Abstract

The modification of the polarization and spectral structure of light by electro-optic modulation with longitudinal effect in crystals of class 4¯2m is analyzed in the frame of a Pauli algebraic and Poincaré geometric approach. The results are generalized, in a vectorial Pauli algebraic form, for any birefringent time-varying device.

© 2008 Optical Society of America

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References

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    [CrossRef]
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  3. V. S. Zapasskii and G. G. Kozlov, “Polarized light in anisotropic medium versus spin in a magnetic field,” Phys. Uspekhi 42, 817-822 (1999).
    [CrossRef]
  4. C. S. Brown and A. Em. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625-1635 (1995).
    [CrossRef]
  5. S. E. Segre, “New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform fully anisotropic medium: a magnetized plasma,” J. Opt. Soc. Am. A 18, 2601-2606 (2001).
    [CrossRef]
  6. N. Gisin, “Solution of the dynamical equations for polarisation dipersion,” Opt. Commun. 86, 371-373 (1991).
    [CrossRef]
  7. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibres,” Opt. Commun. 142, 119-125 (1997).
    [CrossRef]
  8. L. Yi and A. Yariv, “Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821-1827 (2000).
    [CrossRef]
  9. S. E. Segre, “Evolution of the polarization state for radiation propagating in a nonuniform birefrigent, optically active and dichroic medium: the case of a magnetized plasma,” J. Opt. Soc. Am. A 17, 95-100 (2000).
    [CrossRef]
  10. M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321-325 (1989).
    [CrossRef]
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  13. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).
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    [CrossRef]
  15. I. P. Kaminow and E. H. Turner, “Electrooptic light modulators,” Appl. Opt. 5, 1612-1628 (1966).
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  16. C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
  17. K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981).
  18. N. W. McLachlan, Bessel Functions for Engineers (Oxford Univ. Press, 1941).
  19. T. Tudor and A. Gheondea, “Pauli algebraic forms of normal and nonnormal operators,” J. Opt. Soc. Am. A 24, 204-210(2007).
    [CrossRef]
  20. L. C. Biedenharn, J. D. Louck, and P. A. Caruthers, “Angular momentum in quantum physics. Theory and applications,” in Encyclopedia of Mathematics and its Applications, G. -C. Rota, ed. (Addison-Wesley, 1981), Vol. 8.

2007 (1)

2005 (1)

2001 (1)

2000 (3)

1999 (1)

V. S. Zapasskii and G. G. Kozlov, “Polarized light in anisotropic medium versus spin in a magnetic field,” Phys. Uspekhi 42, 817-822 (1999).
[CrossRef]

1997 (1)

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibres,” Opt. Commun. 142, 119-125 (1997).
[CrossRef]

1995 (1)

C. S. Brown and A. Em. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

1991 (1)

N. Gisin, “Solution of the dynamical equations for polarisation dipersion,” Opt. Commun. 86, 371-373 (1991).
[CrossRef]

1989 (1)

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321-325 (1989).
[CrossRef]

1985 (1)

H. Kubo and R. Nagata, “Vector representation of behavior of polarized light in a weakly inhomogeneous medium with birefringence and dichroism. II. Evolution of polarization states,” J. Opt. Soc. Am. B 2, 30-34 (1985).
[CrossRef]

1966 (1)

1949 (1)

1948 (1)

Azzam, R. M. A.

Bak, A. Em.

C. S. Brown and A. Em. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

Biedenharn, L. C.

L. C. Biedenharn, J. D. Louck, and P. A. Caruthers, “Angular momentum in quantum physics. Theory and applications,” in Encyclopedia of Mathematics and its Applications, G. -C. Rota, ed. (Addison-Wesley, 1981), Vol. 8.

Billings, B. H.

Blum, K.

K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Brown, C. S.

C. S. Brown and A. Em. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Caruthers, P. A.

L. C. Biedenharn, J. D. Louck, and P. A. Caruthers, “Angular momentum in quantum physics. Theory and applications,” in Encyclopedia of Mathematics and its Applications, G. -C. Rota, ed. (Addison-Wesley, 1981), Vol. 8.

Dong, H.

Gheondea, A.

Gisin, N.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibres,” Opt. Commun. 142, 119-125 (1997).
[CrossRef]

N. Gisin, “Solution of the dynamical equations for polarisation dipersion,” Opt. Commun. 86, 371-373 (1991).
[CrossRef]

Huttner, B.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibres,” Opt. Commun. 142, 119-125 (1997).
[CrossRef]

Jones, R. C.

Kaminow, I. P.

Kitano, M.

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321-325 (1989).
[CrossRef]

Kozlov, G. G.

V. S. Zapasskii and G. G. Kozlov, “Polarized light in anisotropic medium versus spin in a magnetic field,” Phys. Uspekhi 42, 817-822 (1999).
[CrossRef]

Kubo, H.

H. Kubo and R. Nagata, “Vector representation of behavior of polarized light in a weakly inhomogeneous medium with birefringence and dichroism. II. Evolution of polarization states,” J. Opt. Soc. Am. B 2, 30-34 (1985).
[CrossRef]

Louck, J. D.

L. C. Biedenharn, J. D. Louck, and P. A. Caruthers, “Angular momentum in quantum physics. Theory and applications,” in Encyclopedia of Mathematics and its Applications, G. -C. Rota, ed. (Addison-Wesley, 1981), Vol. 8.

McLachlan, N. W.

N. W. McLachlan, Bessel Functions for Engineers (Oxford Univ. Press, 1941).

Nagata, R.

H. Kubo and R. Nagata, “Vector representation of behavior of polarized light in a weakly inhomogeneous medium with birefringence and dichroism. II. Evolution of polarization states,” J. Opt. Soc. Am. B 2, 30-34 (1985).
[CrossRef]

Ning, G.

Segre, S. E.

Shum, P.

Tudor, T.

Turner, E. H.

Yabuzaki, T.

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321-325 (1989).
[CrossRef]

Yan, M.

Yariv, A.

Yi, L.

Zapasskii, V. S.

V. S. Zapasskii and G. G. Kozlov, “Polarized light in anisotropic medium versus spin in a magnetic field,” Phys. Uspekhi 42, 817-822 (1999).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (2)

H. Kubo and R. Nagata, “Vector representation of behavior of polarized light in a weakly inhomogeneous medium with birefringence and dichroism. II. Evolution of polarization states,” J. Opt. Soc. Am. B 2, 30-34 (1985).
[CrossRef]

L. Yi and A. Yariv, “Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821-1827 (2000).
[CrossRef]

Opt. Commun. (2)

N. Gisin, “Solution of the dynamical equations for polarisation dipersion,” Opt. Commun. 86, 371-373 (1991).
[CrossRef]

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibres,” Opt. Commun. 142, 119-125 (1997).
[CrossRef]

Opt. Eng. (1)

C. S. Brown and A. Em. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321-325 (1989).
[CrossRef]

Phys. Uspekhi (1)

V. S. Zapasskii and G. G. Kozlov, “Polarized light in anisotropic medium versus spin in a magnetic field,” Phys. Uspekhi 42, 817-822 (1999).
[CrossRef]

Other (5)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981).

N. W. McLachlan, Bessel Functions for Engineers (Oxford Univ. Press, 1941).

L. C. Biedenharn, J. D. Louck, and P. A. Caruthers, “Angular momentum in quantum physics. Theory and applications,” in Encyclopedia of Mathematics and its Applications, G. -C. Rota, ed. (Addison-Wesley, 1981), Vol. 8.

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Figures (5)

Fig. 1
Fig. 1

Modulating crystal.

Fig. 2
Fig. 2

Oscillation of the Poincaré vector of the modulated light when the incident light is polarized at 45 ° with the induced principal axes of the crystal.

Fig. 3
Fig. 3

Oscillation of the Poincaré vector of the modulated light for some general state of the incident light.

Fig. 4
Fig. 4

(a) Suppression of the odd order harmonics ( V = 0 ) and (b) of the even order harmonics ( V 0 = V λ / 4 ) in the spectrum of the intensity of the modulated light.

Fig. 5
Fig. 5

Oscillation of the Poincaré vector of the modulated light for a general modulating crystal of Poincaré axis n.

Equations (55)

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n x = n 0 1 2 n 0 3 r 63 E , n y = n 0 + 1 2 n 0 3 r 63 E ,
± ϕ ( t ) = ± Φ 0 ± Γ sin Ω t ,
Φ 0 = π λ 0 n 0 3 r 63 V 0 , Γ = π λ 0 n 0 3 r 63 V m ,
2 ϕ ( t ) = 2 ( Φ 0 + Γ sin Ω t )
σ 1 = | P x P x | | P y P y | , σ 2 = | P 45 ° P 45 ° | | P 45 ° P 45 ° | , σ 3 = | R R | | L L | ,
| P x ( 2 ϕ ) = e i ϕ n · σ ,
n ( 1 , 0 , 0 )
σ ( σ 1 , σ 2 , σ 3 )
| P x ( t ) = e i ϕ ( t ) σ 1 = e i ( Φ 0 + Γ sin Ω t ) σ 1 = σ 0 cos ( Φ 0 + Γ sin Ω t ) i σ 1 sin ( Φ 0 + Γ sin Ω t ) ,
| S ( t 2 ) = ( t 2 , t 1 ) | S ( t 1 ) ,
( t 2 , t 1 ) = e i [ Φ 0 + Γ sin Ω ( t 2 t 1 ) ] σ 1 .
i ( t ) t = H ( t ) ( t ) ,
( t ) = e i ( Φ 0 + Γ sin Ω t ) σ 1 ,
H ( t ) = Γ Ω σ 1 cos Ω t .
ρ = 1 2 ( σ + 0 s · σ ) ,
ρ o ( t ) = ( t ) ρ i ( t ) ,
s i ( 0 , 1 , 0 ) ,
ρ i = 1 2 ( σ 0 + σ 2 ) .
ρ o ( t ) = 1 2 ( σ 0 cos ϕ i σ 1 sin ϕ ) ( σ 0 + σ 2 ) ( σ 0 cos ϕ + i σ 1 sin ϕ ) = 1 2 ( σ 0 + σ 2 cos 2 ϕ + σ 3 sin 2 ϕ ) = 1 2 [ σ 0 + σ 2 cos 2 ( Φ 0 + Γ sin Ω t ) + σ 3 sin 2 ( Φ 0 + Γ sin Ω t ) ] .
s o ( 0 , cos 2 ϕ , sin 2 ϕ ) = s o [ 0 , cos 2 ( Φ 0 + Γ sin Ω t ) , sin 2 ( Φ 0 + Γ sin Ω t ) ] .
s o ( 0 , cos 2 Φ 0 , sin 2 Φ 0 ) .
ε ( t ) = ϕ ( t ) = ( Φ + 0 Γ sin Ω t ) .
s o = s o [ 0 , sin ( 2 Γ sin Ω t ) , cos ( 2 Γ sin Ω t ) ] ,
s o = s o [ 0 , cos ( 2 Γ sin Ω t ) , sin ( 2 Γ sin Ω t ) ] ,
ρ o = 1 2 { σ 0 + σ 2 [ cos 2 Φ 0 cos ( 2 Γ sin Ω t ) sin 2 Φ 0 sin ( 2 Γ sin Ω t ) ] + σ 3 [ sin 2 Φ 0 cos ( 2 Γ sin Ω t ) + cos 2 Φ 0 sin ( 2 Γ sin Ω t ) ] } = 1 2 [ σ 0 + ( σ 2 cos 2 Φ 0 + σ 3 sin 2 Φ 0 ) cos ( 2 Γ sin Ω t ) + ( - σ 2 sin 2 Φ 0 + σ 3 cos 2 Φ 0 ) sin ( 2 Γ sin Ω t ) ] .
cos ( Γ sin θ ) = J 0 ( Γ ) + 2 k = 1 J 2 k ( Γ ) cos 2 k θ , sin ( Γ sin θ ) = 2 k = 1 J 2 k 1 ( Γ ) sin ( 2 k 1 ) θ ,
ρ o = 1 2 { σ 0 + ( σ 2 cos 2 Φ 0 + σ 3 sin 2 Φ 0 ) [ J 0 ( 2 Γ ) + 2 k = 1 J 2 k ( 2 Γ ) cos 2 k Ω t ] ( σ 2 sin 2 Φ 0 - σ 3 cos 2 Φ 0 ) 2 k = 1 J 2 k 1 ( 2 Γ ) sin ( 2 k 1 ) Ω t } .
P | P 45 = 1 2 ( σ 0 σ 2 ) ,
ρ o = P | P 45 ° ρ o P | P 45 ° ,
ρ o = 1 8 ( σ 0 σ 2 ) ( σ 0 + σ 2 cos 2 ϕ + σ 3 sin 2 ϕ ) ( σ 0 σ 2 ) = 1 8 [ 2 σ 0 ( 1 cos 2 ϕ ) 2 σ 2 ( 1 cos 2 ϕ ) ] = 1 2 ( σ 0 σ 2 ) sin 2 ϕ ,
ρ o = 1 2 ( σ 0 σ 2 ) sin 2 ( Φ 0 + Γ sin Ω t ) .
ρ o = 1 2 ( σ 0 + σ 2 ) cos 2 ( Φ 0 + Γ sin Ω t ) .
I | | = cos 2 ( Φ 0 + Γ sin Ω t ) ,
I = sin 2 ( Φ 0 + Γ sin Ω t ) .
I , = 1 2 { 1 ± cos 2 Φ 0 [ J 0 ( 2 Γ ) + 2 k = 1 J 2 k ( 2 Γ ) cos 2 k Ω t ] sin 2 Φ 0 [ 2 k = 1 J 2 k 1 ( 2 Γ ) sin ( 2 k 1 ) Ω t ] } .
I , = 1 2 { 1 ± J o ( 2 Γ ) ± 2 k = 1 J 2 k ( 2 Γ ) cos 2 k Ω t } .
I | | , = 1 2 { 1 2 k = 1 J 2 k 1 ( 2 Γ ) sin ( 2 k 1 ) Ω t } .
I , = 1 2 { 1 J 0 ( 2 Γ ) 2 k = 1 J 2 k ( 2 Γ ) cos 2 k Ω t } .
n ( t ) = e i ϕ ( t ) n · σ = σ 0 cos ϕ ( t ) i n · σ sin ϕ ( t ) .
| S o ( t ) = n ( t ) | S i = e i ϕ ( t ) n · σ | S i
ρ o ( t ) = n ( t ) ρ i ( t ) n ( t ) ,
ρ = 1 2 ( σ 0 + s · σ ) .
( a · σ ) ( b · σ ) = a · b + i ( a × b ) · σ ,
ρ o = 1 2 ( σ 0 cos ϕ i n · σ sin ϕ ) ( σ 0 + s i · σ ) × ( σ 0 cos ϕ + i n · σ sin ϕ ) = 1 2 [ σ 0 + s i · σ ( cos 2 ϕ sin 2 ϕ ) + 2 ( n × s i ) · σ sin ϕ cos ϕ + 2 n · s i ( n · σ ) sin 2 ϕ ] = 1 2 [ σ 0 + s i · σ cos 2 ϕ + ( n × s i ) · σ sin 2 ϕ + 2 n · s i ( n · σ ) sin 2 ϕ ] .
ρ o = 1 2 ( σ 0 + s o · σ ) ,
s o = s i cos 2 ϕ + n × s i sin 2 ϕ + 2 ( n · s i ) n sin 2 ϕ .
s 0 = R n ( δ ) s i ,
R n ( δ ) = cos δ + ( 1 cos δ ) n ( n · ) + sin δ ( n × ) .
δ = 2 ϕ ( t ) = 2 Φ 0 + 2 Γ sin Ω t ,
s ˙ = H × s ,
H = 2 H χ ,
H ( t ) = Γ Ω n · σ cos Ω t .
2 H = 2 Γ Ω cos Ω t ,
ρ o = 1 2 [ σ 0 + σ 2 cos 2 ( Φ 0 + Γ sin Ω t ) + σ 3 sin 2 ( Φ 0 + Γ sin Ω t ) ] ,
s 0 = s i cos 2 ( Φ 0 + Γ sin Ω t ) + n × s i sin 2 ( Φ 0 + Γ sin Ω t ) .

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