Abstract

The quaternion formalism has been used to derive new systems of equations that describe transformation of the polarization of light in inhomogeneous birefringent media. In quaternion algebra the problem of parametric representation of the unitary transformation matrix reduces to the problem of formulation of the quaternion in trigonometric form. It is shown that this can be done in 30 different ways and that to each trigonometric form corresponds its own system of transformation equations. The six simplest systems of transformation equations have been derived.

© 2001 Optical Society of America

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References

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  1. A. R. Stokes, The Theory of the Optical Properties of Inhomogeneous Materials (Spon, London, 1963).
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  3. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  4. P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
  5. P. McIntyre, A. W. Snyder, “Light propagation in twisted anisotropic media: application to photoreceptors,” J. Opt. Soc. Am. 68, 149–157 (1978).
    [CrossRef] [PubMed]
  6. H. K. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
    [CrossRef]
  7. H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414–1421 (1986).
    [CrossRef]
  8. L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
    [CrossRef]
  9. J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.
  10. H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
    [CrossRef]
  11. H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
    [CrossRef]
  12. F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).
  13. R. D. Mindlin, L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89–95 (1949).
    [CrossRef]
  14. A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).
  15. R. C. Jones, “A new calculus for the treatment of optical systems. VII,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  16. C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. 61, 1207–1213 (1971).
    [CrossRef]
  17. H. Takenaka, “A unified formalism for polarization optics by using group theory. I (Theory),” Jpn. J. Appl. Phys. 12, 226–231 (1973).
    [CrossRef]
  18. S. Li, “Jones-matrix analysis with Pauli matrices: application to ellipsometry,” J. Opt. Soc. Am. A 17, 920–926 (2000).
    [CrossRef]
  19. J. Cernosek, “Simple geometrical method for analysis of elliptical polarization,” J. Opt. Soc. Am. 61, 324–327 (1971).
    [CrossRef]
  20. C. K. Lee, C. P. Hu, “The geometric meaning method—quantitative determination of polarization,” in Proceedings of the Seventh International Conference on Experimental Stress Analysis (Israel Institute of Technology, Haifa, Israel, 1982), pp. 431–432.
  21. F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, Washington, D.C., 1962).
  22. A. S. Marathay, “Operator formalism in the theory of partial polarization,” J. Opt. Soc. Am. 55, 969–980 (1965).

2000

1999

1998

1997

1986

1978

1973

H. Takenaka, “A unified formalism for polarization optics by using group theory. I (Theory),” Jpn. J. Appl. Phys. 12, 226–231 (1973).
[CrossRef]

1971

1966

H. K. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[CrossRef]

1965

1962

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).

1949

R. D. Mindlin, L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89–95 (1949).
[CrossRef]

1948

1843

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).

Aben, H.

Aben, H. K.

H. K. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[CrossRef]

Ainola, L.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Cernosek, J.

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).

Goodman, L. E.

R. D. Mindlin, L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89–95 (1949).
[CrossRef]

Hu, C. P.

C. K. Lee, C. P. Hu, “The geometric meaning method—quantitative determination of polarization,” in Proceedings of the Seventh International Conference on Experimental Stress Analysis (Israel Institute of Technology, Haifa, Israel, 1982), pp. 431–432.

Jones, R. C.

Josepson, J.

H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
[CrossRef]

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

Kuske, A.

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).

Lee, C. K.

C. K. Lee, C. P. Hu, “The geometric meaning method—quantitative determination of polarization,” in Proceedings of the Seventh International Conference on Experimental Stress Analysis (Israel Institute of Technology, Haifa, Israel, 1982), pp. 431–432.

Li, S.

Marathay, A. S.

McIntyre, P.

Mindlin, R. D.

R. D. Mindlin, L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89–95 (1949).
[CrossRef]

Murnaghan, F. D.

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, Washington, D.C., 1962).

Neumann, F. E.

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).

Snyder, A. W.

Stokes, A. R.

A. R. Stokes, The Theory of the Optical Properties of Inhomogeneous Materials (Spon, London, 1963).

Takenaka, H.

H. Takenaka, “A unified formalism for polarization optics by using group theory. I (Theory),” Jpn. J. Appl. Phys. 12, 226–231 (1973).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).

Whitney, C.

Abh. Kön. Akad. Wiss. Berlin

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).

Appl. Opt.

Exp. Mech.

H. K. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[CrossRef]

J. Appl. Phys.

R. D. Mindlin, L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89–95 (1949).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

H. Takenaka, “A unified formalism for polarization optics by using group theory. I (Theory),” Jpn. J. Appl. Phys. 12, 226–231 (1973).
[CrossRef]

Optik (Stuttgart)

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).

Other

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

A. R. Stokes, The Theory of the Optical Properties of Inhomogeneous Materials (Spon, London, 1963).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

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

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).

C. K. Lee, C. P. Hu, “The geometric meaning method—quantitative determination of polarization,” in Proceedings of the Seventh International Conference on Experimental Stress Analysis (Israel Institute of Technology, Haifa, Israel, 1982), pp. 431–432.

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, Washington, D.C., 1962).

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Equations (128)

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dEdz=VE,
E=E1E2,
V=-i 12 C(ε1-ε2)dϕdz-dϕdzi 12 C(ε1-ε2).
V=-V*,
E=UE0,
U*=U-1,
dUdz=VU.
dU*dz=U*V*.
U*dUdz+dU*dz U=U*VU+U*V*U.
ddz (U*U)=0,
U*U=C,
σ0=1001.
U*U=σ0.
|det U| =1,
U=α0+iα1α2+iα3-α2+iα3α0-iα1,
α02+α12+α22+α32=1.
σ1=i00-i,σ2=01-10,σ3=0ii0.
U=α0σ0+α1σ1+α2σ2+α3σ3.
σ0σj=σjσ0=σj,σj2=-σ0,
σ1σ2=-σ2σ1=σ3,σ2σ3=-σ3σ2=σ1,
σ3σ1=-σ1σ3=σ2.
Q¯=α0σ0-α1σ1-α2σ2-α3σ3.
N(Q)=QQ¯=QQ¯=α02+α12+α22+α32
N(Q1Q2)=N(Q1)N(Q2).
Q-1=Q¯.
dQdz Q¯=W,
W=-12 C(ε1-ε2)σ1+dϕdz σ2.
(α0σ0+α1σ1+α2σ2+α3σ3)(α0σ0-α1σ1-α2σ2
-α3σ3)=-12 C(ε1-ε2)σ1+dϕdz σ2.
α0α0+α1α1+α2α2+α3α3=0,
-α1α0+α0α1-α3α2+α2α3=-12 C(ε1-ε2),
-α2α0+α3α1+α0α2-α1α3=dϕdz,
-α3α0-α2α1+α1α2+α0α3=0.
α12α0α1+α32α2α3=12 C(ε1-ε2),
α22α0α2+α12α3α1=-dϕdz,
α32α0α3+α22α1α2=0.
{(α02+αj2)+(αk2+αl2)}=1,
α0=cos ρ1cos ρ2,
αj=cos ρ1sin ρ2,
αk=sin ρ1cos ρ3,
αl=sin ρ1sin ρ3.
-π<ρ1π,-π2<ρ2π2,-π2<ρ3π2.
{α02+[αj2+(αk2+α12)]}=1.
α0=cos ρ1,
αj=sin ρ1cos ρ2,
αk=sin ρ1sin ρ2cos ρ3,
αl=sin ρ1sin ρ2sin ρ3,
α0=cos ρ1,
αj=sin ρ1sin ρ2,
αk=sin ρ1cos ρ2cos ρ3,
αl=sin ρ1cos ρ2sin ρ3.
{[α02+(αj2+αk2)]+αl2}=1.
α0=cos ρ1cos ρ2,
αj=cos ρ1sin ρ2cos ρ3,
αk=cos ρ1sin ρ2sin ρ3,
αl=sin ρ1.
{[(α02+αj2)+αk2]+αl2}=1,
α0=cos ρ1cos ρ2cos ρ3,
αj=cos ρ1cos ρ2sin ρ3,
αk=cos ρ1sin ρ2,
αl=sin ρ1.
Qj=α0σ0+αjσj(j=1, 2, 3).
Qj(μ)=(cos μ)σ0+(sin μ)σj.
Qj-1(μ)=Qj(-μ),
Qj(μ)Qj(ν)=Qj(μ+ν).
U1=exp(iμ)00exp(-iμ),
U2=cos μsin μ-sin μcos μ,
U3=cos μi sin μi sin μcos μ.
U3(μ)=U2-π4U1(μ)U2π4
U3(μ)=U1-π4U2(μ)U1-π4,
Qjk=α0σ0+αjσj+αkσk,
α02+αj2+αk2=1.
[α02+(αj2+αk2)]=1,
α0=cosν1,
αj=sinν1cosν2,
αk=sinν1sinν2.
[(α02+αj2)+αk2]=1,
α0=cosν1cosν2,
αj=cosν1sinν2,
αk=sinν1.
Qjk(1)=Ql-jlkν22Qj(ν1)Qljlkν22,
Qjk(2)=Qjν22Qk(ν1)Qjν22.
Qjkl(1)=Qj-kjlρ32Qjρ22Qk(ρ1)Qjρ22Qjkjlρ32.
Qjkl(1)=Qjρ22-kjlρ32Qk(ρ1)Qjρ22+kjlρ32.
Qjkl(2)=Qj-jlkρ32Ql-jlkρ22Qj(ρ1)×Qljlkρ22Qjjlkρ32,
Qjkl(3)=Qjjlkρ32 Ql-jlkρ22Qk(ρ1)×Qljklρ22Qj-jklρ32,
Qjkl(4)=Qk-kjlρ32Qjρ22Ql(ρ1)×Qjρ22Qkkjlρ22,
Qjkl(5)=Qjρ32Qkρ22Ql(ρ1)Qkρ22Qjρ32.
12 (ρ2+ρ3)-12 (cos 2ρ1)(ρ3-ρ2)=-12 C(ε1-ε2),
[cos(ρ2+ρ3)]ρ1-12 [sin 2ρ1sin(ρ2+ρ3)](ρ3-ρ2)
=dϕdz,
[sin(ρ2+ρ3)]ρ1+12 [sin 2ρ1cos(ρ2+ρ3)](ρ3-ρ2)
=0.
λ=ρ2+ρ32,κ=ρ3-ρ22,θ=ρ1
λ+(tan 2λ cot 2θ)θ=-12 C(ε1-ε2),
1cos 2λ θ=dϕdz.
Q123(1)=Q1(λ)Q2(θ)Q1(-κ).
1cos 2κ θ=-12 C(ε1-ε2),
κ+(tan 2κ cot 2θ)θ=-dϕdz.
Q213(1)=Q2(-κ)Q1(θ)Q2(λ).
(cos 2λ)θ-(tan 2θ sin 2λ)λ=-12 C(ε1-ε2),
(sin 2λ)θ+(tan 2θ cos 2λ)λ=-dϕdz.
Q312(1)=Q3(λ)Q1(θ)Q3(-κ).
1sin 2λ θ=-12 C(ε1-ε2),
λ-(cot 2λ cot 2θ)θ=-dϕdz.
Q231(1)=Q2(λ)Q3(θ)Q2(-κ).
κ-(cot 2θ cot 2κ)θ=12 C(ε1-ε2),
1sin 2κ θ=-dϕdz.
Q132(1)=Q1(-κ)Q3(θ)Q1(λ).
(sin κ)θ+(tan 2θ cos 2κ)κ=-12 C(ε1-ε2),
-(cos κ)θ+(tan 2θ sin 2κ)κ=-dϕdz.
Q321(1)=Q3(-κ)Q2(θ)Q3(λ).
(cos ρ2)ρ1-12 (sin 2ρ1sin ρ2)ρ2-(sin2 ρ1sin2 ρ2)ρ3
=-12 C(ε1-ε2),
(sin ρ2sin ρ3)ρ1-(sin ρ1)(sin ρ1cos ρ3-cos ρ1cos ρ2sin ρ3)ρ2+(sin ρ1sin ρ2)×(sin ρ1cos ρ2sin ρ3+cos ρ1cos ρ3)ρ3
=dϕdz,
(sin ρ2cos ρ3)ρ1+(sin ρ1)(cos ρ1cos ρ2cos ρ3+sin ρ1sin ρ3)ρ2-(sin ρ1sin ρ2)(cos ρ1sin ρ3
-sin ρ1cos ρ2cos ρ3)ρ3=0.
Q132(2)=Q1-ρ32Q2-ρ22Q1(ρ1)Q2ρ22Q1ρ32.
(sin ρ2)ρ1-12 (sin 2ρ1cos ρ2)ρ2+(cos2 ρ1cos2 ρ2)ρ3
=-12 C(ε1-ε2),
(cos2 ρ1cos ρ2cos ρ3)ρ1-(cos ρ1cos ρ3)(cos ρ1+sin ρ1sin ρ2)ρ2+(cos ρ1cos ρ2sin ρ3)(sin ρ1
-cos ρ1sin ρ2)ρ3=-dϕdz,
(cos ρ2cos ρ3)ρ1+(cos ρ1cos ρ3)(cos ρ1+sin ρ1sin ρ2)ρ2+(cos ρ1cos ρ2sin ρ3)(sin ρ1+cos ρ1sin ρ2)ρ3
=0.
Q123(5)=Q1ρ32Q2ρ22Q3(ρ1)Q2ρ22Q3ρ32.

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