Abstract

This paper demonstrates that numerous calculations involving polarization transformations can be condensed by employing suitable geometric algebra formalism. For example, to describe polarization mode dispersion and polarization-dependent loss, both the material birefringence and differential loss enter as bivectors and can be combined into a single symmetric quantity. Their frequency and distance evolution, as well as that of the Stokes vector through an optical system, can then each be expressed as a single compact expression, in contrast to the corresponding Mueller matrix formulations. The intrinsic advantage of the geometric algebra framework is further demonstrated by presenting a simplified derivation of generalized Stokes parameters that include the electric field phase. This procedure simultaneously establishes the tensor transformation properties of these parameters.

© 2014 Optical Society of America

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References

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  1. N. J. Frigo and F. Bucholtz, “Geometrical representation of optical propagation phase,” J. Lightwave Technol. 27, 3283–3293 (2009).
    [CrossRef]
  2. N. J. Frigo, F. Bucholtz, and C. V. McLaughlin, “Polarization in phase-modulated optical links: Jones and generalized Stokes-space analysis,” J. Lightwave Technol. 31, 1503–1511 (2013).
    [CrossRef]
  3. M. Karlsson and M. Peterson, “Quaternion approach to PMD and PDL phenomena in optical fiber systems,” J. Lightwave Technol. 22, 1137–1146 (2004).
    [CrossRef]
  4. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
    [CrossRef]
  5. M. Reimer and D. Yevick, “A Clifford algebra analysis of polarization mode dispersion and polarization-dependent loss,” IEEE Photon. Technol. Lett. 18, 734–736 (2006).
    [CrossRef]
  6. C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).
  7. W. Baylis, ed. Clifford (Geometric) Algebras with Applications in Physics, Mathematics and Engineering (Birkhauser, 1996).
  8. J. W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory (Wiley, 2011).
  9. E. Collett, Polarized Light in Fiber Optics (The PolaWave Group, 2003).
  10. D. Han, Y. S. Kim, and M. E. Noz, “Stokes paramaters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
    [CrossRef]
  11. J. N. Damask, “Properties of polarization-dependent loss and polarization mode dispersion,” in Polarization Optics in Telecommunications (Springer, 2004), Chap. 8.
  12. M. Reimer and D. Yevick, “Mueller matrix description of polarization mode dispersion and polarization-dependent loss,” J. Opt. Soc. Am. A 23, 1503–1508 (2006).
    [CrossRef]
  13. M. Karlsson, “The connection between polarization calculus and four-dimensional rotations,” arXiv:1303.1836v1 (2013).
  14. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2002), Appendix A, p. 601.
  15. D. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).

2013 (1)

2009 (1)

2006 (2)

M. Reimer and D. Yevick, “Mueller matrix description of polarization mode dispersion and polarization-dependent loss,” J. Opt. Soc. Am. A 23, 1503–1508 (2006).
[CrossRef]

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

2004 (1)

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef]

1997 (1)

D. Han, Y. S. Kim, and M. E. Noz, “Stokes paramaters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Arthur, J. W.

J. W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory (Wiley, 2011).

Bucholtz, F.

Collett, E.

E. Collett, Polarized Light in Fiber Optics (The PolaWave Group, 2003).

Damask, J. N.

J. N. Damask, “Properties of polarization-dependent loss and polarization mode dispersion,” in Polarization Optics in Telecommunications (Springer, 2004), Chap. 8.

Doran, C.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).

Frigo, N. J.

Goldstein, H.

H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2002), Appendix A, p. 601.

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef]

Griffiths, D.

D. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).

Han, D.

D. Han, Y. S. Kim, and M. E. Noz, “Stokes paramaters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Karlsson, M.

M. Karlsson and M. Peterson, “Quaternion approach to PMD and PDL phenomena in optical fiber systems,” J. Lightwave Technol. 22, 1137–1146 (2004).
[CrossRef]

M. Karlsson, “The connection between polarization calculus and four-dimensional rotations,” arXiv:1303.1836v1 (2013).

Kim, Y. S.

D. Han, Y. S. Kim, and M. E. Noz, “Stokes paramaters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef]

Lasenby, A.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).

McLaughlin, C. V.

Noz, M. E.

D. Han, Y. S. Kim, and M. E. Noz, “Stokes paramaters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Peterson, M.

Poole, C.

H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2002), Appendix A, p. 601.

Reimer, M.

M. Reimer and D. Yevick, “Mueller matrix description of polarization mode dispersion and polarization-dependent loss,” J. Opt. Soc. Am. A 23, 1503–1508 (2006).
[CrossRef]

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

Safko, J.

H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2002), Appendix A, p. 601.

Yevick, D.

M. Reimer and D. Yevick, “Mueller matrix description of polarization mode dispersion and polarization-dependent loss,” J. Opt. Soc. Am. A 23, 1503–1508 (2006).
[CrossRef]

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

IEEE Photon. Technol. Lett. (1)

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

J. Lightwave Technol. (3)

J. Opt. Soc. Am. A (1)

Phys. Rev. E (1)

D. Han, Y. S. Kim, and M. E. Noz, “Stokes paramaters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef]

Other (8)

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).

W. Baylis, ed. Clifford (Geometric) Algebras with Applications in Physics, Mathematics and Engineering (Birkhauser, 1996).

J. W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory (Wiley, 2011).

E. Collett, Polarized Light in Fiber Optics (The PolaWave Group, 2003).

J. N. Damask, “Properties of polarization-dependent loss and polarization mode dispersion,” in Polarization Optics in Telecommunications (Springer, 2004), Chap. 8.

M. Karlsson, “The connection between polarization calculus and four-dimensional rotations,” arXiv:1303.1836v1 (2013).

H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2002), Appendix A, p. 601.

D. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).

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Tables (3)

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Table 1. Geometric Algebra Element Construction

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Table 2. Even Subalgebra of Cl(1,3) and Isomorphic Components in Cl(0,3)

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Table 3. Jones Matrices and Corresponding Rotors

Equations (66)

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[M1,M2]=12(M1M2M2M1)=M1×M2,
M¯=r˜Mr,
r=exp(φ2γ2γ3)=cos(φ/2)+γ2γ3sin(φ/2),
e¯=Ue.
S¯=MS,
S=γ0S0+γ1S1+γ2S2+γ3S3,
(cos(θ/2)sin(θ/2)eiδsin(θ/2)eiδcos(θ/2))
r=(θ,δ)=exp(θ2(γ1γ2cosδ+γ1γ3sinδ))=cos(θ2)+sin(θ2)(γ1γ2cosδ+γ1γ3sinδ)
(exp(iφ/2)00exp(φ/2))
rφ(φ)=exp(φ2γ3γ2)=cos(φ2)+γ3γ2sin(φ2)
(exp(α/2)00exp(α/2))
rα(α)=exp(α2γ0γ1)=cosh(α2)+γ0γ1sinh(α2)
S¯=r˜Sr.
S¯ω=r˜ωrS¯+S¯r˜rω.
S¯ω=r˜ωrS¯S¯r˜ωr.
r˜ωr=12Ω,
S¯ω=[Ω,S¯]=Ω×S¯.
Ω=K1γ1γ0+K2γ2γ0+K3γ3γ0+K1γ2γ3+K2γ3γ1+K3γ1γ2.
γ0(I00I),γ1(0σ1σ10),γ2(0σ2σ20),γ3(0σ3σ30),
σ1=(1001),σ2=(0110),σ3=(0ii0),I=(1001),
Ω(iK⃗σ⃗K⃗σ⃗K⃗σ⃗iK⃗σ⃗),
S¯(S0IS⃗σ⃗S⃗σ⃗S0I).
S0ω=K⃗S⃗S⃗ω=K⃗×S⃗+S0K⃗.
S¯(z+Δz)=r˜(θ,δ)r˜α(αΔz)r˜φ(βΔz)r˜(θ,δ)×S¯(z)r(θ,δ)r˜φ(βΔz)rα(αΔz)r(θ,δ).
S¯z=Bβα×S¯,
Bβα=β+α=γ2γ3βcosθ+γ3γ1βsinθcosδ+γ3γ1βsinθsinδ+γ1γ0αcosθ+γ2γ0αsinθcosδ+γ3γ0αsinθsinδ,
Ωz=Bβαω+Bβα×Ω,
SA=F1γ1γ0+F2γ2γ0+F3γ3γ0+G1γ3γ2+G2γ1γ3+G3γ2γ1=F+IG,
F⃗=(2Re(exey)Re(ex2ey2)Im(ex2+ey2))TG⃗=(2Im(exey)Im(ex2ey2)Re(ex2+ey2))T.
Γt=Γtγ0Γtγ02Γ0=Γtγ0Γtγ0γ0Γt+Γtγ0.
Γz=β×Γ+α×γ012((Γ×γ0)α+α(Γ×γ0))Γ.
αv=αcosθγ1+αsinθcosδγ2+αsinθsinδγ3=αxγ1+αyγ2+αzγ3,
Γz=β×Γ+αv12(Γαv+αvΓ)Γ,
Γω=Ωr×Γ+Λv12(ΓΛv+ΛvΓ)Γ,
Ωt=r˜Ωsr,
Ωtω=r˜Ωsωr,Ωtωω=r˜Ωsωωr+Ωt×Ωtω.
Ωt=Ω2+r˜2Ω1r2.
Ωt=Ω2ω+r˜2Ω1ωr2+Ω2×Ωt.
r˜ω=12Ωr˜,
r˜=exp(j=0Nj(ω))r˜0.
N1(ω)=12ω0ωdω1Ω(ω1)N2(ω)=14ω0ωdω1ω0ω1dω2Ω(ω1)×Ω(ω2).
N1(ω)=12(ΩrΔω+Δω22!Ω˙r+Δω33!Ω¨r+ΛΔω+Δω22!Λ˙+Δω33!Λ¨)N2(ω)=14Δω33!(Ωr×Ω˙r+Ωr×Λ˙+Λ×Ω˙r+Λ×Λ˙),
S¯=12(S+γ0Sγ0)+a2(Sγ0Sγ0).
C=(exey)(eyex)=(exeyex2ey2eyex)
UCU1.
C=(exex*exey*eyex*eyey*).
C=12(F⃗σ⃗+iG⃗σ⃗).
Q=(χρρ*χ*).
(χχ*ρρ*(χ*ρ+χρ*)i(χρ*ρχ*)(χρ+χ*ρ*)12(χ2ρ*2+χ*2ρ2)i2(χ2ρ*2χ*2+ρ2)i(χρχ*ρ*)i2(χ2ρ*2χ*2ρ2)12(χ2+ρ*2+χ*2+ρ2)).
J=(exey*eyex*),
J¯=UJ=(χρρ*χ*)(exey*eyex*),
E¯s=REs=R(S1F1G1S2F2G2S3F3G3),
e¯4,out=(U00W)(eeoth).
e¯4e¯4=(U00W)e4e4(U00W).
e4e4=(exex*exey*exeyex2ex*eyeyey*ey2exeyex*ey*ey*2eyey*exey*ex*2ex*ey*ex*eyexex*).
F¯1=F1F¯2=F2coshα+G3sinhαF¯3=F3coshαG2sinhαG¯1=G1G¯2=G2coshαF3sinhαG¯3=G3coshα+F2sinhα.
sμv=(0F1F2F3F10G3G2F2G30G1F3G2G10),
S¯μν=MλμMηνSλη.
S¯A=MSAMT.
(S02+S12+S22+S322S0S12S0S22S0S32S0S1S02+S12S22S322S1S22S1S32S0S22S1S2S02S12+S22S322S2S32S0S32S1S32S2S3S02S12S22+S32).
S¯ωA=MωM1S¯A+S¯A(MωM1)T.
MωM1=(0Λ⃗TΛ⃗Ω⃗r×)
(0F⃗¯ωTF⃗¯ωG⃗¯ω×)=(0Λ⃗TΛ⃗Ω⃗r×)(0F⃗¯TF⃗¯G⃗¯×)+(0F⃗¯TF⃗¯G⃗¯×)(0Λ⃗TΛ⃗Ω⃗r×).
F⃗ω=Ω⃗r×F⃗Λ⃗×G⃗G⃗ω=Ω⃗r×G⃗+Λ⃗×F⃗,
MzM1=(0α⃗Tα⃗β⃗×),
F⃗z=β⃗×F⃗α⃗×G⃗G⃗z=β⃗×G⃗+α⃗×F⃗.

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