Abstract

Birefringent crystals are extensively used to manipulate polarized light. The generalized transfer matrix developed allows efficient calculation of the full polarization state of light transmitted through and reflected by a stack of arbitrarily many discrete layers of isotropic and birefringent materials at any frequency and angle of incidence. The matrix of a uniaxial birefringent crystal with arbitrary rotation is calculated, along with its reduction to the matrix of an isotropic medium. This method is of great practical importance where tight control of systematic effects is needed in optical systems employing birefringent crystals, one example being wave plates used by cosmic microwave background polarimetry with wide field-of-view telescopes.

© 2013 Optical Society of America

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Corrections

Thomas Essinger-Hileman, "Transfer matrix for treating stratified media including birefringent crystals: erratum," Appl. Opt. 53, 4264-4265 (2014)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-53-19-4264

References

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  1. D. H. Goldstein, Polarized Light (CRC, 2011).
  2. J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).
  3. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  4. S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. A 41, 137–144 (1955).
  5. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).
  6. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  7. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).
  8. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  9. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  10. T. M. Essinger-Hileman, “Probing inflationary cosmology: the atacama B-mode search (ABS),” Ph.D. thesis (Princeton University, 2011).
  11. Note that this ordering of the Pauli matrices is different from that often used in the literature: σI=(1001)σQ=(100−1)σU=(0110)σV=(0−ii0).
  12. W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
    [CrossRef]
  13. S. A. Bryan, T. E. Montroy, and J. E. Ruhl, “Modeling dielectric half-wave plates for cosmic microwave background polarimetry using a Mueller matrix formalism,” Appl. Opt. 49, 6313–6323 (2010).
    [CrossRef]
  14. H. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983).
  15. G. Szivessy, Handbuch der Physik, Vol. 20 (Julius Springer, 1928), Chap. 11, p. 715.

2010 (1)

2007 (1)

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

1979 (1)

1972 (1)

1955 (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. A 41, 137–144 (1955).

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).

Berreman, D. W.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bryan, S. A.

Chen, H.

H. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983).

Contaldi, C. R.

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Crill, B. P.

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Essinger-Hileman, T. M.

T. M. Essinger-Hileman, “Probing inflationary cosmology: the atacama B-mode search (ABS),” Ph.D. thesis (Princeton University, 2011).

Goldstein, D. H.

D. H. Goldstein, Polarized Light (CRC, 2011).

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

Jones, W. C.

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Kisner, T. S.

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Kliger, D. S.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Lange, A. E.

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Lewis, J. W.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

MacTavish, C. J.

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Montroy, T. E.

S. A. Bryan, T. E. Montroy, and J. E. Ruhl, “Modeling dielectric half-wave plates for cosmic microwave background polarimetry using a Mueller matrix formalism,” Appl. Opt. 49, 6313–6323 (2010).
[CrossRef]

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Netterfield, C. B.

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. A 41, 137–144 (1955).

Randall, C. E.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Ruhl, J. E.

S. A. Bryan, T. E. Montroy, and J. E. Ruhl, “Modeling dielectric half-wave plates for cosmic microwave background polarimetry using a Mueller matrix formalism,” Appl. Opt. 49, 6313–6323 (2010).
[CrossRef]

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

Szivessy, G.

G. Szivessy, Handbuch der Physik, Vol. 20 (Julius Springer, 1928), Chap. 11, p. 715.

Tinbergen, J.

J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Yeh, P.

Ann. Phys. (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés,” Ann. Phys. 5, 596–640 (1950).

Appl. Opt. (1)

Astron. Astrophys. (1)

W. C. Jones, T. E. Montroy, B. P. Crill, C. R. Contaldi, T. S. Kisner, A. E. Lange, C. J. MacTavish, C. B. Netterfield, and J. E. Ruhl, “Instrumental and analytic methods for bolometric polarimetry,” Astron. Astrophys. 470, 771–785 (2007).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. A 41, 137–144 (1955).

Other (9)

H. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, 1983).

G. Szivessy, Handbuch der Physik, Vol. 20 (Julius Springer, 1928), Chap. 11, p. 715.

D. H. Goldstein, Polarized Light (CRC, 2011).

J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

T. M. Essinger-Hileman, “Probing inflationary cosmology: the atacama B-mode search (ABS),” Ph.D. thesis (Princeton University, 2011).

Note that this ordering of the Pauli matrices is different from that often used in the literature: σI=(1001)σQ=(100−1)σU=(0110)σV=(0−ii0).

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Figures (1)

Fig. 1.
Fig. 1.

Geometry of the rays used to calculate the generalized transfer matrix for the uniaxial crystal. The two polarization states defined by the plane of incidence, the s and p waves, are mixed inside the uniaxial crystal into the ordinary and extraordinary waves.

Equations (31)

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P=EE=IσI+QσQ+UσU+VσV,
Mij=12Tr(σiJσjJ),
((EipErp)n12cosθ1n1(Eip+Erp)n12(Eis+Ers)(EisErs)(n1cosθ1))=T(Etpn32cosθ3n3Etpn32EtsEts(n3cosθ3)).
α=(t11n32cosθ3+t12n3)/(n12cosθ1),β=(t13n32+t14n3cosθ3)/(n12cosθ1),γ=(t21n32cosθ3+t22n3)/n1,δ=(t23n32+t24n3cosθ3)/n1,η=(t31n32cosθ3+t32n3)/n12,κ=(t33n32+t34n3cosθ3)/n12,ρ=(t41n32cosθ3+t42n3)/(n1cosθ1),σ=(t43n32+t44n3cosθ3)/(n1cosθ1),Γ=[(α+γ)(κ+σ)(β+δ)(η+ρ)]1.
(EtpEts)=(J11tJ12tJ21tJ22t)(EipEis)=2Γ(κ+σβδηρα+γ)(EipEis).
(ErpErs)=(J11rJ12rJ21rJ22r)(EipEis),
J11r=Γ((γα)(κ+σ)(δβ)(η+ρ))J12r=Γ(αδγβ)J21r=Γ(ησρκ)J22r=Γ((α+γ)(κσ)(β+δ)(ηρ)).
ε=R(χ)(ne2000no2000no2)R(χ),
R(χ)=(cosχsinχ0sinχcosχ0001).
|H|=1n|D|.
H=1nk^×D.
v=vo=c/no,
v=cn=(vo2cos2ψ+ve2sin2ψ)1/2=c(cos2ψno2+sin2ψne2)1/2.
nosinθ=nsinθ=n1sinθ1.
n=(cos2ψno2+sin2ψne2)1/2=none[(ne2no2)sin2θcos2χ+no2]1/2,
sinθ=n1sinθ1none[(ne2no2)sin2θcos2χ+no2]1/2.
sinθ=nonesinθ1[no2ne2+n12(no2ne2)sin2θ1cos2χ]1/2.
k^=(sinθ,0,cosθ);k^=(sinθ,0,cosθ).
D^=D|D|=αk^×ξ^=α(sinχcosθcosχcosθsinχsinθ).
D^=D/|D|=αk^×D^=α(cosχcosθcosθsinχ[sinθsinθ+cosθcosθ]cosχcosθsinθ).
δ=n˜otcosθ,δ=n˜tcosθ,
DiII=DtIexp(ik0δ),DiII=DtIexp(ik0δ).
DrII=DrIIexp(ik0δ),DrII=DrIIexp(ik0δ).
DtI=|DtI|(D^tI(x),D^tI(y),D^tI(z)),DtI=|DtI|(D^tI(x),D^tI(y),D^tI(z)),HtI=1n|DtI|(H^tI(x),H^tI(y),H^tI(z)),HtI=1n|DtI|(H^tI(x),H^tI(y),H^tI(z)),
DrII=|DrII|(D^tI(x),D^tI(y),D^tI(z)),DrII=|DrII|(D^tI(x),D^tI(y),D^tI(z)),HrII=1n|DrII|(H^tI(x),H^tI(y),H^tI(z)),HrII=1n|DrII|(H^tI(x),H^tI(y),H^tI(z)).
ΦI=(D^tI(x)D^tI(x)D^tI(x)D^tI(x)1nH^tI(y)1nH^tI(y)1nH^tI(y)1nH^tI(y)D^tI(y)D^tI(y)D^tI(y)D^tI(y)1nH^tI(x)1nH^tI(x)1nH^tI(x)1nH^tI(x)),
P=(exp(Δ)0000exp(Δ)0000exp(Δ)0000exp(Δ)),
ΛI=ΦI(PΦI)1ΛII.
D^tI=(sinχcosθcosχcosθ,sinχsinθ)/[cos2θ+sin2θsin2χ]1/2D^tI=(cosχcosθcosθ,sinχ(sinθsinθ+cosθcosθ,cosχcosθsinθ)/[cos2χcos2θ+sin2χcos2(θθ)]1/2H^tI=(cos2θcosχ,sinχ,cosθsinθcosχ)/[cos2θcos2χ+sin2χ]1/2H^tI=(cos(θθ)cosθsinχ,cosθcosχ,cos(θθ)sinθsinχ)/[cos2(θθ)sin2χ+cos2θcos2χ]1/2.
(cosk0δinsink0δcosθ00isink0δncosθcosk0δ0000cosk0δinsink0δcosθ00isink0δcosθncosk0δ).
σI=(1001)σQ=(1001)σU=(0110)σV=(0ii0).

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