Abstract

The generation of light beams carrying orbital angular momentum (OAM) has been greatly advanced with the emergence of the recently reported integrated optical vortex emitters. Generally, optical vortices emitted by these devices possess cylindrically symmetric states of polarization and spiral phase fronts, and they can be defined as cylindrical vector vortices (CVVs). Using the radiation of angularly arranged dipoles to model the CVVs, these beams as hybrid modes of two circularly polarized scalar vortices are theoretically demonstrated to own well-defined total angular momentum. Moreover, the effect of spin–orbit interactions of angular momentum is identified in the CVVs when the size of the emitting structure varies. This effect results in the diminishing spin component of angular momentum and purer OAM states at large structure radii.

© 2014 Optical Society of America

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References

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  1. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
    [CrossRef]
  2. N. K. Fontaine, C. R. Doerr, and L. Buhl, “Efficient multiplexing and demultiplexing of free-space orbital angular momentum using photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical DigestLos Angeles, California, March4–8, 2012 (Optical Society of America, 2012), paper OTu1l.2.
  3. Z. Zhao, J. Wang, S. Li, and A. E. Willner, Opt. Lett. 38, 932 (2013).
    [CrossRef]
  4. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef]
  5. J. Zhu, X. Cai, Y. Chen, and S. Yu, Opt. Lett. 38, 1343 (2013).
    [CrossRef]
  6. W. Nasalski, Opt. Lett. 38, 809 (2013).
    [CrossRef]
  7. W. Nasalski, Appl. Phys. B 115, 155 (2014).
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    [CrossRef]
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    [CrossRef]
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  16. J. H. Crichton and P. L. Marston, Electron. J. Diff. Eqns. Conf. 04, 37 (2000).
  17. S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
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    [CrossRef]

2014 (1)

W. Nasalski, Appl. Phys. B 115, 155 (2014).
[CrossRef]

2013 (4)

2012 (1)

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

2010 (1)

S. M. Barnett, J. Mod. Opt. 57, 1339 (2010).
[CrossRef]

2002 (2)

2000 (1)

J. H. Crichton and P. L. Marston, Electron. J. Diff. Eqns. Conf. 04, 37 (2000).

1998 (1)

M. V. Berry, Proc. SPIE 3487, 6 (1998).
[CrossRef]

1994 (2)

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

S. van Enk and G. Nienhuis, Europhys. Lett. 25, 497 (1994).
[CrossRef]

1992 (1)

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

1956 (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Allen, L.

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Barnett, S. M.

S. M. Barnett, J. Mod. Opt. 57, 1339 (2010).
[CrossRef]

S. M. Barnett, J. Opt. B 4, S7 (2002).
[CrossRef]

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Belyi, V. N.

V. N. Belyi, N. A. Khilo, S. N. Kurilkina, and N. S. Kazak, J. Opt. 15, 044018 (2013).
[CrossRef]

Berry, M. V.

M. V. Berry, Proc. SPIE 3487, 6 (1998).
[CrossRef]

Biener, G.

Bomzon, Z.

Buhl, L.

N. K. Fontaine, C. R. Doerr, and L. Buhl, “Efficient multiplexing and demultiplexing of free-space orbital angular momentum using photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical DigestLos Angeles, California, March4–8, 2012 (Optical Society of America, 2012), paper OTu1l.2.

Cai, X.

J. Zhu, X. Cai, Y. Chen, and S. Yu, Opt. Lett. 38, 1343 (2013).
[CrossRef]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Chen, Y.

Crichton, J. H.

J. H. Crichton and P. L. Marston, Electron. J. Diff. Eqns. Conf. 04, 37 (2000).

Doerr, C. R.

N. K. Fontaine, C. R. Doerr, and L. Buhl, “Efficient multiplexing and demultiplexing of free-space orbital angular momentum using photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical DigestLos Angeles, California, March4–8, 2012 (Optical Society of America, 2012), paper OTu1l.2.

Fontaine, N. K.

N. K. Fontaine, C. R. Doerr, and L. Buhl, “Efficient multiplexing and demultiplexing of free-space orbital angular momentum using photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical DigestLos Angeles, California, March4–8, 2012 (Optical Society of America, 2012), paper OTu1l.2.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2006).

Hasman, E.

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Kazak, N. S.

V. N. Belyi, N. A. Khilo, S. N. Kurilkina, and N. S. Kazak, J. Opt. 15, 044018 (2013).
[CrossRef]

Khilo, N. A.

V. N. Belyi, N. A. Khilo, S. N. Kurilkina, and N. S. Kazak, J. Opt. 15, 044018 (2013).
[CrossRef]

Kliener, V.

Kurilkina, S. N.

V. N. Belyi, N. A. Khilo, S. N. Kurilkina, and N. S. Kazak, J. Opt. 15, 044018 (2013).
[CrossRef]

Li, S.

Marston, P. L.

J. H. Crichton and P. L. Marston, Electron. J. Diff. Eqns. Conf. 04, 37 (2000).

Nasalski, W.

W. Nasalski, Appl. Phys. B 115, 155 (2014).
[CrossRef]

W. Nasalski, Opt. Lett. 38, 809 (2013).
[CrossRef]

Nienhuis, G.

S. van Enk and G. Nienhuis, Europhys. Lett. 25, 497 (1994).
[CrossRef]

O’Brien, J. L.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2006).

Sorel, M.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Spreeuw, R.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Strain, M. J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Thompson, M. G.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

van Enk, S.

S. van Enk and G. Nienhuis, Europhys. Lett. 25, 497 (1994).
[CrossRef]

Wang, J.

Z. Zhao, J. Wang, S. Li, and A. E. Willner, Opt. Lett. 38, 932 (2013).
[CrossRef]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Willner, A. E.

Woerdman, J.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Yu, S.

J. Zhu, X. Cai, Y. Chen, and S. Yu, Opt. Lett. 38, 1343 (2013).
[CrossRef]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Zhao, Z.

Zhu, J.

J. Zhu, X. Cai, Y. Chen, and S. Yu, Opt. Lett. 38, 1343 (2013).
[CrossRef]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Appl. Phys. B (1)

W. Nasalski, Appl. Phys. B 115, 155 (2014).
[CrossRef]

Electron. J. Diff. Eqns. Conf. (1)

J. H. Crichton and P. L. Marston, Electron. J. Diff. Eqns. Conf. 04, 37 (2000).

Europhys. Lett. (1)

S. van Enk and G. Nienhuis, Europhys. Lett. 25, 497 (1994).
[CrossRef]

J. Mod. Opt. (1)

S. M. Barnett, J. Mod. Opt. 57, 1339 (2010).
[CrossRef]

J. Opt. (1)

V. N. Belyi, N. A. Khilo, S. N. Kurilkina, and N. S. Kazak, J. Opt. 15, 044018 (2013).
[CrossRef]

J. Opt. B (1)

S. M. Barnett, J. Opt. B 4, S7 (2002).
[CrossRef]

Opt. Commun. (1)

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (1)

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).

Proc. SPIE (1)

M. V. Berry, Proc. SPIE 3487, 6 (1998).
[CrossRef]

Science (1)

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
[CrossRef]

Other (3)

N. K. Fontaine, C. R. Doerr, and L. Buhl, “Efficient multiplexing and demultiplexing of free-space orbital angular momentum using photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical DigestLos Angeles, California, March4–8, 2012 (Optical Society of America, 2012), paper OTu1l.2.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2006).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

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Figures (4)

Fig. 1.
Fig. 1.

Angular-dipole groups as the basic constructing elements of the CVV emitters: APADs (red solid arrows) and RPADs (blue dashed arrows).

Fig. 2.
Fig. 2.

Numerically calculated ratio of the far-field AM fluxes to the energy flux of CVVs emitted by an APAD group as a function of topological charge. q=36, R=3.9μm, and λ=1.55μm. The ratios of the TAM fluxes (×) and the OAM flux (○) to the energy flux are presented as the difference between the ratios and the topological charge; the spin components are illustrated by the + marks.

Fig. 3.
Fig. 3.

Numerically calculated ratios of the AM fluxes to the energy flux of the angular-dipole-emitted CVVs. The spin (+), orbital (○) components, and total angular momentum flux (×) as a function of the normalized radius (λ=1.55μm) are illustrated as (a) APADs, =1; (b) APADs, =2; (c) RPADs, =1; and (d) RPADs, =2.

Fig. 4.
Fig. 4.

Representation of spin-to-orbital AM conversion of APAD-emitted beam when =1, from (a) a circularly polarized beam when at the critical radius to (b) a vectorial vortex beam with well-defined OAM when Rn1.

Equations (12)

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Eρ,A=(j)Aν2Φ(ρ,ζ)ρ2+ζ2exp(jφ)(J+1+J1),
Eφ,A=(j)+1Aν2Φ(ρ,ζ)ρ2+ζ2exp(jφ)(J+1J1),
Eζ,A=(j)Aν2ρζΦ(ρ,ζ)(ρ2+ζ2)3/2exp(jφ)(J+1+J1),
ET,A=[Ex,AEy,A]T=(j)Aν2Φ(ρ,ζ)ρ2+ζ2×{J1ej(1)φ[1j]T+J+1ej(+1)φ[1j]T}.
S1,A=2J1J+1cos2φ/(J12+J+12),
S2,A=2J1J+1sin2φ/(J12+J+12),
S3,A=(J12J+12)/(J12+J+12).
SζA=0[ρdρ(J12J+12)/(ρ2+ζ2)]0[ρdρ(J12+J+12)/(ρ2+ζ2)],
LζA=0{ρdρ[(1)J12+(+1)J+12]/(ρ2+ζ2)}0[ρdρ(J12+J+12)/(ρ2+ζ2)].
SζA=/ν2+3(142)/8ν4+1(42+3)/8ν2+3(164+56215)/128ν4·.
MS,zz=12μ0ωRe[jρdφdρ(ExBx*+EyBy*)],
ML,zz=14μ0ωRe[jρdφdρ(Bx*Eyφ+EyBx*φExBy*φ+By*Exφ)],

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