Abstract

Guided by the aim to construct light fields with spin-like orbital angular momentum (OAM), that is light fields with a uniform and intrinsic OAM density, we investigate the OAM of strictly periodic arrays of optical vortices with rectangular symmetry. We find that the OAM per unit cell depends on the choice of unit cell and can even change sign when the unit cell is translated. This is the case even if the OAM in each unit cell is intrinsic, that is independent of the choice of measurement axis. We show that spin-like OAM can be found only if the OAM per unit cell vanishes. Our results are applicable to the z component of the angular momentum of any x- and y-periodic momentum distribution in the xy plane, and can also be applied to other periodic light beams and arrays of rotating solids or liquids.

© 2006 Optical Society of America

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References

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  1. J. H. Poynting, "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light," Proc. R. Soc. London, Ser. A 82, 560-567 (1909).
    [CrossRef]
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  3. A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053,601 (2002).
    [CrossRef]
  4. M. Berry, "Paraxial beams of spinning light," in Singular Optics, M. S. Soskin, ed., vol. 3487 of Proc. SPIE, pp. 1-5 (SPIE - the International Society for Optical Engineering, Bellingham, Wash., USA, 1998).
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    [CrossRef]
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    [CrossRef]
  7. R. M. Jenkins, J. Banerji, and A. R. Davies, "The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides," J. Opt. A: Pure Appl. Opt. 3, 527-532 (2001).
    [CrossRef]
  8. A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002).
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    [CrossRef]
  11. K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Optics Express 12, 1144-1149 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144</a>.
    [CrossRef] [PubMed]
  12. M. R. Dennis and J. H. Hannay, "Saddle points in the chaotic analytic function and Ginibre characteristic polynomial," J. Phys. A: Math. Gen. 36, 3379-3383 (2003).
    [CrossRef]
  13. J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, "Observation of Vortex Lattices in Bose-Einstein Condensates," Science 292, 476-479 (2001).
    [CrossRef] [PubMed]
  14. R. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991).
  15. P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, England, 1992).
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    [CrossRef]
  17. D. L. Boiko, G. Guerrero, and E. Kapon, "Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays," Optics Express 12, 2597-2602 (2005). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597</a>.
    [CrossRef]
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    [CrossRef]

J. Fluid Mech. (1)

J. Sommeria, "Experimental study of the two-dimensional inverse energy cascade in a square box," J. Fluid Mech. 170, 139-168 (1986).
[CrossRef]

J. Mod. Opt. (1)

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, "Momentum paradox in a vortex core," J. Mod. Opt. 52, 1135-1144 (2005).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

R. M. Jenkins, J. Banerji, and A. R. Davies, "The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides," J. Opt. A: Pure Appl. Opt. 3, 527-532 (2001).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A: Math. Gen. (1)

M. R. Dennis and J. H. Hannay, "Saddle points in the chaotic analytic function and Ginibre characteristic polynomial," J. Phys. A: Math. Gen. 36, 3379-3383 (2003).
[CrossRef]

Opt. Commun. (3)

F. S. Roux, "Optical vortex density limitation," Opt. Commun. 223, 31-37 (2003).
[CrossRef]

I. Dana and I. Freund, "Vortex-lattice wave fields," Opt. Commun. 136, 93-113 (1997).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
[CrossRef]

Optics Express (2)

K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Optics Express 12, 1144-1149 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144</a>.
[CrossRef] [PubMed]

D. L. Boiko, G. Guerrero, and E. Kapon, "Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays," Optics Express 12, 2597-2602 (2005). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597</a>.
[CrossRef]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053,601 (2002).
[CrossRef]

Progr. Opt. (1)

K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).

R. Soc. London (1)

J. H. Poynting, "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light," Proc. R. Soc. London, Ser. A 82, 560-567 (1909).
[CrossRef]

Science (1)

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, "Observation of Vortex Lattices in Bose-Einstein Condensates," Science 292, 476-479 (2001).
[CrossRef] [PubMed]

SPIE (1)

M. Berry, "Paraxial beams of spinning light," in Singular Optics, M. S. Soskin, ed., vol. 3487 of Proc. SPIE, pp. 1-5 (SPIE - the International Society for Optical Engineering, Bellingham, Wash., USA, 1998).

Other (3)

R. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991).

P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, England, 1992).

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

Supplementary Material (4)

» Media 1: MOV (196 KB)     
» Media 2: MOV (44 KB)     
» Media 3: MOV (188 KB)     
» Media 4: MOV (262 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Rotation of spin (left) and orbital angular momentum (OAM, right) states. The left plot shows the local electric field vector (red arrows) in a number of positions in a circularly polarised plane wave. The right plot shows a colour representation of the local phase (red → green → blue → red represents a full 2π phase cycle) of an m = 1 optical vortex. Whereas the spin state rotates around every point, the OAM state rotates around the beam axis. The rotation is indicated here by white arrows; it can clearly be seen in the movies of the time evolution over one optical period of spin (48 KB) and OAM (204 KB), which are contained in the additional multimedia material [Media 1] [Media 2].

Fig. 2.
Fig. 2.

Example of the field ψ(x,y) of a rectangular vortex array. The graph on the left shows the intensity cross-section, coloured according to its phase (using the phase-to-colour mapping described in Fig. 1); the graph on the right shows the phase cross-section. Neighbouring vortices have charges with alternating signs (see phase plot); the intensity is concentrated around the vortices with a positive charge. In the left graph, red arrows indicate the transverse momentum density in the beam, which corresponds to a rotation about the vortex centres. This can be clearly seen in the movie (200 KB) showing the time evolution of the phase-coloured intensity distribution, which is contained in the additional multimedia material. The function ψ(x,y) was calculated using Eq. (5). Specific unit cells that serve as examples throughout this paper are indicated as squares marked ‘1’ to ‘3’. [Media 3]

Fig. 3.
Fig. 3.

OAM per photon in units of h̄, l, of different unit cells of the vortex array shown in Fig. 2. Each unit cell is a square of side length 2π with its lower left corner positioned at coordinates (x 0,y 0). The values of l range from approximately -2.93 (in the corners) to 0.96 (in the centre). The points corresponding to the sample unit cells highlighted in Fig. 2 are marked ‘1’ to ‘3’.

Fig. 4.
Fig. 4.

Unit cells A = A 1 + A 2 + A 3 + A 4 and B = B 1 + B 2 + B 3 + B 4.

Fig. 5.
Fig. 5.

Schematic of the transverse momentum density in different unit cells of the array shown in Fig. 2. Solid arrows indicate the momentum density in the beam. Comparison of cells ‘1’ and ‘2’ shows that in different unit cells the bulk of the momentum density can be at different radii from the cell’s centre (as the OAM in each unit cell is intrinsic and therefore independent of the axis position, we might as well calculate it with respect to a symmetric choice of axis position, namely the centre of the unit cell), leading to a larger OAM density according to Eq. (2), and that the circulation of the momentum density with respect to the axis (dotted arrow in ‘2’) can even be reversed, resulting in OAM of the opposite sign. The respective OAM per photon in unit cells ‘1’ and ‘2’ is 0.96h̄ and -2.93h̄. In cell ‘3’ the OAM in each of the two parts of the cell (separated by the dotted white line) is intrinsic and equal and opposite that in the other part (the circulation of the momentum density in opposite directions is again indicated by dotted arrows), so the OAM in the whole cell is 0.

Fig. 6.
Fig. 6.

OAM per photon in units of h̄, l, per unit cell on propagation through one Talbot period, τ, calculated for the unit cells highlighted in Fig. 2 of the beam given at z = 0 by Eq. (5). The curves corresponding to cells ‘1’, ‘2’ and ‘3’ are respectively drawn solid, dashed and dotted. Insets show the intensity cross-sections across the beam at z = 0, τ/8, τ/4 and τ/2; the position of the unit cells is marked. The additional multimedia material contains a movie (268 KB) showing the evolution of the beam cross-section in unit cell ’1’ over one Talbot period. [Media 4]

Fig. 7.
Fig. 7.

Finite-size optical-vortex array. The centre of the array is covered by unit cells (shown with reduced contrast). An area at the edge of the beam (full contrast) cannot be covered by whole unit cells. The momentum density in the edge is indicated by white arrows; solid arrows indicate the actual momentum density, the dashed arrow indicated the “trend”.

Fig. 8.
Fig. 8.

Example of the dependence of the OAM per photon in a unit cell on its shape. The OAM per photon in the unit cell on the left is -2.94h̄, that in the unit cell on the right is -6.84h̄ (note that the four parts of the unit cell are not connected). The actual unit cell is shown in full contrast; some of the surrounding beam is also shown, in reduced contrast. Like in Fig. 5, the larger OAM per photon in the cell on the right is again due to the fact that the bulk of the momentum density is at a larger radius from the cell’s centre.

Equations (30)

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p ( r ) = Im ( ψ * ψ ) ,
ω ( r ) = ( r × p ( r ) ) z xp x yp x .
Ω A = A ω ( r ) d x d y
l = Ω A ( A ψ 2 d x d y ) .
ψ x y = ( sin x + i sin y ) ( cos x 2 cos y 2 ) 4 .
ψ ( x + nX , y + mY ) = ψ x y ,
P = ( P x , P y ) = D p d x d y = 0 ,
D ( ( r R ) × p ) z d x d y = D ( r × p ) z d x dy .
Ω A = 0 Y 0 X ( r × p ) z d x dy
Ω B = y 0 y 0 + Y x 0 x 0 + X ( r × p ) z d x dy
Ω B = Ω A + X A 3 + A 4 p y d x d y Y A 3 + A 4 p x d x d y
= Ω A + X 0 x 0 0 Y p y d y d x Y 0 y 0 0 X p X d x d y .
0 Y p y d y = 0
0 X p x d x = 0
Ω A = 0 Y 0 X xp y yp x d x d y = 0 X x ( 0 Y p y d y ) d x 0 Y y ( 0 X p x d x ) d y ,
0 X 0 Y Ω ( x 0 , y 0 ) d x 0 d y 0 ,
0 X 0 Y Ω ( x 0 , y 0 ) d x 0 d y 0 = 2 0 X 0 Y ( x 0 P y y 0 P x ) d x 0 d y 0 ,
0 X 0 Y Ω ( x 0 , y 0 ) d x 0 d y 0 = 0 .
ψ x y = m , n a m , n exp ( 2 πi ( mx X + ny Y ) ) ,
ω x y = 2 π Re m , n , n , n a m , n * a m , n ( nx Y my X ) exp ( 2 π i ( ( m m ) x X + ( n n ) y Y ) ) .
p = 2 π m , n ( mY , nX ) a m , n 2 .
Ω ( x c , y c ) = y c Y 2 y c + Y 2 x c X 2 x c + X 2 ω x y d x d y
= r c × P + Im m . n a m , n ( nX 2 m m a m , n * ( 1 ) m m exp ( 2 π i ( m m ) x c X ) m m mY 2 n n a m , n * ( 1 ) n n exp ( 2 π i ( n n ) y c Y ) n n ) .
exp ( i ( k x x + k y y ) ) exp ( i ( k x x + k y y + k z z ) ) ,
k z ( k 2 k x 2 k y 2 ) 1 2 k ( k x 2 + k y 2 ) ( 2 k ) ,
ψ prop ( x , y , z ) = m , n a m , n exp ( 2 πi ( mx X + ny Y ( m 2 X 2 + n 2 Y 2 ) ) ( πz k ) ) .
ψ prop ( x , y , z ) = m , n a m , n exp ( 2 πi ( mx X + ny Y ( m 2 M 2 + n 2 N 2 ) ( π kZ 2 ) z ) ) .
a m , n a m , n exp ( 2 πi ( M 2 m 2 + N 2 n 2 ) z τ ) .
ψ prop ( x , y , z + τ 2 ) = ψ prop ( x ± X 2 , y ± Y 2 , z ) .
Ω ( x c , y c ) z = 1 2 Im m , n 0 a m , n ( a m , n * n m X 2 exp ( 4 π i mx c X ) a m , n * m n Y 2 exp ( 4 πiny c Y ) ) .

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