Abstract

We introduce a 3×3 matrix for the study of unitary optical systems. This 3×3 matrix is a submatrix of the 4×4 Mueller matrix. The elements of this 3×3 matrix are real, and thus complex-number calculations can be avoided. The 3×3 matrix is useful for illustrating the polarization state of an optical system. One can also use it to derive the conditions for linear and circular polarization output for a general optical system. New characterization methods for unitary optical systems are introduced. It is shown that the trajectory of the Stokes vector on a Poincaré sphere is either a circle or an ellipse as the optical system or input polarizer is rotated. One can use this characteristic circle or ellipse to measure the equivalent optical retardation and rotation of any lossless optical system.

© 2001 Optical Society of America

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  1. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  2. H. Hurwitz, R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalent theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  3. R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. 143, 165–169 (1990).
    [CrossRef]
  4. V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
    [CrossRef]
  5. H. S. Kwok, “Parameter space representation of liquid crystal displays in transmission and reflection,” J. Appl. Phys. 80, 3687–3693 (1996).
    [CrossRef]
  6. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [CrossRef]
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1989).
  8. S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, New York, 1994).
  9. E. P. Raynes, R. J. A. Tough, “The guiding of plane polarized light by twisted liquid crystal layers,” Mol. Cryst. Liq. Cryst. Lett. 2, 139–145 (1985).
  10. H. Goldstein, Classical Mechanics (Addison-Wesley, Boston, Mass., 1980).
  11. R. Mirman, Group Theory—An Intuitive Approach (World Scientific, Singapore, 1995).
  12. E. Collett, Polarized Light—Fundamentals and Applications (Marcel Dekker, New York, 1993).
  13. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  14. A. Schonhofer, H. G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
    [CrossRef]
  15. C. Brosseau, Fundamentals of Polarized Light—A Statistical Optics Approach (Wiley, New York, 1998).
  16. G. Streng, Linear Algebra and Its Applications (Harcourt Brace Jovanovich, San Diego, Calif., 1986).
  17. S. T. Tang, H. S. Kwok, “A new method for measuring liquid crystal cell parameters,” in 1998 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1998), pp. 552–555.
  18. S. T. Tang, H. S. Kwok, “A new method of designing LCDs with optimal optical properties,” in 1999 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1999), pp. 195–197.
  19. S. T. Tang, H. S. Kwok, “Optically optimized transmittive and reflective bistable twisted nematic liquid crystal displays,” J. Appl. Phys. 87, 632–637 (2000).
    [CrossRef]
  20. Z. Z. Zhuang, Y. J. Kim, J. Patel, “Achromatic linear polarization rotator using twisted nematic liquid crystals,” Appl. Phys. Lett. 76, 3995–3997 (2000).
    [CrossRef]
  21. C. H. Gooch, H. A. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
    [CrossRef]
  22. C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid crystal structure with twist angles ⩽90°,” J. Phys. D 8, 1575–1584 (1975).
    [CrossRef]

2000 (2)

S. T. Tang, H. S. Kwok, “Optically optimized transmittive and reflective bistable twisted nematic liquid crystal displays,” J. Appl. Phys. 87, 632–637 (2000).
[CrossRef]

Z. Z. Zhuang, Y. J. Kim, J. Patel, “Achromatic linear polarization rotator using twisted nematic liquid crystals,” Appl. Phys. Lett. 76, 3995–3997 (2000).
[CrossRef]

1996 (2)

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

H. S. Kwok, “Parameter space representation of liquid crystal displays in transmission and reflection,” J. Appl. Phys. 80, 3687–3693 (1996).
[CrossRef]

1990 (1)

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. 143, 165–169 (1990).
[CrossRef]

1987 (1)

A. Schonhofer, H. G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
[CrossRef]

1985 (1)

E. P. Raynes, R. J. A. Tough, “The guiding of plane polarized light by twisted liquid crystal layers,” Mol. Cryst. Liq. Cryst. Lett. 2, 139–145 (1985).

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

1975 (1)

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid crystal structure with twist angles ⩽90°,” J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

1974 (1)

C. H. Gooch, H. A. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

1941 (2)

Aben, H.

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1989).

Bagini, V.

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1989).

Borgh, R.

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light—A Statistical Optics Approach (Wiley, New York, 1998).

Chandrasekhar, S.

S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, New York, 1994).

Collett, E.

E. Collett, Polarized Light—Fundamentals and Applications (Marcel Dekker, New York, 1993).

Frezza, F.

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Boston, Mass., 1980).

Gooch, C. H.

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid crystal structure with twist angles ⩽90°,” J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

C. H. Gooch, H. A. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

Gori, F.

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

Hurwitz, H.

Jones, R. C.

Kim, Y. J.

Z. Z. Zhuang, Y. J. Kim, J. Patel, “Achromatic linear polarization rotator using twisted nematic liquid crystals,” Appl. Phys. Lett. 76, 3995–3997 (2000).
[CrossRef]

Kuball, H. G.

A. Schonhofer, H. G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
[CrossRef]

Kwok, H. S.

S. T. Tang, H. S. Kwok, “Optically optimized transmittive and reflective bistable twisted nematic liquid crystal displays,” J. Appl. Phys. 87, 632–637 (2000).
[CrossRef]

H. S. Kwok, “Parameter space representation of liquid crystal displays in transmission and reflection,” J. Appl. Phys. 80, 3687–3693 (1996).
[CrossRef]

S. T. Tang, H. S. Kwok, “A new method of designing LCDs with optimal optical properties,” in 1999 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1999), pp. 195–197.

S. T. Tang, H. S. Kwok, “A new method for measuring liquid crystal cell parameters,” in 1998 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1998), pp. 552–555.

Mirman, R.

R. Mirman, Group Theory—An Intuitive Approach (World Scientific, Singapore, 1995).

Mukunda, N.

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. 143, 165–169 (1990).
[CrossRef]

Patel, J.

Z. Z. Zhuang, Y. J. Kim, J. Patel, “Achromatic linear polarization rotator using twisted nematic liquid crystals,” Appl. Phys. Lett. 76, 3995–3997 (2000).
[CrossRef]

Raynes, E. P.

E. P. Raynes, R. J. A. Tough, “The guiding of plane polarized light by twisted liquid crystal layers,” Mol. Cryst. Liq. Cryst. Lett. 2, 139–145 (1985).

Santarsiero, M.

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

Schettini, G.

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

Schonhofer, A.

A. Schonhofer, H. G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. 143, 165–169 (1990).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Spagnolo, G. S.

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

Streng, G.

G. Streng, Linear Algebra and Its Applications (Harcourt Brace Jovanovich, San Diego, Calif., 1986).

Tang, S. T.

S. T. Tang, H. S. Kwok, “Optically optimized transmittive and reflective bistable twisted nematic liquid crystal displays,” J. Appl. Phys. 87, 632–637 (2000).
[CrossRef]

S. T. Tang, H. S. Kwok, “A new method for measuring liquid crystal cell parameters,” in 1998 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1998), pp. 552–555.

S. T. Tang, H. S. Kwok, “A new method of designing LCDs with optimal optical properties,” in 1999 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1999), pp. 195–197.

Tarry, H. A.

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid crystal structure with twist angles ⩽90°,” J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

C. H. Gooch, H. A. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

Tough, R. J. A.

E. P. Raynes, R. J. A. Tough, “The guiding of plane polarized light by twisted liquid crystal layers,” Mol. Cryst. Liq. Cryst. Lett. 2, 139–145 (1985).

Zhuang, Z. Z.

Z. Z. Zhuang, Y. J. Kim, J. Patel, “Achromatic linear polarization rotator using twisted nematic liquid crystals,” Appl. Phys. Lett. 76, 3995–3997 (2000).
[CrossRef]

Appl. Phys. Lett. (1)

Z. Z. Zhuang, Y. J. Kim, J. Patel, “Achromatic linear polarization rotator using twisted nematic liquid crystals,” Appl. Phys. Lett. 76, 3995–3997 (2000).
[CrossRef]

Chem. Phys. (1)

A. Schonhofer, H. G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
[CrossRef]

Electron. Lett. (1)

C. H. Gooch, H. A. Tarry, “Optical characteristics of twisted nematic liquid-crystal films,” Electron. Lett. 10, 2–4 (1974).
[CrossRef]

Eur. J. Phys. (1)

V. Bagini, R. Borgh, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

J. Appl. Phys. (2)

H. S. Kwok, “Parameter space representation of liquid crystal displays in transmission and reflection,” J. Appl. Phys. 80, 3687–3693 (1996).
[CrossRef]

S. T. Tang, H. S. Kwok, “Optically optimized transmittive and reflective bistable twisted nematic liquid crystal displays,” J. Appl. Phys. 87, 632–637 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. D (1)

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid crystal structure with twist angles ⩽90°,” J. Phys. D 8, 1575–1584 (1975).
[CrossRef]

Mol. Cryst. Liq. Cryst. Lett. (1)

E. P. Raynes, R. J. A. Tough, “The guiding of plane polarized light by twisted liquid crystal layers,” Mol. Cryst. Liq. Cryst. Lett. 2, 139–145 (1985).

Opt. Commun. (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Phys. Lett. (1)

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. 143, 165–169 (1990).
[CrossRef]

Other (10)

C. Brosseau, Fundamentals of Polarized Light—A Statistical Optics Approach (Wiley, New York, 1998).

G. Streng, Linear Algebra and Its Applications (Harcourt Brace Jovanovich, San Diego, Calif., 1986).

S. T. Tang, H. S. Kwok, “A new method for measuring liquid crystal cell parameters,” in 1998 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1998), pp. 552–555.

S. T. Tang, H. S. Kwok, “A new method of designing LCDs with optimal optical properties,” in 1999 Society for Information Display International Symposium Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1999), pp. 195–197.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1989).

S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, New York, 1994).

H. Goldstein, Classical Mechanics (Addison-Wesley, Boston, Mass., 1980).

R. Mirman, Group Theory—An Intuitive Approach (World Scientific, Singapore, 1995).

E. Collett, Polarized Light—Fundamentals and Applications (Marcel Dekker, New York, 1993).

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

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Figures (9)

Fig. 1
Fig. 1

Unitary optical system between two polarizers (Pols). All angles are relative to the x axis. Mi are arbitrary unitary optical elements.

Fig. 2
Fig. 2

Reflective optical system with unitary optical elements Mi and a single front polarizer.

Fig. 3
Fig. 3

Solution curves of Eq. (27) are circles in ϕδ space. Circles correspond to the order of solution; the smallest circle is for N=1, and so on.

Fig. 4
Fig. 4

LP2 solutions span the whole ϕδ space. The curves shown are the first two orders of solutions with different input polarizer angle α. The input polarizer angle begins at 10°–80° from left to right.

Fig. 5
Fig. 5

CP solutions with Eq. (29) as α is varied. Curves are for different orders of the solution. No solution could be obtained for ϕ>δ.

Fig. 6
Fig. 6

For the CP solutions, one can plot the input polarizer angle α as a function of the twist angle. The three curves here represent the three curves in Fig. 5.

Fig. 7
Fig. 7

Experimental setup for the Stokes parameter method of cell parameter measurement. S and S are the Stokes vectors before and after the Gooch–Tarry nematic LC cell.

Fig. 8
Fig. 8

Examples of characteristic circles on the S1S2 plane: c1, first minimum twisted nematic cell (ϕ=90°, dΔn=0.5 μm); c2, second minimum twisted nematic cell (ϕ=90°, dΔn=1.0 μm); c3, standard twisted nematic LC cell (ϕ=240°, dΔn=0.85 μm). Wavelength, 632.8 nm.

Fig. 9
Fig. 9

Characteristic ellipses for the same LC cells as in Fig. 8.

Tables (2)

Tables Icon

Table 1 Conditions for Obtaining Linearly or Circularly Polarized Output for General Mueller and Jones Calculus

Tables Icon

Table 2 Relationship of Characteristic Parameters of the Unitary Optical System to Various Matrix Elements and to LC Cell Parameters

Equations (41)

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MM=T  (MJMJ*)  T-1,
T=1000100-101100i-i0,
MJMJ*m11m11*m11m12*m12m11*m12m12*m11m21*m11m22*m12m21*m12m22*m21m11*m21m12*m22m11*m22m12*m21m21*m21m22*m22m21*m22m22*,
MJ=a+ib-c+idc+ida-ib,
a=cos β cos ϕ+ϕβsin β sin ϕ,
b=-δβsin β cos ϕ,
c=cos β sin ϕ-ϕβsin β cos ϕ,
d=-δβsin β sin ϕ,
MM=100001-2(c2+d2)2(bd-ac)-2(ad+bc)02(ac+bd)1-2(b2+c2)2(ab-cd)02(ad-bc)-2(ab+cd)1-2(b2+d2)=10000ABC0DEF0GHK.
S=S0S1S2S3,
MM=ABCDEFGHK.
S=S1S2S3.
WP(Γ)=1000cos 2Γ-sin 2Γ0sin 2Γcos 2Γ.
R(θ)=cos 2θ-sin 2θ0sin 2θcos 2θ0001.
Sout=MMSin.
T=0.5+0.5(cos 2γsin 2γ0)  M  cos 2αsin 2α0,
Pol(γ)=121cos 2γsin 2γ0cos 2γcos2 2γcos 2γ sin 2γ0sin 2γcos 2γ sin 2γsin2 2γ00000,
R=0.5+0.5(cos 2α sin 2α0)  MPM  cos 2αsin 2α0,
Mp=M11M21-M31M12M22-M32-M13-M23M33.
MP=OMTO-1
O=10001000-1.
cos 2γsin 2γ0=ABCDEFGHK cos 2αsin 2α0.
G cos 2α+H sin 2α=0.
G=H=0,
tan 2α=-G/H.
00±1=ABCDEFGHK cos 2αsin 2α0
ADGBEHCFK 00±1=cos 2αsin 2α0.
tan 2α=H/G,
K=0.
sin β=0,
tan 2αLP2=ϕ/β tan β,
δβsin β=12,
tan 2αCP=-β/ϕ cot β.
M=R(χ)WP(Γ, ψ),
S=R(χ)  R(θ)  WP(Γ)  R(-θ)  S.
S=R(χ)  cos2 2θ+sin2 2θ cos 2Γsin 2θ cos 2θ(1-cos 2Γ)sin 2Γ sin 2θ=R(χ)  S.
(S1-cos2 Γ)2+S22=sin4 Γ.
S=R(χ)  1000cos 2Γ-sin 2θ0sin 2Γcos 2Γ cos 2θsin 2θ0=R(χ)  S,
S1=cos 2θ,
S2=cos 2Γ sin 2θ.
SI2+S2cos 2Γ2=1,

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