Abstract

We calculated the Fresnel paraxial propagator in a birefringent plate having topological charge q at its center, named “q-plate.” We studied the change of the beam transverse profile when it traverses the plate. An analytical closed form of the beam profile propagating in the q-plate can be found for many important specific input beam profiles. We paid particular attention to the plate having a topological unit charge and found that if small losses due to reflection, absorption, and scattering are neglected, the plate can convert the photon spin into orbital angular momentum with up to 100% efficiency provided the thickness of the plate is less than the Rayleigh range of the incident beam.

© 2009 Optical Society of America

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References

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  1. G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef] [PubMed]
  2. G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
    [CrossRef]
  3. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pasko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
    [CrossRef] [PubMed]
  4. A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  5. L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  6. L. Marrucci, C. Manzo, and D. Paparo, Appl. Phys. Lett. 88, 221102 (2006).
    [CrossRef]
  7. G. F. Calvo and A. Picón, Opt. Lett. 32, 838 (2007).
    [CrossRef] [PubMed]
  8. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
    [CrossRef] [PubMed]

2007 (3)

2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

2004 (1)

2002 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Barnett, S. M.

Calvo, G. F.

Courtial, J.

Franke-Arnold, S.

Gibson, G.

Karimi, E.

Mair, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

Marrucci, L.

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Padgett, M. J.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

Pasko, V.

Piccirillo, B.

Picón, A.

Santamato, E.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Vasnetsov, M.

Vaziri, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Welhs, G.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Zito, G.

Appl. Phys. Lett. (1)

L. Marrucci, C. Manzo, and D. Paparo, Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Beating of SAM (dashed curve) and OAM (solid curve) as a function of the optical retardation Δ n z λ while a circularly polarized input beam propagates in the 1-plate. (a) For LG 00 and (b) LG 01 as an input beam. We used the following data: n o = 1.5 , n e = 1.7 , and w 0 = 50 λ .

Fig. 2
Fig. 2

Intensity profile for (a) full STOC and (b) no STOC in the 1-plate. Solid and dashed curves are simulated by [7] and our theory, respectively. The input beam assumed the TEM 00 .

Fig. 3
Fig. 3

Intensity profile in the far-field beyond the 1-plate after free-air propagation. (a) No STOC, (b) full STOC.

Equations (12)

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f ( r ) + f ( r ) r + ( k 0 2 ( n o 2 γ 2 ) μ 2 r 2 ) f ( r ) = ν g ( r ) r 2 ,
g ( r ) + g ( r ) r + ( k 0 2 ( n e 2 γ 2 ) μ 2 r 2 ) g ( r ) = ν f ( r ) r 2 ,
f o ( r ) + f o ( r ) r + ( k 0 2 β 2 μ 2 r 2 ) f o ( r ) = ν g o ( r ) r 2 ,
g o ( r ) + g o ( r ) r + ( k 0 2 β 2 Λ 2 μ 2 r 2 ) g o ( r ) = ν f o ( r ) r 2 ,
E ( r , ϕ , z ) = 1 2 0 ρ d ρ 0 2 π d ψ R ̂ ( q ϕ ) { ( K o + K e ) 1 ̂ + ( K o K e ) σ ̂ z } R ̂ ( q ψ ) E ( ρ , ψ , 0 ) ,
K μ ( m ) o , e ( r , ρ ; z ) = ( i n o , e k 0 2 π z ) i μ ( m ) J μ ( m ) ( k 0 n o , e r ρ z ) × e i k 0 n o , e ( r 2 + ρ 2 ) 2 z i k 0 n o , e z .
[ E + E ] = e i ( l ϕ k 0 n o z ) [ K μ + K μ + e 2 i q ϕ K μ e 2 i q ϕ K μ + + ] [ a b ] ,
HyGG p m ( ρ , ζ ) = C p m ζ p 2 ( ζ + i ) ( 1 + m + p 2 ) ρ m × e i ρ 2 ( ζ + i ) F 1 1 ( p 2 , 1 + m ; ρ 2 ζ ( ζ + i ) ) ,
C p m = i m + 1 2 p + m + 1 π Γ ( p + m + 1 ) Γ ( 1 + m + p 2 ) Γ ( m + 1 ) ,
S z ( z ) = 1 ω R [ e i k 0 Δ n z ( b 2 I l μ + , μ + ( z ) a 2 I l μ , μ ( z ) ) ] ,
L z ( z ) + q ω S z ( z ) = 1 ω ( ( l q ) a 2 + ( l + q ) b 2 ) ,
I p , m ( ζ ) = 2 p + m + 1 Γ 2 ( p 2 + m + 1 ) Γ ( m + 1 ) Γ ( p + m + 1 ) χ p 2 ( ζ ) ( n o n e 2 n o n e i ( n e n o ) ζ ) p + m + 1 F 1 2 ( p 2 , p 2 ; m + 1 ; χ ( ζ ) ) ,

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