Abstract

In a cascaded system comprising a combination of oblique retarders, the effect of optical rotation is observed in addition to the overall retardation. This shows that the combined system does not behave as a pure oblique retarder. Analyzing such a general system using Pauli Spin matrices, it is shown that the effect of optical rotation may be completely annulled through the use of a suitably oriented retarder at the output of the cascaded system. The analysis assumes monochromaticity of the illuminating light beam.

© 2012 Optical Society of America

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References

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  1. R. C. Jones, “A new calculus for the treatment of optical systems III. The Sohncke theory of optical activity,” J. Opt. Soc. Am. 31, 500–503 (1941).
    [CrossRef]
  2. H. Hurwitz and R. C. Jones, “A new calculus for the treatment of optical systems II. Proof of three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  3. S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).
  4. Md. Tahir, K. Bhattacharya, and A. K. Chakraborty, “Use of Dirac matrices in polarization optics,” Optik 121, 1840–1844 (2010).
    [CrossRef]
  5. M. Reimer and D. Yevick, “A Clifford algebra analysis of polarization-mode dispersion and polarization-dependent loss,” Photon. Technol. Lett. 18, 734–736 (2006).
    [CrossRef]
  6. J. N. Damask, “The spin-vector calculus of polarization,” in Polarization Optics in Telecommunications (Springer, 2004), pp. 52–61.

2010

Md. Tahir, K. Bhattacharya, and A. K. Chakraborty, “Use of Dirac matrices in polarization optics,” Optik 121, 1840–1844 (2010).
[CrossRef]

2006

M. Reimer and D. Yevick, “A Clifford algebra analysis of polarization-mode dispersion and polarization-dependent loss,” Photon. Technol. Lett. 18, 734–736 (2006).
[CrossRef]

1955

S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).

1941

Bhattacharya, K.

Md. Tahir, K. Bhattacharya, and A. K. Chakraborty, “Use of Dirac matrices in polarization optics,” Optik 121, 1840–1844 (2010).
[CrossRef]

Chakraborty, A. K.

Md. Tahir, K. Bhattacharya, and A. K. Chakraborty, “Use of Dirac matrices in polarization optics,” Optik 121, 1840–1844 (2010).
[CrossRef]

Damask, J. N.

J. N. Damask, “The spin-vector calculus of polarization,” in Polarization Optics in Telecommunications (Springer, 2004), pp. 52–61.

Hurwitz, H.

Jones, R. C.

Pancharatnam, S.

S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).

Reimer, M.

M. Reimer and D. Yevick, “A Clifford algebra analysis of polarization-mode dispersion and polarization-dependent loss,” Photon. Technol. Lett. 18, 734–736 (2006).
[CrossRef]

Tahir, Md.

Md. Tahir, K. Bhattacharya, and A. K. Chakraborty, “Use of Dirac matrices in polarization optics,” Optik 121, 1840–1844 (2010).
[CrossRef]

Yevick, D.

M. Reimer and D. Yevick, “A Clifford algebra analysis of polarization-mode dispersion and polarization-dependent loss,” Photon. Technol. Lett. 18, 734–736 (2006).
[CrossRef]

J. Opt. Soc. Am.

Optik

Md. Tahir, K. Bhattacharya, and A. K. Chakraborty, “Use of Dirac matrices in polarization optics,” Optik 121, 1840–1844 (2010).
[CrossRef]

Photon. Technol. Lett.

M. Reimer and D. Yevick, “A Clifford algebra analysis of polarization-mode dispersion and polarization-dependent loss,” Photon. Technol. Lett. 18, 734–736 (2006).
[CrossRef]

Proc. Indian Acad. Sci.

S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955).

Other

J. N. Damask, “The spin-vector calculus of polarization,” in Polarization Optics in Telecommunications (Springer, 2004), pp. 52–61.

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Figures (3)

Fig. 1.
Fig. 1.

Configuration of the three birefringent plates.

Fig. 2.
Fig. 2.

Graphical solution for test case 1.

Fig. 3.
Fig. 3.

Graphical solution for test case 2.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

L=C(δ3,ϕ3)C(δ2,ϕ2)C(δ1,ϕ1).
L=C(δ3,ϕ2)C(δ2,ϕ1)C(δ1,0).
C(δ,ϕ)=|cosδ2+isinδ2cos2ϕisinδ2sin2ϕisinδ2sin2ϕcosδ2isinδ2cos2ϕ|.
C(Δ,ψ)=|cosδ32+isinδ32cos2ϕ2isinδ32sin2ϕ2isinδ32sin2ϕ2cosδ32isinδ32cos2ϕ2||cosδ22+isinδ22cos2ϕ1isinδ22sin2ϕ1isinδ22sin2ϕ1cosδ22isinδ22cos2ϕ1||eiδ1/200eiδ1/2|=|PQRS|.
P=[cosδ12cosδ22cosδ32cosδ12sinδ22sinδ32cos(2ϕ22ϕ1)sinδ12cosδ22sinδ32cos2ϕ2sinδ12sinδ22cosδ32cos2ϕ1]+i[cosδ12cosδ22sinδ32cos2ϕ2+cosδ12sinδ22cosδ32cos2ϕ1+sinδ12cosδ22cosδ32sinδ12sinδ22sinδ32cos(2ϕ22ϕ1)]
=a+ib,
Q=[cosδ12sinδ22sinδ32sin(2ϕ22ϕ1)+sinδ12sinδ22cosδ32sin2ϕ1+sinδ12cosδ22sinδ32sin2ϕ2]+i[cosδ12sinδ22cosδ32sin2ϕ1+cosδ12cosδ22sinδ32sin2ϕ2sinδ12sinδ22sinδ32sin(2ϕ22ϕ1)]
=c+id,
R=[cosδ12sinδ22sinδ32sin(2ϕ22ϕ1)+sinδ12sinδ22cosδ32sin2ϕ1+sinδ12cosδ22sinδ32sin2ϕ2]+i[cosδ12sinδ22cosδ32sin2ϕ1+cosδ12cosδ22sinδ32sin2ϕ2sinδ12sinδ22sinδ32sin(2ϕ22ϕ1)]
=c+id,
S=[cosδ12cosδ22cosδ32cosδ12sinδ22sinδ32cos(2ϕ22ϕ1)sinδ12cosδ22sinδ32cos2ϕ2sinδ12sinδ22cosδ32cos2ϕ1]i[cosδ12cosδ22sinδ32cos2ϕ2+cosδ12sinδ22cosδ32cos2ϕ1+sinδ12cosδ22cosδ32sinδ12sinδ22sinδ32cos(2ϕ22ϕ1)]
=aib.
L=|PQRS|=|a+ibc+idc+idaib|.
σ0=|1001|,σ1=|1001|,σ2=|0110|,σ3=|0ii0|.
σiσj+σjσi=0(i=1,2,3),
σiσjσjσi=2iσk(i=1,2,3andcycle),
σi2=σ0,
σiσ0=σ0σi=σi(i=1,2,3),
σ1σ2=iσ3,
σ2σ1=iσ3,
σ1σ3=iσ2,
σ3σ1=iσ2,
σ2σ3=iσ1,
σ3σ2=iσ1.
L=|PQRS|=a|1001|+ib|1001|+id|0110|+(ic)|0ii0|
=l0σ0+l1σ1+l2σ2+l3σ3=i=03liσi,
l0=a,l1=ib,l2=id,l3=ic.
C(δ,ϕ)=cosδ2|1001|+isinδ2cos2ϕ|1001|+isinδ2sin2ϕ|0110|
=A0σ0+A1σ1+A2σ2+A3σ3,
A0=cosδ2,A1=isinδ2cos2ϕ,A2=isinδ2sin2ϕ,A3=0.
R(θ)=|cosθsinθsinθcosθ|=σ0cosθiσ3sinθ.
M=α0σ0+α1σ1+α2σ2+α3σ3.
D=ML=(α0σ0+α1σ1+α2σ2+α3σ3)(l0σ0+l1σ1+l2σ2+l3σ3)=(α0l0+α1l1+α2l2+α3l3)σ0+(α0l1+α1l0+iα3l2iα2l3)σ1+(α0l2+α2l0+iα1l3iα3l1)σ2+(α0l3+α3l0+iα2l1iα1l2)σ3.
α0l3+α3l0+i(α2l1α1l2)=0.
icα0+aα3+i(ibα2idα1)=0
(aα3bα2+dα1)icα0=0.
i(dsinδ42cos2ϕ4bsinδ42sin2ϕ4ccosδ42)=0.
L=|0.4915+0.4590i0.4131+0.6141i0.4131+0.6141i0.49150.4590i|.
D=|0.1092+0.5469i0.8301i0.8301i0.10920.5469i|.
L=|0.8201+0.3666i0.0942+0.4292i0.0942+0.4292i0.82010.3666i|.
D=|0.8643+0.4226i0.2728i0.2728i0.86430.4226i|.

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