Abstract

The close relation between the processes of paraxial diffraction and coherent diffusion is reflected in the similarity between their shape-preserving solutions, notably the Gaussian modes. Differences between these solutions enter only for high-order modes. Here we experimentally study the behavior of shape-preserving high-order modes of coherent diffusion, known as “elegant” modes, and contrast them with the nonshape-preserving evolution of the corresponding “standard” modes of optical diffraction. Diffusion of the light field is obtained by mapping it onto the atomic coherence field of a diffusing vapor in a storage-of-light setup. The growth of the elegant mode fits well the theoretical expectations.

© 2013 Optical Society of America

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2012 (1)

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

2010 (1)

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

2009 (1)

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, Nat. Phys. 5, 665 (2009).
[CrossRef]

2008 (2)

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).
[CrossRef]

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

2007 (1)

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).
[CrossRef]

2003 (1)

2001 (1)

1998 (1)

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

1992 (1)

K. B. MacAdam, A. Steinbach, and C. Weiman, Am. J. Phys. 60, 1098 (1992).
[CrossRef]

1986 (1)

1985 (1)

1977 (2)

1973 (1)

Borghi, R.

Buchler, B. C.

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

Davidson, N.

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, Nat. Phys. 5, 665 (2009).
[CrossRef]

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).
[CrossRef]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).
[CrossRef]

Felsen, L. B.

Firstenberg, O.

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, Nat. Phys. 5, 665 (2009).
[CrossRef]

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).
[CrossRef]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).
[CrossRef]

Fredkin, D. R.

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

Fukumitsu, O.

Higginbottom, D. B.

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

Hosseini, M.

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

Kuga, T.

Lam, P. K.

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

London, P.

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, Nat. Phys. 5, 665 (2009).
[CrossRef]

MacAdam, K. B.

K. B. MacAdam, A. Steinbach, and C. Weiman, Am. J. Phys. 60, 1098 (1992).
[CrossRef]

Mukae, T.

Pinel, O.

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

Porras, M.

Pratesi, R.

Pugatch, R.

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).
[CrossRef]

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).
[CrossRef]

Rancic, M.

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

Ron, A.

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, Nat. Phys. 5, 665 (2009).
[CrossRef]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).
[CrossRef]

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).
[CrossRef]

Ronchi, L.

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

Santarsiero, M.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

Shin, S. Y.

Shuker, M.

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, Nat. Phys. 5, 665 (2009).
[CrossRef]

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).
[CrossRef]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).
[CrossRef]

Siegman, A. E.

A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
[CrossRef]

A. E. Siegman, Lasers (University Science, 1986).

Sparkes, B. M.

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

Steinbach, A.

K. B. MacAdam, A. Steinbach, and C. Weiman, Am. J. Phys. 60, 1098 (1992).
[CrossRef]

Takenaka, T.

Torii, Y.

Umeki, T.

Weiman, C.

K. B. MacAdam, A. Steinbach, and C. Weiman, Am. J. Phys. 60, 1098 (1992).
[CrossRef]

Yankelev, D.

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

Yokota, M.

Yoshikawa, Y.

Zauderer, E.

Am. J. Phys. (1)

K. B. MacAdam, A. Steinbach, and C. Weiman, Am. J. Phys. 60, 1098 (1992).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Nat. Phys. (1)

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, Nat. Phys. 5, 665 (2009).
[CrossRef]

Phys. Rev. A (2)

O. Firstenberg, M. Shuker, R. Pugatch, D. R. Fredkin, N. Davidson, and A. Ron, Phys. Rev. A 77, 043830 (2008).
[CrossRef]

D. B. Higginbottom, B. M. Sparkes, M. Rancic, O. Pinel, M. Hosseini, P. K. Lam, and B. C. Buchler, Phys. Rev. A 86, 023801 (2012).
[CrossRef]

Phys. Rev. Lett. (3)

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).
[CrossRef]

O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, Phys. Rev. Lett. 105, 183602 (2010).
[CrossRef]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (University Science, 1986).

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Figures (5)

Fig. 1.
Fig. 1.

Scheme of the experimental setup. A probe pulse is reflected from a spatial light modulator (SLM), which imprints the desired phase pattern. A 4f telescope images the SLM plane to the middle of the Rb cell, with an iris filtering high spatial frequencies. Polarizing beam splitters are used to combine the pump with the probe before the cell and to separate them afterward. The cell is surrounded by three layers of magnetic shielding (MS) and by a set of Helmholtz coils (HC) for controlling the magnetic field and setting the quantization axis. The retrieved probe pulse after storage is imaged with a single lens to a CCD.

Fig. 2.
Fig. 2.

(a) Images of the Gaussian and HG1,1 modes after diffusion, for different storage durations. The intensity maps are normalized. (b) Normalized intensity profiles of the Gaussian and HG1,1 modes after diffusion. Solid curves are the analytic expressions for Gaussian and HG1,1 modes.

Fig. 3.
Fig. 3.

Comparison between (a) standard and (d) eHG2,2 modes and (b), (e) after diffraction for 150 cm or (c), (f) after diffusion for 30 μs. It can be seen that the standard mode is shape preserving only under diffraction, while the elegant is shape preserving only under diffusion.

Fig. 4.
Fig. 4.

Cross sections (of the images from Fig. 3) around the maximum intensity of (a) standard and (b) elegant HG2,2 modes (blue circles) and after diffraction for 150 cm (green solid curve) and diffusion for 30 μs (red dashed curve). On the left are the theoretical profiles, and on the right the experimental ones. The cross sections are scaled according to the fitted or calculated waist, for the original and the diffracted/diffused results.

Fig. 5.
Fig. 5.

(a) Images of the eHG2,2 mode after diffusion, for different storage durations. The intensity maps are normalized. (b) Normalized intensity cross section after diffusion of the eHG2,2 mode, for different storage durations. Solid curve is the analytic expression for eHG2,2 mode [Eq. (3)].

Equations (3)

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En,msHG(x,y,z;w0)Hn(2xw(z))Hm(2yw(z))e(x˜2+y˜2).
En,meHG(x,y,z;w0)Hn(x˜)Hm(y˜)e(x˜2+y˜2).
Er(x,y;τ)=eγτs(τ)(n+m+1)En,meHG(x,y,w0s(τ)),

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