Abstract

For the first time to the best of our knowledge, we derive expressions for coordinates of the trajectory on the Poincaré sphere that represent polarization evolution in an arbitrary beam of completely polarized light. Our work substantially extends the mapping function of the Poincaré sphere, and opens up new possibilities for its use in optics. In particular, the obtained expressions allow one to visualize the results of the finite-difference time- domain modeling of light propagation through birefringent crystals, including simulations of polarization rotation experienced by ultrashort pulses in nonlinear media.

© 2010 Optical Society of America

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References

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  1. H. Poincaré, Théorie Mathématique de la Lumiére II(Gauthiers-Villars, 1892).
  2. G. B. Malykin, Radiophys. Quantum Electron. 40, 175(1997).
    [CrossRef]
  3. B. A. Robson, The Theory of Polarization Phenomena(Oxford U. Press, 1974).
  4. G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik (Springer Verlag, 1961).
  5. W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).
  6. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  7. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  8. T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).
    [CrossRef]
  9. W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Norstrand, 1964).
  10. R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  11. G. M. Fikhtengolts, The Fundamentals of Mathematical Analysis (Pergamon, 1965), Vol. I.

2000

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

1999

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

1997

G. B. Malykin, Radiophys. Quantum Electron. 40, 175(1997).
[CrossRef]

1990

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

1977

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

1974

B. A. Robson, The Theory of Polarization Phenomena(Oxford U. Press, 1974).

1965

G. M. Fikhtengolts, The Fundamentals of Mathematical Analysis (Pergamon, 1965), Vol. I.

1964

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Norstrand, 1964).

1962

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

1961

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik (Springer Verlag, 1961).

1892

H. Poincaré, Théorie Mathématique de la Lumiére II(Gauthiers-Villars, 1892).

Azzam, R.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Ballard, S. S.

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Norstrand, 1964).

Bashara, N.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Born, M.

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

Brabec, T.

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

Fikhtengolts, G. M.

G. M. Fikhtengolts, The Fundamentals of Mathematical Analysis (Pergamon, 1965), Vol. I.

Kliger, D. S.

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

Krausz, F.

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

Lewis, J. W.

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

Malykin, G. B.

G. B. Malykin, Radiophys. Quantum Electron. 40, 175(1997).
[CrossRef]

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumiére II(Gauthiers-Villars, 1892).

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik (Springer Verlag, 1961).

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik (Springer Verlag, 1961).

Randall, C. E.

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

Robson, B. A.

B. A. Robson, The Theory of Polarization Phenomena(Oxford U. Press, 1974).

Shurcliff, W. A.

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Norstrand, 1964).

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

Wolf, E.

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

Radiophys. Quantum Electron.

G. B. Malykin, Radiophys. Quantum Electron. 40, 175(1997).
[CrossRef]

Rev. Mod. Phys.

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

Other

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Norstrand, 1964).

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

G. M. Fikhtengolts, The Fundamentals of Mathematical Analysis (Pergamon, 1965), Vol. I.

B. A. Robson, The Theory of Polarization Phenomena(Oxford U. Press, 1974).

G. N. Ramachandran and S. Ramaseshan, in Handbuch der Physik (Springer Verlag, 1961).

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

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 U. Press, 1999).

H. Poincaré, Théorie Mathématique de la Lumiére II(Gauthiers-Villars, 1892).

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Figures (3)

Fig. 1
Fig. 1

Polarization ellipse and the Poincaré sphere.

Fig. 2
Fig. 2

Trajectories on the Poincaré sphere representing instantaneous polarization forms for two Lissajous patterns: E x = cos 3 ξ and E y = sin 4 ξ on the left; E x = cos ξ and E y = sin ( 2 ξ + π / 4 ) on the right; ξ [ 0 , 2 π ) .

Fig. 3
Fig. 3

Polarization pattern of a few-cycle pulse and its representation on the Poincaré sphere.

Equations (13)

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x = cos 2 ψ cos 2 χ , y = sin 2 ψ cos 2 χ , z = sin 2 χ ,
F ( E x , E y ) ( E x cos φ + E y sin φ ) 2 a 2 + ( E x sin φ E y cos φ ) 2 b 2 1 = 0 ,
f ( ξ ) d E y d E x = E y E x ,
g ( ξ ) d 2 E y d E x 2 = E y ′′ E x E x ′′ E y ( E x ) 3 ,
d E y d E x = F / E x F / E y ,
d 2 E y d E x 2 = [ 2 2 F E x E y F E x F E y 2 F E x 2 ( F E y ) 2 2 F E y 2 ( F E x ) 2 ] ( F E y ) 3 .
a = ( f E x E y ) ( E x cos φ + E y sin φ ) f cos φ sin φ ,
b = ( f E x E y ) ( E x sin φ E y cos φ ) f sin φ + cos φ .
φ = 1 2 tan 1 [ 2 g E x E y + 2 f ( f E x E y ) g ( E x 2 E y 2 ) + ( 1 f 2 ) ( f E x E y ) ] .
ψ = φ ( π / 4 ) [ 1 sign ( a b ) ] sign φ ,
χ = sign f tan 1 ( a / b ) sign ( b a ) .
E x = A x sin ( β ξ + ϕ x ) , E y = A y sin ( β ξ + ϕ y ) ,
tan 2 ψ = 2 A x A y cos Δ A x 2 A y 2 , sin 2 χ = 2 A x A y sin Δ A x 2 + A y 2 .

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