. R. Jacobson, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 821, 157 (1987).
[CrossRef]

Southwell, W. H.

J. K. Moyle, W. J. Gunning, and W. H. Southwell, "Microstructure modeling: scattering and form birefringence in dielectric thin films," in Modeling of Optical Thin Films, M. R. Jacobson, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 821, 157 (1987).
[CrossRef]

Taber, D.

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

Wilson, P. W.

I. J. Hodgekinson and P. W. Wilson, "Microstructuralinduced anisotropy in thin films for optical applications," CRC Crit. Rev. Solid State Mater. Sci. 15, 27 (1988).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 705–708.

Wu, Q.

Yariv, A.

See, for example, A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–208, and references therein.

Yeh, P.

See, for example, A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–208, and references therein.

See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 118–138, and references therein.

P. Yeh, W. J. Gunning, J. P. Eblen, Jr., and M. Khoshnevisan, "Compensator for liquid crystal displays having two types of layers with different refractive indices alternating," U.S. patent 5,196,953 (March 23, 1993).

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

Appl. Opt. (1)

CRC Crit. Rev. Solid State Mater. Sci. (1)

I. J. Hodgekinson and P. W. Wilson, "Microstructuralinduced anisotropy in thin films for optical applications," CRC Crit. Rev. Solid State Mater. Sci. 15, 27 (1988).
[CrossRef]

Thin Solid Films (1)

D. E. Aspnes, "Optical properties of thin films," Thin Solid Films 89, 249–262 (1982).
[CrossRef]

Other (8)

J. K. Moyle, W. J. Gunning, and W. H. Southwell, "Microstructure modeling: scattering and form birefringence in dielectric thin films," in Modeling of Optical Thin Films, M. R. Jacobson, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 821, 157 (1987).
[CrossRef]

F. Horowitz, "Structure-induced optical anisotropy in thin films," Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1983).

P. Yeh, W. J. Gunning, J. P. Eblen, Jr., and M. Khoshnevisan, "Compensator for liquid crystal displays having two types of layers with different refractive indices alternating," U.S. patent 5,196,953 (March 23, 1993).

J. P. Eblen, Jr., W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, and M. Khoshnevisan, "Birefringent compensators for normally white TN-LCDs," in SID 94 Digest (Society for Information Display, San Jose, 1994), pp. 245–248.

J. P. Eblen, Jr., W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, and L. Hale, "Thin-film birefringent devices based on form birefringence," in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2262, 234 (1994).
[CrossRef]

See, for example, P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 118–138, and references therein.

See, for example, A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–208, and references therein.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 705–708.

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Figures (4)

Fig. 1
Fig. 1

Reflection at an interface between a homogeneous material and a semi-infinite periodic layered medium.

Fig 2
Fig 2

Reflectivity (R = |r|2) as a function of the period of the layered medium The parameters are chosen as wavelength λ = 0 5 μm, indices of refraction n1 = 1.5 and n2 = 2.7, relative thickness of each layer a/b = 1, and incident angle 0° (normal incidence).

Fig. 3
Fig. 3

Reflectivities R = |r|2 as functions of the incident angle [ θ = sin 1 ( k 1 z / β 2 + k 1 z 2 ) ] for both TE and TM waves at various periods, Λ → 0, Λ = 0.05 μm, and Λ = 0.09 μm. Other parameters are the same as those in Fig. 2.

Fig. 4
Fig. 4

Dielectric constant profile of two gratings.

Tables (1)

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Table 1 Birefringence and Crystal Symmetry

Equations (31)

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n ( z ) = { n 1 z < 0 n 2 m Λ < z < m Λ + b ( m = 0 , 1 , 2 , ) , n 1 m Λ + b < z < ( m + 1 ) Λ ( m = 0 , 1 , 2 , )
E ( z < 0 ) = [ a 0 p ˆ 1 exp ( i k 1 z z ) + b 0 p ˆ 2 exp ( i k 1 z z ) ] × exp [ i ( ω t β x ) ] ,
E ( z > 0 ) = E K ( z ) exp ( i K z ) exp [ i ( ω t β x ) ] ,
K = ( 1 / Λ ) cos 1 [ ( A + D ) / 2 ] ,
( a 0 b 0 ) = [ B exp ( i K Λ ) A ] ,
r = b 0 a 0 = exp ( i K Λ ) A B .
K TE 2 n o 2 + β 2 n o 2 = ω 2 c 2 ( TE ) ,
K TM 2 n o 2 + β 2 n e 2 = ω 2 c 2 ( TM ) ,
n o 2 = a Λ n 1 2 + b Λ n 2 2 ,
1 n e 2 = a Λ 1 n 1 2 + b Λ 1 n 2 2 .
K TE 2 = k 1 z 2 a Λ + k 2 z 2 b Λ ( TE ) ,
K TM 2 = k 1 z 2 a Λ ( a Λ + n 2 2 n 1 2 b Λ ) + k 2 z 2 b Λ ( b Λ + n 1 2 n 2 2 a Λ ) ( TM ) .
r TE = k 1 z K TE k 1 z + K TE ( TE ) ,
r TM = n 1 2 K TM n o 2 k 1 z n 1 2 K TM + n o 2 k 1 z ( TM ) ,
x x = y y = z z = 0 + 1 cos K z ,
= [ o 0 0 0 o 0 0 0 e ] ,
D x = x x E x = x x E x
E z = z z 1 D z = z z 1 D z ,
o = x x = 0 ,
e = 1 z z 1 = 0 2 1 2 .
= 0 + [ 0 0 0 0 0 0 0 0 δ ] ,
δ 1 2 / ( 2 0 ) .
x x = y y = z z = 0 + 1 cos K 1 z + 2 cos K 2 y .
= 0 + [ 0 0 0 0 0 0 0 0 δ 1 ] + [ 0 0 0 0 δ 2 0 0 0 0 ] ,
δ 1 1 2 / ( 2 0 ) ,
δ 2 2 2 / ( 2 0 ) .
0 + 1 p ( K 1 ) + 1 p ( K 2 ) + 1 p ( K 3 ) ,
= 0 + [ δ 1 0 0 0 δ 1 0 0 0 δ 1 ] ,
0 + 1 p ( K 1 ) + 1 p ( K 2 ) + 1 p ( K 3 ) + 2 p ( K z ) ,
= 0 + [ 0 0 0 0 0 0 0 0 δ 2 ] + m = 0 2 [ cos ( m π / 3 ) sin ( m π / 3 ) 0 sin ( m π / 3 ) cos ( m π / 3 ) 0 0 0 0 ] × [ δ 1 0 0 0 0 0 0 0 0 ] [ cos ( m π / 3 ) sin ( m π / 3 ) 0 sin ( m π / 3 ) cos ( m π / 3 ) 0 0 0 0 ] ,
= 0 + [ 3 δ 1 / 2 0 0 0 3 δ 1 / 2 0 0 0 δ 2 ] ,

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