Abstract

Ladder operators are introduced in SU(2) parameter space to analyze the space-variant Pancharatnam–Berry (PB) phase. The spin and orbital angular momentum transfers to a wave passing through the PB phase optical elements are understood in terms of the PB phase. Space-variant PB phase structures are identified in polarization grating and cholesteric liquid crystal, providing an explanation for the beam propagation direction alteration. A simultaneous occurrence of the wavefront shaping and the beam propagation direction alteration is demonstrated experimentally.

© 2008 Optical Society of America

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References

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  1. M. V. Berry, “The adiabatic phase and Pancharatnam's phase for polarized light,” J. Mod. Opt. 34, 1401-1407 (1987).
    [CrossRef]
  2. Z. Bomzon, A. Niv, G. Biener, V. Kleiner, and E. Hasmana, “Nondiffracting periodically space-variant polarization beams with subwavelength gratings,” Appl. Phys. Lett. 80, 3685-3687 (2002).
    [CrossRef]
  3. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  4. L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Appl. Phys. Lett. 88, 221102 (2006).
    [CrossRef]
  5. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875-1877 (2002).
    [CrossRef]
  6. P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).
  7. B. Piccirillo and E. Santamato, “Light angular momentum flux and forces in birefringent inhomogeneous media,” Phys. Rev. E 69, 056613 (2004).
    [CrossRef]
  8. L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579-588 (1984).
    [CrossRef]
  9. H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys. 94, 1298-1303 (2003).
    [CrossRef]
  10. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-125 (1936).
    [CrossRef]
  11. P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, 1993).
  12. S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, 1992).
    [CrossRef]
  13. H. Choi, J. H. Woo, J. W. Wu, D.-W. Kim, T.-K. Lim, and S. H. Song, “Holographic inscription of helical wavefronts in a liquid crystal polarization grating,” Appl. Phys. Lett. 94, 141112 (2007).
    [CrossRef]

2007 (1)

H. Choi, J. H. Woo, J. W. Wu, D.-W. Kim, T.-K. Lim, and S. H. Song, “Holographic inscription of helical wavefronts in a liquid crystal polarization grating,” Appl. Phys. Lett. 94, 141112 (2007).
[CrossRef]

2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

2004 (1)

B. Piccirillo and E. Santamato, “Light angular momentum flux and forces in birefringent inhomogeneous media,” Phys. Rev. E 69, 056613 (2004).
[CrossRef]

2003 (1)

H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys. 94, 1298-1303 (2003).
[CrossRef]

2002 (2)

Z. Bomzon, A. Niv, G. Biener, V. Kleiner, and E. Hasmana, “Nondiffracting periodically space-variant polarization beams with subwavelength gratings,” Appl. Phys. Lett. 80, 3685-3687 (2002).
[CrossRef]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875-1877 (2002).
[CrossRef]

1999 (1)

P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam's phase for polarized light,” J. Mod. Opt. 34, 1401-1407 (1987).
[CrossRef]

1984 (1)

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579-588 (1984).
[CrossRef]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-125 (1936).
[CrossRef]

Appl. Phys. Lett. (3)

Z. Bomzon, A. Niv, G. Biener, V. Kleiner, and E. Hasmana, “Nondiffracting periodically space-variant polarization beams with subwavelength gratings,” Appl. Phys. Lett. 80, 3685-3687 (2002).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

H. Choi, J. H. Woo, J. W. Wu, D.-W. Kim, T.-K. Lim, and S. H. Song, “Holographic inscription of helical wavefronts in a liquid crystal polarization grating,” Appl. Phys. Lett. 94, 141112 (2007).
[CrossRef]

J. Appl. Phys. (1)

H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys. 94, 1298-1303 (2003).
[CrossRef]

J. Mod. Opt. (2)

P. Hariharan, S. Mujumdar, and H. Ramachandran, “A simple demonstration of the Pancharatnam phase as a geometric phase,” J. Mod. Opt. 46, 1443-1446 (1999).

M. V. Berry, “The adiabatic phase and Pancharatnam's phase for polarized light,” J. Mod. Opt. 34, 1401-1407 (1987).
[CrossRef]

Opt. Acta (1)

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579-588 (1984).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-125 (1936).
[CrossRef]

Phys. Rev. E (1)

B. Piccirillo and E. Santamato, “Light angular momentum flux and forces in birefringent inhomogeneous media,” Phys. Rev. E 69, 056613 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Other (2)

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, 1993).

S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, 1992).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

(a) and (c) Diffracted beam is right-circularly polarized and has the orbital helicity of + 2 when a left-circularly polarized TEM 00 beam is incident on the grating. (b) and (d) Diffracted beam is left-circularly polarized and has the orbital helicity of 2 when a right-circularly polarized TEM 00 beam is incident on the grating.

Equations (15)

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χ ( + ) = 1 2 [ 1 + i ] , χ ( ) = 1 2 [ 1 i ] .
S ̂ ± = S ̂ z ± i S ̂ x .
S ̂ ± χ ( ) = + χ ( ± ) .
M = [ M 0 + M z M x i M y M x + i M y M 0 M z ] = M 0 I + M y S ̂ y + M z i M x S ̂ + + M z + i M x S ̂ .
P ( θ ) = R ( θ ) [ 1 0 0 0 ] R ( + θ ) = 1 2 I + e i 2 θ 2 S ̂ + + e + i 2 θ 2 S ̂ .
P ( θ ) χ ( + ) + P ( θ ) χ ( ) 2 = 1 + cos 2 θ .
M ( x ) = [ cos γ ( x ) sin γ ( x ) sin γ ( x ) cos γ ( x ) ] = e + i γ ( x ) S ̂ + e i γ ( x ) S ̂ + ,
Ψ = s z , l , k l m , k m .
Ψ = s z = , l = 2 z ̂ , k = k l m , k m .
Ψ = s z = , l = l , k = k z ̂ + G x ̂ l m = l m + 2 z ̂ , k m = k m G x ̂ .
Ψ ϵ ( z ) Ψ = ϵ a 2 Ψ [ cos 2 q 0 z sin 2 q 0 z sin 2 q 0 z cos 2 q 0 z ] Ψ = ϵ a 2 [ Ψ e + i 2 q 0 z S ̂ Ψ + Ψ e i 2 q 0 z S ̂ + Ψ ] .
Ψ = s z = , l = l , k = k 2 q 0 z ̂ l m = l m 2 z ̂ ,
k m = k m + 2 q 0 z ̂ = s z = , l = l ,
k = q 0 z ̂ l m = l m , k m = k m + 2 q 0 z ̂ .
Ψ = s z = , l = 2 , k = k z ̂ + 2 π Λ x ̂ l m = l m , k m = k m 2 π Λ x ̂ .

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