Abstract

We propose a handy scheme to generate a vectorial elliptic hollow beam by using a short dual-mode elliptic hollow fiber, study its propagation characteristics in free space, and calculate its total angular momentum at different propagating distances. Our study shows that the resulting elliptic hollow beam has a self-focusing effect in near field, an arbitrary polarization, and a position-dependent fractional angular momentum and that the directions of the major and minor axes of the elliptical intensity profile will be interchanged after the self-focusing plane.

© 2011 Optical Society of America

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References

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  1. J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier Science, 2003), pp. 119–204.
    [CrossRef]
  2. J. C. Gutiérrez-Vega, Laser Beam Shaping IX 7062, 706207(2008).
  3. C. Yeh, J. Appl. Phys. 33, 3235 (1962).
    [CrossRef]
  4. J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
    [CrossRef]
  5. F. A. Alhargan, ACM Trans. Math. Softw. 26, 390 (2000).
    [CrossRef]
  6. R. K. Luneberg, Mathematical Theory of Optics (U. of California, 1964).
  7. Y. I. Shin, K. Kim, J. A. Kim, H. R. Noh, W. Jhe, K. Oh, and U. C. Paek, Opt. Lett. 26, 119 (2001).
    [CrossRef]
  8. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  9. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
    [CrossRef]
  10. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
    [CrossRef] [PubMed]

2008 (1)

J. C. Gutiérrez-Vega, Laser Beam Shaping IX 7062, 706207(2008).

2006 (1)

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

2001 (2)

2000 (1)

F. A. Alhargan, ACM Trans. Math. Softw. 26, 390 (2000).
[CrossRef]

1998 (1)

J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
[CrossRef]

1992 (1)

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

1962 (1)

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[CrossRef]

Alhargan, F. A.

F. A. Alhargan, ACM Trans. Math. Softw. 26, 390 (2000).
[CrossRef]

Allen, L.

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

Andersen, M. F.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Beijersbergen, M. W.

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

Chávez-Cerda, S.

Cladé, P.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Gao, W.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier Science, 2003), pp. 119–204.
[CrossRef]

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, Laser Beam Shaping IX 7062, 706207(2008).

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
[CrossRef]

Helmerson, K.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Jhe, W.

Y. I. Shin, K. Kim, J. A. Kim, H. R. Noh, W. Jhe, K. Oh, and U. C. Paek, Opt. Lett. 26, 119 (2001).
[CrossRef]

J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
[CrossRef]

Kim, J. A.

Kim, K.

Y. I. Shin, K. Kim, J. A. Kim, H. R. Noh, W. Jhe, K. Oh, and U. C. Paek, Opt. Lett. 26, 119 (2001).
[CrossRef]

J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
[CrossRef]

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (U. of California, 1964).

Natarajan, V.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

New, G. H. C.

Noh, H. R.

Oh, K.

Paek, U. C.

Phillips, W. D.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Ryu, C.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Shim, W.

J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
[CrossRef]

Shin, Y. I.

Spreeuw, R. J. C.

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

Vaziri, A.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Woerdman, J. P.

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

Yeh, C.

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[CrossRef]

Yin, J.

J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
[CrossRef]

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier Science, 2003), pp. 119–204.
[CrossRef]

Zhu, Y.

J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
[CrossRef]

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier Science, 2003), pp. 119–204.
[CrossRef]

ACM Trans. Math. Softw. (1)

F. A. Alhargan, ACM Trans. Math. Softw. 26, 390 (2000).
[CrossRef]

J. Appl. Phys. (1)

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[CrossRef]

Laser Beam Shaping IX (1)

J. C. Gutiérrez-Vega, Laser Beam Shaping IX 7062, 706207(2008).

Opt. Eng. (Bellingham, Wash.) (1)

J. Yin, K. Kim, W. Shim, Y. Zhu, and W. Jhe, Opt. Eng. (Bellingham, Wash.) 37, 2277 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (1)

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Other (2)

R. K. Luneberg, Mathematical Theory of Optics (U. of California, 1964).

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier Science, 2003), pp. 119–204.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Generation scheme of the dual-mode output elliptic hollow beam. GB, Gaussian beam; EHF, elliptic hollow fiber; vEHB, vectorial elliptic hollow beam. (b) Elliptic coordinate system.

Fig. 2
Fig. 2

Transmission of the two modes in the EHF. (a), (b) Intensity and polarization distributions of a single HE e 11 and HE o 11 mode. (c) TAM per photon of a single mode and dual mode vary with the length of the EHF.

Fig. 3
Fig. 3

Propagation characteristics of the EHB and its self-focusing effect. (a)–(c) Transverse intensity profiles and polarization distributions of the EHB at z = 0 , 200 μm , and 5 mm . (d) Longitudinal propagation of the EHB. (e) Dependence of the beam radius of the EHB on z.

Fig. 4
Fig. 4

TAM per photon of the vEHB in (a) near and (b) far field. Dashed lines show the invariable TAM in the whole cross section. Black squares are the numerically calculated results, and solid curves are the fitting curves of the TAM.

Equations (5)

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H z = { A e I e 1 ( ξ , q 1 e ) Q e 1 ( η , q 1 e ) e j ( ω t β e z ) + A o I o 1 ( ξ , q 1 o ) Q o 1 ( η , q 1 o ) e j ( ω t β o z ) 0 ξ ξ 1 [ B e J e 1 ( ξ , q 2 e ) + C e Y e 1 ( ξ , q 2 e ) ] S e 1 ( η , q 2 e ) e j ( ω t β e z ) + [ B o J o 1 ( ξ , q 2 o ) + C o Y o 1 ( ξ , q 2 o ) ] S o 1 ( η , q 2 o ) e j ( ω t β o z ) ξ 1 < ξ ξ 2 D e K e 1 ( ξ , q 3 e ) Q e 1 ( η , q 3 e ) e j ( ω t β e z ) + D o K o 1 ( ξ , q 3 o ) Q o 1 ( η , q 3 o ) e j ( ω t β o z ) ξ 2 < ξ < ,
E z = { A e I o 1 ( ξ , q 1 e ) Q o 1 ( η , q 1 e ) e j ( ω t β e z ) + A o I e 1 ( ξ , q 1 o ) Q e 1 ( η , q 1 o ) e j ( ω t β o z ) 0 ξ ξ 1 [ B e J o 1 ( ξ , q 2 e ) + C e Y o 1 ( ξ , q 2 e ) ] S o 1 ( η , q 2 e ) e j ( ω t β e z ) + [ B o J e 1 ( ξ , q 2 o ) + C o Y e 1 ( ξ , q 2 o ) ] S e 1 ( η , q 2 o ) e j ( ω t β o z ) ξ 1 < ξ ξ 2 D e K o 1 ( ξ , q 3 e ) Q o 1 ( η , q 3 e ) e j ( ω t β e z ) + D o K e 1 ( ξ , q 3 o ) Q e 1 ( η , q 3 o ) e j ( ω t β o z ) ξ 2 < ξ < ,
E ˜ j ( ξ , η , z ) = z i λ 0 2 π 0 E ˜ j ( ξ 1 , η 1 , 0 ) e i k ρ ρ 2 ( 1 1 i k ρ ) h 2 ( cosh 2 ξ 1 cos 2 η 1 ) d ξ 1 d η 1 ,
E ˜ z ( ξ , η , z ) = 1 i λ 0 2 π 0 [ h h u ( cosh ξ 1 sinh ξ sin η 1 sin η + sinh ξ 1 cosh ξ cos η 1 cos η cosh ξ 1 sinh ξ 1 ) E ˜ ξ ( ξ 1 , η 1 , 0 ) + h h v ( sinh ξ 1 sinh ξ cos η 1 sin η cosh ξ 1 cosh ξ sin η 1 cos η + cos η 1 sin η 1 ) E ˜ η ( ξ 1 , η 1 , 0 ) + z E ˜ z ( ξ 1 , η 1 , 0 ) e i k ρ ρ 2 ( 1 1 i k ρ ) h 2 ( cosh 2 ξ 1 cos 2 η 1 ) d ξ 1 d η 1 ,
δ J z = ω r × p φ d x d y c p z d x d y = ω r × E × H φ d x d y c E × H z d x d y ,

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