Abstract

We describe an approach to determining both the angular and the radial modal content of a scalar optical beam in terms of optical angular momentum modes. A modified Mach–Zehnder interferometer that incorporates a spatial rotator to determine the angular modes and an optical realization of the fractional Hankel transform (fHT) to determine the radial modes is analyzed. Varying the rotation angle and the order of the fHT produces a two-dimensional (2D) interferogram from which we extract the modal coefficients by simple 2D Fourier analysis.

© 2011 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  2. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
    [CrossRef]
  3. S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
    [CrossRef]
  4. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]
  5. R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
    [CrossRef] [PubMed]
  6. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
    [CrossRef]
  7. T. Alieva and M. J. Bastiaans, Opt. Lett. 24, 1206 (1999).
    [CrossRef]
  8. V. Namias, IMA J. Appl. Math. 26, 187 (1980).
    [CrossRef]
  9. L. Yu, Y. Lu, X. Zeng, M. Huang, M. Chen, W. Huang, and Z. Zhu, Opt. Lett. 23, 1158 (1998).
    [CrossRef]
  10. V. Namias, IMA J. Appl. Math. 25, 241 (1980).
    [CrossRef]
  11. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).
  12. A. Sahin, H. M. Ozaktas, and D. Mendlovic, Appl. Opt. 37, 2130 (1998).
    [CrossRef]
  13. J. A. Rodrigo, T. Alieva, and M. L. Calvo, Opt. Express 17, 4976 (2009).
    [CrossRef] [PubMed]
  14. W. H. Peeters, E. J. K. Verstegen, and M. P. van Exter, Phys. Rev. A 76, 042302 (2007).
    [CrossRef]

2010 (1)

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

2009 (1)

2008 (1)

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

2007 (1)

W. H. Peeters, E. J. K. Verstegen, and M. P. van Exter, Phys. Rev. A 76, 042302 (2007).
[CrossRef]

2006 (1)

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

2004 (1)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

1999 (1)

1998 (2)

1992 (1)

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

1980 (2)

V. Namias, IMA J. Appl. Math. 26, 187 (1980).
[CrossRef]

V. Namias, IMA J. Appl. Math. 25, 241 (1980).
[CrossRef]

Alieva, T.

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

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

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Barnett, S. M.

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Bastiaans, M. J.

Beijersbergen, M. W.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

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

Berkhout, G. C. G.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Calvo, M. L.

Chen, M.

Courtial, J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Huang, M.

Huang, W.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Lavery, M. P. J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Lu, Y.

Mendlovic, D.

Namias, V.

V. Namias, IMA J. Appl. Math. 26, 187 (1980).
[CrossRef]

V. Namias, IMA J. Appl. Math. 25, 241 (1980).
[CrossRef]

Ozaktas, H. M.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, Appl. Opt. 37, 2130 (1998).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Padgett, M. J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Peeters, W. H.

W. H. Peeters, E. J. K. Verstegen, and M. P. van Exter, Phys. Rev. A 76, 042302 (2007).
[CrossRef]

Rodrigo, J. A.

Sahin, A.

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

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]

van Exter, M. P.

W. H. Peeters, E. J. K. Verstegen, and M. P. van Exter, Phys. Rev. A 76, 042302 (2007).
[CrossRef]

Verstegen, E. J. K.

W. H. Peeters, E. J. K. Verstegen, and M. P. van Exter, Phys. Rev. A 76, 042302 (2007).
[CrossRef]

Woerdman, J. P.

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

Yu, L.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Zambrini, R.

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

Zeng, X.

Zhu, Z.

Appl. Opt. (1)

IMA J. Appl. Math. (2)

V. Namias, IMA J. Appl. Math. 26, 187 (1980).
[CrossRef]

V. Namias, IMA J. Appl. Math. 25, 241 (1980).
[CrossRef]

Laser Photon. Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (2)

W. H. Peeters, E. J. K. Verstegen, and M. P. van Exter, Phys. Rev. A 76, 042302 (2007).
[CrossRef]

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

Phys. Rev. Lett. (3)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Other (2)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

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Figures (2)

Fig. 1
Fig. 1

Optical configurations for (a) an angular mode analyzer and (b) a joint angular–radial mode analyzer.

Fig. 2
Fig. 2

Examples of calculated 2D radial and angular spectra corresponding to the setup in Fig. 1b. The first column displays the beam intensity I ( ρ , θ ) = | E ( ρ , θ ) | 2 ; each panel is 25.6 × 25.6 of the Gaussian mode standard deviation. The second column displays the 2D interferogram P ( α , φ ) and the third column displays the angular–radial modal spectrum | c p | 2 , the 2D FT of P ( α , φ ) . All panels are normalized to the peak value.

Equations (14)

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u p ( ρ , θ ) = A p ρ | | L p | | ( ρ 2 ) e ρ 2 / 2 e i θ ,
E ( ρ , θ ) = , p c p u p ( ρ , θ ) ,
P ( φ ) = 1 + 0 0 2 π E ( ρ , θ ) E * ( ρ , θ φ ) ρ d ρ d θ = 1 + | c 0 | 2 cos ( φ ) ,
E ( ρ , θ ; φ ) = R ( θ , θ ; φ ) E ( ρ , θ ) d θ = E ( ρ , θ φ ; 0 ) ,
R ( θ , θ ; φ ) = δ ( θ θ φ ) = 1 2 π e i φ e i θ e i θ .
K 0 ( ρ , ρ ; α ) u 0 p ( ρ ) ρ d ρ = e i p α u 0 p ( ρ ) .
K 0 ( ρ , ρ ; α ) = p e i p α u 0 p ( ρ ) u 0 p * ( ρ ) .
K 0 ( ρ , ρ ; α ) = i e i a sin a exp ( i ρ 2 + ρ 2 2 tan a ) J 0 ( ρ ρ sin a ) ,
P ( α ) = 1 + 0 0 2 π E * ( ρ , θ ) F α { E ( ρ , θ ) } ρ d ρ d θ = 1 + p | c 0 p | 2 cos ( p α ) .
K ( ρ , ρ ; α ) = e i ( 1 + ) ( π 2 a ) sin a exp ( i ρ 2 + ρ 2 2 tan a ) J ( ρ ρ sin a ) .
K ( ρ , ρ ; α ) u p ( ρ , θ ) ρ d ρ = e i p α u p ( ρ , θ ) ,
K ( ρ , ρ ; α ) = p e i p α u p ( ρ , θ ) u p * ( ρ , θ ) .
E ( ρ , θ ; α , φ ) = p c p e i ( φ + p α ) u p ( ρ , θ ) ,
P ( α , φ ) = 1 + 0 0 2 π E ( ρ , θ ) E * ( ρ , θ ; α , φ ) ρ d ρ d θ = 1 + , p | c p | 2 cos ( p α + φ ) .

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