Abstract

Orthogonal sets of 2D transverse modes are key to controlling the spatial degree of freedom of light in a classical or quantum context. In contrast to the azimuthal part, which is easily accessible using orbital angular momentum modes, control of the radial part is more difficult. We show here that simple sets of orthogonal binary sequences, the Walsh functions, provide a workable solution for exploration of the radial space with phase-only spatial light modulation. We demonstrate this by measuring “sequency” quantum correlations between different radial Walsh modes of spatially entangled photon pairs and for numerically optimized versions thereof.

© 2013 Optical Society of America

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  1. J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
    [CrossRef]
  2. J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990).
    [CrossRef]
  3. C. H. Monken, P. H. S. Ribeiro, and S. Pádua, Phys. Rev. A 57, 3123 (1998).
    [CrossRef]
  4. M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. K. G. Beauchamp, Walsh Functions and Their Applications (Academic, 1975).
  21. S. G. Tzafestas, Walsh Functions in Signal and Systems Analysis and Design (Van Nostrand Reinhold, 1985).
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2012 (5)

F. Miatto, H. L. Pires, S. Barnett, and M. Exter, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, Phys. Rev. Lett. 108, 173604 (2012).
[CrossRef]

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

C. Stockbridge, Y. Lu, J. Moore, S. Hoffman, R. Paxman, K. Toussaint, and T. Bifano, Opt. Express 20, 15086 (2012).
[CrossRef]

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, Opt. Express 20, 23589 (2012).
[CrossRef]

2011 (3)

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, Phys. Rev. A 83, 060302 (2011).
[CrossRef]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

F. M. Miatto, A. M. Yao, and S. M. Barnett, Phys. Rev. A 83, 033816 (2011).
[CrossRef]

2009 (1)

2005 (2)

M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

S. P. Walborn, S. Pádua, and C. H. Monken, Phys. Rev. A 71, 053812 (2005).
[CrossRef]

2004 (1)

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

2003 (1)

2002 (1)

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, Phys. Rev. A 65, 033823 (2002).
[CrossRef]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

1998 (1)

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, Phys. Rev. A 57, 3123 (1998).
[CrossRef]

1990 (1)

J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990).
[CrossRef]

1989 (1)

J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
[CrossRef]

1988 (1)

1986 (1)

1977 (1)

M. De and L. Hazra, Opt. Acta 24, 221 (1977).
[CrossRef]

1969 (2)

H. Harmuth, IEEE Spectrum 6, 82 (1969).
[CrossRef]

J. E. Gibbs and H. A. Gebbie, Nature 224, 1012 (1969).
[CrossRef]

1923 (1)

J. L. Walsh, Am. J. Math. 45, 5 (1923).
[CrossRef]

Agnew, M.

Allen, L.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Barnett, S.

F. Miatto, H. L. Pires, S. Barnett, and M. Exter, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

Barnett, S. M.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

F. M. Miatto, A. M. Yao, and S. M. Barnett, Phys. Rev. A 83, 033816 (2011).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Beauchamp, K. G.

K. G. Beauchamp, Walsh Functions and Their Applications (Academic, 1975).

Bifano, T.

Bobrov, I. B.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, Phys. Rev. A 83, 060302 (2011).
[CrossRef]

Boyd, R. W.

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, Opt. Express 20, 23589 (2012).
[CrossRef]

M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Cottrell, D. M.

Davis, J. A.

De, M.

M. De and L. Hazra, Opt. Acta 24, 221 (1977).
[CrossRef]

Dennis, M. R.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

Eberly, J. H.

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

Eliel, E. R.

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, Phys. Rev. Lett. 108, 173604 (2012).
[CrossRef]

Exter, M.

F. Miatto, H. L. Pires, S. Barnett, and M. Exter, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

Fickler, R.

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, arXiv:1306.0096 (2013).

Forbes, A.

Franke-Arnold, S.

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Franson, J. D.

J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
[CrossRef]

Gebbie, H. A.

J. E. Gibbs and H. A. Gebbie, Nature 224, 1012 (1969).
[CrossRef]

Gibbs, J. E.

J. E. Gibbs and H. A. Gebbie, Nature 224, 1012 (1969).
[CrossRef]

Giovannini, D.

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

Guha, A.

Harmuth, H.

H. Harmuth, IEEE Spectrum 6, 82 (1969).
[CrossRef]

H. Harmuth, Transmission of Information by Orthogonal Functions (Springer, 1969).

Hazra, L.

M. De and L. Hazra, Opt. Acta 24, 221 (1977).
[CrossRef]

Hazra, L. N.

Hoffman, S.

Howell, J. C.

M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Huber, M.

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, arXiv:1306.0096 (2013).

Ivanov, D. P.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, Phys. Rev. A 83, 060302 (2011).
[CrossRef]

Jack, B.

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

Kalinkin, A. A.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, Phys. Rev. A 83, 060302 (2011).
[CrossRef]

Khan, I. A.

M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Krenn, M.

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, arXiv:1306.0096 (2013).

Kulik, S. P.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, Phys. Rev. A 83, 060302 (2011).
[CrossRef]

Lapkiewicz, R.

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, arXiv:1306.0096 (2013).

Law, C. K.

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

Leach, J.

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, Opt. Express 20, 23589 (2012).
[CrossRef]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

Lilly, R. A.

Löffler, W.

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, Phys. Rev. Lett. 108, 173604 (2012).
[CrossRef]

Lu, Y.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

McLaren, M.

Miatto, F.

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

F. Miatto, H. L. Pires, S. Barnett, and M. Exter, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

Miatto, F. M.

F. M. Miatto, A. M. Yao, and S. M. Barnett, Phys. Rev. A 83, 033816 (2011).
[CrossRef]

Monken, C. H.

S. P. Walborn, S. Pádua, and C. H. Monken, Phys. Rev. A 71, 053812 (2005).
[CrossRef]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, Phys. Rev. A 57, 3123 (1998).
[CrossRef]

Moore, J.

O’Sullivan-Hale, M. N.

M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

Padgett, M.

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

Padgett, M. J.

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, Opt. Express 20, 23589 (2012).
[CrossRef]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Pádua, S.

S. P. Walborn, S. Pádua, and C. H. Monken, Phys. Rev. A 71, 053812 (2005).
[CrossRef]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, Phys. Rev. A 57, 3123 (1998).
[CrossRef]

Paxman, R.

Pires, H. L.

F. Miatto, H. L. Pires, S. Barnett, and M. Exter, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

Ramelow, S.

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, arXiv:1306.0096 (2013).

Rarity, J. G.

J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990).
[CrossRef]

Ribeiro, P. H. S.

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, Phys. Rev. A 57, 3123 (1998).
[CrossRef]

Romero, J.

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

Roux, F. S.

Salakhutdinov, V. D.

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, Phys. Rev. Lett. 108, 173604 (2012).
[CrossRef]

Stockbridge, C.

Straupe, S. S.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, Phys. Rev. A 83, 060302 (2011).
[CrossRef]

Taber, D. B.

Tapster, P. R.

J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990).
[CrossRef]

Toussaint, K.

Tzafestas, S. G.

S. G. Tzafestas, Walsh Functions in Signal and Systems Analysis and Design (Van Nostrand Reinhold, 1985).

Valadéz, K. O.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Walborn, S. P.

S. P. Walborn, S. Pádua, and C. H. Monken, Phys. Rev. A 71, 053812 (2005).
[CrossRef]

Walsh, J. L.

J. L. Walsh, Am. J. Math. 45, 5 (1923).
[CrossRef]

Wang, F.

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Yao, A. M.

F. M. Miatto, A. M. Yao, and S. M. Barnett, Phys. Rev. A 83, 033816 (2011).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, arXiv:1306.0096 (2013).

Am. J. Math. (1)

J. L. Walsh, Am. J. Math. 45, 5 (1923).
[CrossRef]

Appl. Opt. (3)

Eur. Phys. J. D (2)

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

F. Miatto, H. L. Pires, S. Barnett, and M. Exter, Eur. Phys. J. D 66, 1 (2012).
[CrossRef]

IEEE Spectrum (1)

H. Harmuth, IEEE Spectrum 6, 82 (1969).
[CrossRef]

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

Nature (2)

J. E. Gibbs and H. A. Gebbie, Nature 224, 1012 (1969).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Opt. Acta (1)

M. De and L. Hazra, Opt. Acta 24, 221 (1977).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (5)

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, Phys. Rev. A 65, 033823 (2002).
[CrossRef]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, Phys. Rev. A 57, 3123 (1998).
[CrossRef]

S. P. Walborn, S. Pádua, and C. H. Monken, Phys. Rev. A 71, 053812 (2005).
[CrossRef]

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, Phys. Rev. A 83, 060302 (2011).
[CrossRef]

F. M. Miatto, A. M. Yao, and S. M. Barnett, Phys. Rev. A 83, 033816 (2011).
[CrossRef]

Phys. Rev. Lett. (6)

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, Phys. Rev. Lett. 108, 173604 (2012).
[CrossRef]

M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef]

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
[CrossRef]

J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990).
[CrossRef]

J. Romero, J. Leach, B. Jack, M. R. Dennis, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. Lett. 106, 100407 (2011).
[CrossRef]

Other (4)

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, arXiv:1306.0096 (2013).

H. Harmuth, Transmission of Information by Orthogonal Functions (Springer, 1969).

K. G. Beauchamp, Walsh Functions and Their Applications (Academic, 1975).

S. G. Tzafestas, Walsh Functions in Signal and Systems Analysis and Design (Van Nostrand Reinhold, 1985).

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Figures (4)

Fig. 1.
Fig. 1.

Comparison of Walsh function and radial Walsh modes. The left panel (a) shows Wk4(x), the original Walsh functions defined in the interval [0,1]. On the right (b), the radial part of the radial Walsh modes fkN(r) are shown, where r=0. Here, pump- and detection-mode waists were set to wp=w=1. It is easy to see that (a) and (b) describe orthogonal functions in 1D and 2D, respectively.

Fig. 2.
Fig. 2.

Experimental setup. The photon pairs produced by SPDC in a PPKTP crystal are imaged in 4f configuration with L1 and L2 onto the SLMs (7.5× magnification). A wide pinhole (PH) is used to select the first diffraction order of the SLM. The SLM surface is again far-field imaged onto the single-mode fiber (SMF) with 10× microscopy objectives (MO). Photons are counted with avalanche photo diodes (APD), and photons which did belong to a single entangled pair are post-selected by coincide detection.

Fig. 3.
Fig. 3.

Quantum correlations of the radial Walsh modes for Walsh sequence index ks and ki for the signal and idler photon, respectively. Shown is the color-coded logarithmic coincidence rate normalized to unity. We find strong correlation between the mode indices; the decrease along the diagonal is due to lower photon production rate and lower detection efficiency for higher radial sequency index. No normalization to single count rates has been done.

Fig. 4.
Fig. 4.

(a) Quantum correlations (logarithmic normalized coincidence count rate) using the numerically optimized modes. We see reduced cross correlations and a wider spectrum compared to the Walsh modes in Fig. 3. (b) Comparison of the phase step positions between Walsh modes (gray circles) and the optimized modes (crosses).

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

C=dϕrdrup(r)us*(r)ui*(r),
C0rdrer2/wp2fksN(r)fkiN(r)δks,ki.

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