Abstract

Numerical instability is usually observed when the propagation matrix method is used to calculate the reflectance and transmittance spectra for the thick one-dimensional inhomogeneous optical structures and media. To remove this numerical instability we applied two procedures, the normalization and the singular-value decomposition, for the propagation matrix and the matrix involved in calculating the matrix of reflection coefficients, respectively. Examples of a cholesteric liquid crystal and a helical structure of ferroelectric liquid crystals with a twist defect show that the modified propagation matrix method is able to accurately calculate the reflectance spectra for thick structures.

© 2008 Optical Society of America

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References

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

2007 (2)

2003 (1)

H. Hoshi, K. Ishikawa, and H. Takezoe, Phys. Rev. E 68, 020701 (2003).
[CrossRef]

2002 (1)

V. I. Kopp and A. Z. Genack, Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

2000 (1)

D. Yang and X. Mi, J. Phys. D 33, 672 (2000).
[CrossRef]

1999 (2)

I. Abdulhalim, J. Opt. A 1, 646 (1999).
[CrossRef]

V. I. Kopp and A. Z. Genack, Proc. SPIE 3623, 71 (1999).
[CrossRef]

1972 (1)

Abdulhalim, I.

I. Abdulhalim, J. Opt. A 1, 646 (1999).
[CrossRef]

Berreman, D. W.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Genack, A. Z.

V. I. Kopp and A. Z. Genack, Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

V. I. Kopp and A. Z. Genack, Proc. SPIE 3623, 71 (1999).
[CrossRef]

Hoshi, H.

H. Hoshi, K. Ishikawa, and H. Takezoe, Phys. Rev. E 68, 020701 (2003).
[CrossRef]

Ishikawa, K.

H. Hoshi, K. Ishikawa, and H. Takezoe, Phys. Rev. E 68, 020701 (2003).
[CrossRef]

Kopp, V. I.

V. I. Kopp and A. Z. Genack, Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

V. I. Kopp and A. Z. Genack, Proc. SPIE 3623, 71 (1999).
[CrossRef]

Lu, Z.

Mi, X.

D. Yang and X. Mi, J. Phys. D 33, 672 (2000).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Takezoe, H.

H. Hoshi, K. Ishikawa, and H. Takezoe, Phys. Rev. E 68, 020701 (2003).
[CrossRef]

Tarasenko, Y. S.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Vytovtov, K. A.

Yang, D.

D. Yang and X. Mi, J. Phys. D 33, 672 (2000).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

J. Opt. A (1)

I. Abdulhalim, J. Opt. A 1, 646 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

D. Yang and X. Mi, J. Phys. D 33, 672 (2000).
[CrossRef]

Phys. Rev. E (1)

H. Hoshi, K. Ishikawa, and H. Takezoe, Phys. Rev. E 68, 020701 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

V. I. Kopp and A. Z. Genack, Phys. Rev. Lett. 89, 033901 (2002).
[CrossRef] [PubMed]

Proc. SPIE (1)

V. I. Kopp and A. Z. Genack, Proc. SPIE 3623, 71 (1999).
[CrossRef]

Other (2)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

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Figures (3)

Fig. 1
Fig. 1

Normalized constant (dashed curves) and condition number (solid curves) for the thicknesses d = 10 p , 20 p , 50 p , and 100 p . The physical parameters of the cholesteric liquid crystal are: p = 0.3 μ m , ϵ e = 3.0 , ϵ o = 2.0 , and n i = n t = 1.516 . A circular polarization light is normally incident.

Fig. 2
Fig. 2

Reflectance spectra of a cholesteric liquid crystal for different thicknesses d = 10 p , 20 p , 50 p , and 100 p after the normalization and SVD procedures.

Fig. 3
Fig. 3

Reflectance spectra of a helical structure of ferroelectric liquid crystals with a π 2 twist defect at d 2 : (a) d = 202 p and (b) d = 250 p . The parameters are: p = 0.35 μ m , n e = 1.7 , n o = 1.5 , tilt angle 23°, and n i = n t = 1.6 .

Equations (13)

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d Ψ ( z ) d z = i k 0 Δ ( z ) Ψ ( z ) ,
P ( z d , z 0 ) P ( z d , z N 1 ) P ( z 1 , z 0 ) .
P ( z d , z 0 ) e i k 0 h Δ ( z N 1 + h 2 ) e i k 0 h Δ ( z 0 + h 2 ) .
r = ( r p p r p s r s p r s s ) = A 1 1 A 2 ,
t = ( t p p t p s t s p t s s ) = B 1 + B 2 r .
A 1 = ( n i ( n t P 12 cos θ t P 22 ) cos θ i ( n t P 11 cos θ t P 21 ) ( n t P 13 cos θ t P 23 ) n i cos θ i ( n t P 14 cos θ t P 24 ) n i ( n t cos θ t P 32 P 42 ) cos θ i ( n t cos θ t P 31 P 41 ) ( n t cos θ t P 33 P 43 ) n i cos θ i ( n t cos θ t P 34 P 44 ) ) ,
A 2 = ( n i ( n t P 12 cos θ t P 22 ) + cos θ i ( n t P 11 cos θ t P 21 ) ( n t P 13 cos θ t P 23 ) + n i cos θ i ( n t P 14 cos θ t P 24 ) n i ( n t cos θ t P 32 P 42 ) + cos θ i ( n t cos θ t P 31 P 41 ) ( n t cos θ t P 33 P 43 ) + n i cos θ i ( n t cos θ t P 34 P 44 ) ) ,
B 1 = ( ( n i P 22 + cos θ i P 21 ) n t ( P 23 + n i cos θ i P 24 ) n t n i P 32 + cos θ i P 31 P 33 + n i cos θ i P 34 ) ,
B 2 = ( ( n i P 22 cos θ i P 21 ) n t ( P 23 n i cos θ i P 24 ) n t n i P 32 cos θ i P 31 P 33 n i cos θ i P 34 ) ,
P ( z d , z 0 ) = P l P l 1 P i P 1 ,
P i = P ( z m + ( i 1 ) m , z m 1 + ( i 1 ) m ) P ( z 1 + ( i 1 ) m , z ( i 1 ) m ) .
C 1 P ̃ 1 = P 1 , C k P ̃ k = P k P ̃ k 1 , 2 k l .
A 1 1 = V ( 1 w 1 0 0 0 ) U t ,

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