Abstract

We develop a theoretical model for the recently reported integrated optical vortex beam emitters that incorporate angular gratings in microring resonators. Using azimuthally polarized dipole oscillators to represent emissions scattered from the grating elements that are located along the inner wall of the ring waveguide, we obtain expressions for far-field components under the paraxial approximation. The results show that the emission is of the form of cylindrical vector Bessel beams with exactly defined optical orbital angular momentum, and can have azimuthal, radial, and longitudinal field components after propagation. The calculation results for field distributions in both near and far zone agree well with the experimental results.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]

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]

2011 (2)

C. R. Doerr and L. L. Buhl, Opt. Lett. 36, 1209 (2011).
[CrossRef]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 53601 (2011).
[CrossRef]

2009 (1)

2007 (1)

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

2004 (1)

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57, 35 (2004).
[CrossRef]

2002 (1)

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

1996 (2)

1994 (1)

M. W. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. White, Opt. Lett. 17, 221 (1992).
[CrossRef]

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

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 53601 (2011).
[CrossRef]

Allen, L.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57, 35 (2004).
[CrossRef]

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

Beijersbergen, M.

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

Beijersbergen, M. W.

M. W. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Buhl, L. L.

Cai, X.

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]

Coerwinkel, R.

M. W. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57, 35 (2004).
[CrossRef]

Doerr, C. R.

Freund, I.

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

Greene, P. L.

Hall, D. G.

Heckenberg, N. R.

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]

Kristensen, M.

M. W. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

McDuff, R.

Milione, G.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 53601 (2011).
[CrossRef]

Molina-Terriza, G.

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

Nolan, D. A.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 53601 (2011).
[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]

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57, 35 (2004).
[CrossRef]

Smith, C. P.

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]

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]

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 53601 (2011).
[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]

Torner, L.

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

Torres, J. P.

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

Von Hippel, A. R.

A. R. Von Hippel, Dielectrics and Waves (Wiley, 1954).

Wang, 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]

White, A.

Woerdman, J.

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

Woerdman, J. P.

M. W. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Yu, S.

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]

Zhan, Q.

Zhu, 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]

Adv. Opt. Photon. (1)

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

Nat. Phys. (1)

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

Opt. Commun. (2)

M. W. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (1)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 53601 (2011).
[CrossRef]

Phys. Today (1)

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57, 35 (2004).
[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 (1)

A. R. Von Hippel, Dielectrics and Waves (Wiley, 1954).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of the dipole model in the normalized cylindrical coordinate for interference analysis.

Fig. 2.
Fig. 2.

Simulated near-zone intensity distribution of different components at ζ=0.4λ/R with l=0: (a) Eφ, (b) Eρ and Eζ, (c) Ex, and (d) Ey.

Fig. 3.
Fig. 3.

Calculated cross sections of the far-field intensity distribution of Eφ (red curves), Eρ (blue curves), and Eζ (green curves) as a function of diffraction angle Θ, with charges: (a) l=0, (b) l=1, (c) l=2, and (d) l=3. Paraxial approximated results are presented (solid curves) as well as the nonparaxial ones (dashed curves).

Fig. 4.
Fig. 4.

Calculated transversal phase distribution of the far field (a) Eφ (l=0), (b) Eρ (l=1), (c) Eρ (l=2), and (d) Eφ (l=3).

Equations (9)

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El(ρ,φ,ζ)=AR3m=1qexp(jνrm)exp(jlφm)×[(ν2rmjνrm21rm3)(r^m×φ^m)×r^m(j2νrm2+2rm3)(r^m·φ^m)·r^m],
Eρ,l(ρ,φ,ζ)=El(ρ,φ,ζ)·ρ^=AR3ν2ζΦ(ρ,ζ)×m=1qexp(jνtanΘcosΔφm)exp(jlφm)sinΔφm,
m=1qexp(jνtanΘcosΔφm)exp(jnΔφm)jnqJn,
Eρ,l(ρ,φ,ζ)=jlAlqR3νρΦ(ρ,ζ)exp(jlφ)Jl,
Eφ,l(ρ,φ,ζ)=jl1AqR3ν2ζΦ(ρ,ζ)exp(jlφ)Jl,
Eζ,l(ρ,φ,ζ)=jlAlqR3νζΦ(ρ,ζ)exp(jlφ)Jl.
|Eρ,l|,|Eζ,l|Jl(νtanΘ),|Eφ,l|Jl(νtanΘ).
Eφ,0(ρ,ζ)=jAqR3ν2ζΦ(ρ,ζ)J1.
ET,l=[Ex,lEy,l]=jlAqR3ν22ζΦ(ρ,ζ)×{Jl1exp[j(l1)φ][1j]+Jl+1exp[j(l+1)φ][1j]}.

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