Abstract

A coherence-based description of the Lyot depolarizer illuminated by polychromatic light of any spectral density distribution is proposed as a generalization of the formulas provided for symmetrical spectra by Burns [J. Lightwave Technol. 1, 475 (1983) [CrossRef]  ] and Mochizuki [Appl. Opt. 23, 3284 (1984) [CrossRef]  ]. The structure of the derived expressions is explained in physical terms, and a numerical comparison with the previous solutions is performed. The results of the numerical analysis show that the proposed description, when applied to any configuration of a two-segment anisotropic depolarizer, is fully equivalent with the Mueller–Stokes calculus for broadband light. Following this consistency, the range of accuracy of the formula by Mochizuki has been verified.

© 2012 Optical Society of America

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References

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  1. D. H. Goldstein and E. Collett, Polarized Light (Dekker, 2003).
  2. P. Shen and J. C. Palais, “Passive single-mode fiber depolarizer,” Appl. Opt. 38, 1686–1691 (1999).
    [CrossRef]
  3. M. Martinelli and J. Palais, “Dual fiber-ring depolarizer,” J. Lightwave Technol. 19, 899–905 (2001).
    [CrossRef]
  4. B. Lyot, “Recherche sur la polarisationde la lumière des planètes et de quelques substances terrestres,” in Annales de l’Observatoire de Paris (Meudon), Tome VIII, Facs. I (1929).
  5. B. H. Billings, “A monochromatic depolarizer,” J. Opt. Soc. Am. 41, 966–968 (1951).
    [CrossRef]
  6. A. P. Loeber, “Depolarization of white light by a birefringent crystal. II. The Lyot depolarizer,” J. Opt. Soc. Am. 72, 650–656 (1982).
    [CrossRef]
  7. W. K. Burns, “Degree of polarization in the Lyot depolarizer,” J. Lightwave Technol. 1, 475–479 (1983).
    [CrossRef]
  8. K. Mochizuki, “Degree of polarization in jointed fibers: the Lyot depolarizer,” Appl. Opt. 23, 3284–3288 (1984).
    [CrossRef]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  10. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [CrossRef]
  11. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  12. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475–1477 (1999).
    [CrossRef]
  13. E. I. Alekseev and E. N. Bazarov, “Theoretical basis of the method for reducing drift of the zero level of the output signal of a fiber-optic gyroscope with the aid of a Lyot depolarizer,” Sov. J. Quantum Electron. 22, 834–839 (1992).
    [CrossRef]
  14. P. L. Makowski and A. W. Domański, “Degree of polarization fading of light passing through birefringent medium with optical axis variation,” Proc. SPIE 7745, 77450J (2010).
    [CrossRef]
  15. J. Sakai, S. Machida, and T. Kimura, “Existence of eigen polarization modes in anisotropic single-mode optical fibers,” Opt. Lett. 6, 496–498 (1981).
    [CrossRef]

2010 (1)

P. L. Makowski and A. W. Domański, “Degree of polarization fading of light passing through birefringent medium with optical axis variation,” Proc. SPIE 7745, 77450J (2010).
[CrossRef]

2003 (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

2001 (1)

1999 (2)

1992 (1)

E. I. Alekseev and E. N. Bazarov, “Theoretical basis of the method for reducing drift of the zero level of the output signal of a fiber-optic gyroscope with the aid of a Lyot depolarizer,” Sov. J. Quantum Electron. 22, 834–839 (1992).
[CrossRef]

1984 (1)

1983 (1)

W. K. Burns, “Degree of polarization in the Lyot depolarizer,” J. Lightwave Technol. 1, 475–479 (1983).
[CrossRef]

1982 (1)

1981 (1)

1951 (1)

Alekseev, E. I.

E. I. Alekseev and E. N. Bazarov, “Theoretical basis of the method for reducing drift of the zero level of the output signal of a fiber-optic gyroscope with the aid of a Lyot depolarizer,” Sov. J. Quantum Electron. 22, 834–839 (1992).
[CrossRef]

Bazarov, E. N.

E. I. Alekseev and E. N. Bazarov, “Theoretical basis of the method for reducing drift of the zero level of the output signal of a fiber-optic gyroscope with the aid of a Lyot depolarizer,” Sov. J. Quantum Electron. 22, 834–839 (1992).
[CrossRef]

Billings, B. H.

Burns, W. K.

W. K. Burns, “Degree of polarization in the Lyot depolarizer,” J. Lightwave Technol. 1, 475–479 (1983).
[CrossRef]

Collett, E.

D. H. Goldstein and E. Collett, Polarized Light (Dekker, 2003).

Domanski, A. W.

P. L. Makowski and A. W. Domański, “Degree of polarization fading of light passing through birefringent medium with optical axis variation,” Proc. SPIE 7745, 77450J (2010).
[CrossRef]

Goldstein, D. H.

D. H. Goldstein and E. Collett, Polarized Light (Dekker, 2003).

Kato, T.

Kimura, T.

Loeber, A. P.

Lyot, B.

B. Lyot, “Recherche sur la polarisationde la lumière des planètes et de quelques substances terrestres,” in Annales de l’Observatoire de Paris (Meudon), Tome VIII, Facs. I (1929).

Machida, S.

Makowski, P. L.

P. L. Makowski and A. W. Domański, “Degree of polarization fading of light passing through birefringent medium with optical axis variation,” Proc. SPIE 7745, 77450J (2010).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinelli, M.

Mochizuki, K.

Oka, K.

Palais, J.

Palais, J. C.

Sakai, J.

Shen, P.

Wolf, E.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Appl. Opt. (2)

J. Lightwave Technol. (2)

W. K. Burns, “Degree of polarization in the Lyot depolarizer,” J. Lightwave Technol. 1, 475–479 (1983).
[CrossRef]

M. Martinelli and J. Palais, “Dual fiber-ring depolarizer,” J. Lightwave Technol. 19, 899–905 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (2)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

Proc. SPIE (1)

P. L. Makowski and A. W. Domański, “Degree of polarization fading of light passing through birefringent medium with optical axis variation,” Proc. SPIE 7745, 77450J (2010).
[CrossRef]

Sov. J. Quantum Electron. (1)

E. I. Alekseev and E. N. Bazarov, “Theoretical basis of the method for reducing drift of the zero level of the output signal of a fiber-optic gyroscope with the aid of a Lyot depolarizer,” Sov. J. Quantum Electron. 22, 834–839 (1992).
[CrossRef]

Other (4)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

B. Lyot, “Recherche sur la polarisationde la lumière des planètes et de quelques substances terrestres,” in Annales de l’Observatoire de Paris (Meudon), Tome VIII, Facs. I (1929).

D. H. Goldstein and E. Collett, Polarized Light (Dekker, 2003).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (7)

Fig. 1.
Fig. 1.

Model of a collimated polychromatic beam: S, arbitrary light source; P, pinhole; L, collimating lens; B, uniform and fully spatially coherent beam with negligible divergence.

Fig. 2.
Fig. 2.

Two-segment crystal depolarizer with an arbitrary twist angle θ>0 and variable segment lengths L1 and L2; α>0 is the azimuth of the incident linearly polarized wave. The xy and xy coordinate systems are matched to the birefringence axes of the crystals. Above, the amplitude decomposition between the fast and slow axes is shown. The segments L1, L2 induce time delays τ1, τ2 between light components parallel to the axes of the corresponding coordinate systems.

Fig. 3.
Fig. 3.

Characteristics of light spectra for numerical experiments. (a) Spectral density distributions: SLD, commercial superluminescent LED; Gss, a symmetrical equivalent of SLD. (b) Corresponding courses of degree of coherence versus time delay τ induced in a quartz crystal of length L (the upper axis); the two marked points refer to the time delays τ1 and τ2 induced by the segments of the commercial 6 mm long quartz depolarizer.

Fig. 4.
Fig. 4.

Comparison for the three models’ predictions for SLD spectrum (superluminescent diode). Extreme values of DoP(α) versus the twist angle θ between the segments are plotted. Depolarizer segments thickness ratio is 12, linear polarization of the incident beam is reconstructed at the joint plane: Δϕ=0.

Fig. 5.
Fig. 5.

Comparison for the three models’ predictions for SLD spectrum (superluminescent diode). Extreme values of DoP(α) are plotted versus the length of the first crystal L1 (upper axis) and corresponding change of retardance Δϕ (lower axis) for the twist angle between the segments: (a) θ=45° and (b) θ=40°. L2=2L1.

Fig. 6.
Fig. 6.

Comparison for the three models’ predictions for Gss spectrum (Gaussian equivalent of SLD). Extreme values of DoP(α) are plotted versus the length of the first crystal L1 (upper axis) and corresponding change of retardance Δϕ (lower axis) for the twist angle between the segments: (a) θ=45° and (b) θ=40°. L2=2L1.

Fig. 7.
Fig. 7.

Comparison for the three models’ predictions for Gss spectrum (Gaussian equivalent of SLD). Global maximum of DoP(α) is plotted versus the twist angle θ between the segments of the depolarizer. The segments’ lengths are (a) L1=2mm, L2=0 and (b) L1=L2=2mm. Linear polarization of the incident beam is reconstructed at the joint plane Δϕ=0.

Tables (1)

Tables Icon

Table 1. Two Sets of Parameters for Numerical Testsa

Equations (25)

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Jij=Ei(qm)*(z,t)Ej(qm)(z,t),
DoP=1(4DetJ)/(TrJ)2,
Γij=Ei*(z,t)Ej(z,t),
DoP=1(4DetΓ)/(TrΓ)2=14(ΓxxΓyy|Γxy|2)/(Γxx+Γyy)2,
Γ12(τ)=E1*(t)E2(t+τ),Γ21(τ)=Γ12*(τ),
γ12(τ)=Γ12(τ)Γ11(0)Γ22(0),γ21(τ)=γ12*(τ),
Γ11(τ)=+G(ω)eiωτdω,
S=[S0,S1,S2,S3]T=[Γxx+Γyy,ΓxxΓyy,Γxy+Γyx,i(ΓyxΓxy)]T=[Γxx+Γyy,ΓxxΓyy,2Re{Γxy},2Im{Γxy}]T,
sout=(0+M(ω)G^(ω)dω)·sin,
DoP=s12+s22+s32s0.
{Ex=E(t+τ1+τ2)cos(α)cos(θ)E(t+τ2)sin(α)sin(θ)Ey=E(t)sin(α)cos(θ)+E(t+τ1)cos(α)sin(θ).
Γxx=Ex*Ex,Γxy=Ex*Ey,Γyx=Γxy*,Γyy=Ey*Ey.
Γxx=I×[cos2(α+θ)+sin(2α)sin(2θ)1γ11(r)(τ1)2],
Γyy=I×[sin2(α+θ)sin(2α)sin(2θ)1γ11(r)(τ1)2],
Γxy=12I×{cos(2α)sin(2θ)γ11*(τ2)+sin(2α)cos2(θ)γ11*(τ1+τ2)sin(2α)sin2(θ)γ11*(τ2τ1)},
Γyx=Γxy*,
DoP={cos2(2α)[cos2(2θ)+sin2(2θ)|γ(τ2)|2]+sin2(2α)[1/2sin2(2θ)|γ(τ1)|2+cos4(θ)|γ(τ1+τ2)|2+sin4(θ)|γ(τ2τ1)|2]sin(4α)sin(2θ)|γ(τ2)|×[cos2(θ)|γ(τ1+τ2)|sin2(θ)|γ(τ2τ1)|]cos(ω0τ1)+1/2sin2(2α)sin2(2θ)|[γ(τ1)|2|γ(τ2τ1)|×|γ(τ1+τ2)|]cos(2ω0τ1)}1/2,
M(ω)=M2(ω)·T(θ)·M1(ω),
ω00=0+ωG2(ω)dω0+G2(ω)dω.
ϕ(τ1)=arg{γ11(τ1)}=arg{Ex*(L1,t)Ey(L1,t)},
ϕω0(τ1)=ϕ(τ1),
Δϕ(τ1)=ϕω0(τ1)ϕω0(τ1(0)),
ϕω0(τ1(0))=2kπ,kN,
L1(Δϕ)=L1(0)+cΔnω0Δϕ,
γ11(τ)=n=1NG(ωn)eiωnτn=1NG(ωn),

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