Abstract

We exploit the giant cross-Kerr nonlinearity of electromagnetically induced transparency (EIT) system in ultracold atoms to implement vortex-based multimode manipulation of stored light at low light levels. Using image-bearing signal light fields with angular intensity profiles, sinusoidal grating structures with phase-only modulation can be azimuthally imprinted on the stored probe light field, where the nonlinear absorption loss can be ignored. Upon retrieval of the probe light, collinearly superimposed vortex modes can be generated in the far field. Considering the finite size of atomic gas, the Fraunhofer diffraction patterns of the retrieved probe fields and their spiral spectra are numerically investigated, where the diffracted vortex modes can be efficiently controlled by tuning the weak signal fields. Our studies not only exhibit a fundamental diffraction phenomenon with angular grating structures in EIT system, but also provide a fascinating opportunity to realize multidimensional quantum information processing for stored light in an all-optical manner.

© 2015 Optical Society of America

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References

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2015 (3)

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

V. Parigi, V. DAmbrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[Crossref] [PubMed]

J. Sheng, J. Wang, M.-A. Miri, D. N. Christodoulides, and M. Xiao, “Observation of discrete diffraction patterns in an optically induced lattice,” Opt. Express 23, 19777–19782 (2015).
[Crossref] [PubMed]

2014 (1)

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photon. 8, 234–238 (2014).
[Crossref]

2013 (3)

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[Crossref] [PubMed]

J. Wu, Y. Liu, D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Light storage based on four-wave mixing and electromagnetically induced transparency in cold atoms,” Phys. Rev. A 87, 013845 (2013).
[Crossref]

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref] [PubMed]

2012 (2)

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-Spectral Orbital Angular Momentum Transfer via Four-Wave Mixing in Rb Vapor,” Phys. Rev. Lett. 108, 243601 (2012).
[Crossref] [PubMed]

L. Zhao, G. Yang, and W. Duan, “Manipulating stored images with phase imprinting at low light levels,” Opt. Lett. 37, 2853–2855 (2012).
[Crossref] [PubMed]

2011 (3)

S. A. Carvalho and L. E. E. de Araujo, “Electromagnetically-induced phase grating: A coupled-wave theory analysis,” Opt. Express 19, 1936–1944 (2011).
[Crossref] [PubMed]

L. Zhao, W. Duan, and S. F. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
[Crossref]

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804 (2011).
[Crossref]

2010 (3)

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81, 011401 (2010).
[Crossref]

L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
[Crossref] [PubMed]

2009 (3)

L. Zhao, T. Wang, and S. F. Yelin, “Two-dimensional all-optical spatial light modulation with high speed in coherent media,” Opt. Lett. 34, 1930–1932 (2009).
[Crossref] [PubMed]

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

D. Moretti, D. Felinto, and J. W. R. Tabosa, “Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

2008 (4)

T. Wang, L. Zhao, L. Jiang, and S. F. Yelin, “Diffusion-induced decoherence of stored optical vortices,” Phys. Rev. A 77, 043815 (2008).
[Crossref]

L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802 (2008).
[Crossref]

P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008).
[Crossref] [PubMed]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008).
[Crossref] [PubMed]

2007 (2)

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[Crossref]

2006 (2)

Y.-F. Chen, C.-Y. Wang, S.-H. Wang, and I. A. Yu, “Low-light-level cross-phase-modulation based on stored light pulses,” Phys. Rev. Lett. 96, 043603 (2006).
[Crossref] [PubMed]

C.-Y. Wang, Y.-F. Chen, S.-C. Lin, W.-H. Lin, P.-C. Kuan, and I. A. Yu, “Low-light-level all-optical switching,” Opt. Lett. 31, 2350–2352 (2006).
[Crossref] [PubMed]

2005 (2)

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13, 873–881 (2005).
[Crossref] [PubMed]

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

2004 (2)

H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. 93, 073601 (2004).
[Crossref] [PubMed]

M. J. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35–40 (2004).
[Crossref]

2003 (4)

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[Crossref] [PubMed]

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).
[Crossref]

2000 (1)

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref] [PubMed]

1999 (2)

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
[Crossref]

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

1998 (1)

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[Crossref]

1996 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

1989 (1)

P. Cullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403 (1989).
[Crossref]

Allen, L.

M. J. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35–40 (2004).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

Arnold, A. S.

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-Spectral Orbital Angular Momentum Transfer via Four-Wave Mixing in Rb Vapor,” Phys. Rev. Lett. 108, 243601 (2012).
[Crossref] [PubMed]

Arnold, C.

V. Parigi, V. DAmbrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[Crossref] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

Beil, F.

G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81, 011401 (2010).
[Crossref]

Brichkov, Y. A.

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series: Special Functions (Nauka, 1983).

Camacho, R. M.

P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008).
[Crossref] [PubMed]

Carrasco, S.

Carvalho, S. A.

Chen, H.-C.

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804 (2011).
[Crossref]

Chen, J.-X.

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804 (2011).
[Crossref]

Chen, Y.-C.

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804 (2011).
[Crossref]

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804 (2011).
[Crossref]

Chen, Y.-F.

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804 (2011).
[Crossref]

Y.-F. Chen, C.-Y. Wang, S.-H. Wang, and I. A. Yu, “Low-light-level cross-phase-modulation based on stored light pulses,” Phys. Rev. Lett. 96, 043603 (2006).
[Crossref] [PubMed]

C.-Y. Wang, Y.-F. Chen, S.-C. Lin, W.-H. Lin, P.-C. Kuan, and I. A. Yu, “Low-light-level all-optical switching,” Opt. Lett. 31, 2350–2352 (2006).
[Crossref] [PubMed]

Christodoulides, D. N.

Courtial, J.

M. J. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35–40 (2004).
[Crossref]

Cullet, P.

P. Cullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403 (1989).
[Crossref]

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref] [PubMed]

DAmbrosio, V.

V. Parigi, V. DAmbrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[Crossref] [PubMed]

Davidson, N.

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008).
[Crossref] [PubMed]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

de Araujo, L. E. E.

Ding, D.-S.

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[Crossref] [PubMed]

J. Wu, Y. Liu, D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Light storage based on four-wave mixing and electromagnetically induced transparency in cold atoms,” Phys. Rev. A 87, 013845 (2013).
[Crossref]

Duan, W.

L. Zhao, G. Yang, and W. Duan, “Manipulating stored images with phase imprinting at low light levels,” Opt. Lett. 37, 2853–2855 (2012).
[Crossref] [PubMed]

L. Zhao, W. Duan, and S. F. Yelin, “Generation of tunable-volume transmission-holographic gratings at low light levels,” Phys. Rev. A 84, 033806 (2011).
[Crossref]

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

Felinto, D.

D. Moretti, D. Felinto, and J. W. R. Tabosa, “Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

Firstenberg, O.

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008).
[Crossref] [PubMed]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

Fleischhauer, M.

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref] [PubMed]

Franke-Arnold, S.

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L. Zhao, T. Wang, and S. F. Yelin, “Two-dimensional all-optical spatial light modulation with high speed in coherent media,” Opt. Lett. 34, 1930–1932 (2009).
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L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802 (2008).
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D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
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Zhu, Y.

H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. 93, 073601 (2004).
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Appl. Phys. Lett. (1)

Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[Crossref]

Nat. Commun. (2)

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[Crossref] [PubMed]

V. Parigi, V. DAmbrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[Crossref] [PubMed]

Nat. Photon. (2)

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photon. 8, 234–238 (2014).
[Crossref]

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

P. Cullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403 (1989).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. A (11)

J. Wu, Y. Liu, D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Light storage based on four-wave mixing and electromagnetically induced transparency in cold atoms,” Phys. Rev. A 87, 013845 (2013).
[Crossref]

L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,” Phys. Rev. A 77, 041802 (2008).
[Crossref]

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804 (2011).
[Crossref]

G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81, 011401 (2010).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8190 (1992).
[Crossref] [PubMed]

T. Wang, L. Zhao, L. Jiang, and S. F. Yelin, “Diffusion-induced decoherence of stored optical vortices,” Phys. Rev. A 77, 043815 (2008).
[Crossref]

D. Moretti, D. Felinto, and J. W. R. Tabosa, “Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 (a) Schematic of the four-level N-type EIT system. (b) Timing diagram of the light fields and atomic coherence for light-storage-based XPM at low light levels. Without losing generality, we assume that the duration of the signal field (ts) equals the storage time of the probe field. (c) Experimental setup to achieve the retrieved probe light with multivortex structures in the far field. The angular intensity profiles in the signal fields can be generated by projecting the intensity masks into the ultracold 87Rb atomic gas, where lens 2 is an imaging lens. Technically, a liquid-crystal spatial light modulator can be used as an amplitude modulator to generate the intensity masks with high flexibility. The stored probe light is modulated in the atomic gas located at the front focal plane of the Fourier lens 1, while the retrieved probe light is detected at the back focal plane in the Fraunhofer diffraction regime. (d) and (e) Two different types of intensity profiles in the signal light fields. Type I: single sinusoidal intensity distribution depending on the azimuthal angle, which can be expressed as Is(φ) = Is0 + Is1 sin. For example, we have n = 3 in (d). Type II: complex sinusoidal intensity distribution composed of two sinusoidal periods, which can be expressed as Is(φ) = Is0 + Is1 sinn1φ + Is2 sinn2φ. For example, we have n1 = 2 and n2 = 3 in (e). These two types of intensity distributions in the signal fields can induce different angular phase grating structures for the stored probe light. BS: beam splitter. PBS: polarizing beam splitter.
Fig. 2
Fig. 2 Calculated Fraunhofer diffraction patterns for n = 3, which clearly show the spatial interference of different collinear vortex modes in two dimensions. The phase factor Φ1 is π/4 in (a), π/2 in (b), 3π/4 in (c), and π in (d). The intensity is normalized and the size of the pictures is 1×1 mm2. The calculation parameters are as follows: the probe wavelength λp = 795 nm, the focal length of the Fourier lens f = 100 mm, and the radius of the atomic gas a = 0.5 mm.
Fig. 3
Fig. 3 Radial intensity distribution of each vortex mode in the Fraunhofer diffraction patterns in Fig. 2. The thick red curves show the diffraction components of = 0 and the dotted curves show the higher-order diffraction components with = ±3, ±6, etc. Different vortex pairs usually correspond to different ring radii for the principal intensity peaks. In the insets of (a), (b), and (d), we show the entire intensity profiles of the = 0 components.
Fig. 4
Fig. 4 Spiral spectra (P) of the Fraunhofer diffraction patterns in Fig. 2, which show the weights of different vortex modes with respect to the phase factor Φ1. Note that the weight of the zeroth-order diffraction component (i.e., = 0) for Φ1 = 3π/4 in (c) is very weak but does not vanish.
Fig. 5
Fig. 5 Comparison between the diffraction properties of the angular sinusoidal phase gratings with n = 3 and n = 5 at the special value of Φ1 = 2.4 rad. We calculate the two dimensional Fraunhofer diffraction patterns [(a) and (c), where the picture size is 1 × 1 mm2], the radial intensity distributions of the vortex modes [(b) and (d)], and the spiral spectra (P) [(e) and (f)] generated by the gratings with n = 3 and n = 5, respectively. The radii r1 and r2 in (a)–(d) show the positions of the principal intensity peaks of the ±1st- and ±2nd-order diffraction components, respectively.
Fig. 6
Fig. 6 Fraunhofer diffraction patterns with respect to different XPM phase factors of Φ1 and Φ2 for the complex angular gratings with n1 = 2 and n2 = 3. The intensity is normalized and the size of the pictures is 1 × 1 mm2.
Fig. 7
Fig. 7 Spiral spectra (P) corresponding to the Fraunhofer diffraction patterns in Fig. 6, reflecting a complex spectral evolution of the diffraction components with respect to the XPM phase factors of Φ1 and Φ2.

Equations (9)

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σ 21 ( ρ ) = g Ω c circ ( ρ a ) E p ,
σ 21 ( ρ , φ , t s ) = σ 21 ( ρ ) exp [ Ω s 2 ( φ ) t s 2 Γ 4 i Δ s ] ,
Φ ( φ ) = Ω s 2 ( φ ) Δ s t s Γ 2 + 4 Δ s 2 and β a ( φ ) = 1 Ω s 2 ( φ ) Γ t s 2 Γ 2 + 4 Δ s 2 ,
E p ( ρ , φ ) = c i r c ( ρ a ) exp [ Φ ( φ ) i β a ( φ ) ] E p .
E p ( r , θ ) = i k p 2 π f 0 2 π 0 E p ( ρ , φ ) exp [ i k p f r ρ cos ( φ θ ) ] ρ d ρ d φ ,
E p ( r , θ ) = i k p E p exp ( i Φ 0 ) 2 π f 0 2 π 0 a exp ( i Φ 1 sin n φ ) exp [ i k p f r ρ cos ( φ θ ) ] ρ d ρ d φ .
E p ( r , θ ) = i k p E p exp ( i Φ 0 ) 2 π f m = + J m ( Φ 1 ) 0 2 π 0 a exp [ i φ i k p f r ρ cos ( φ θ ) ] ρ d ρ d φ = i k p E p exp ( i Φ 0 ) f m = + ( i ) J m ( Φ 1 ) exp ( i θ ) 0 a J ( k p f r ρ ) ρ d ρ = E p exp ( i Φ 0 ) m = + ( i ) + 1 J m ( Φ 1 ) exp ( i θ ) k p a 2 f ( + 2 ) ! ( k p a r 2 f ) × 1 F 2 [ + 2 2 , + 4 2 , + 1 ; ( k p a r 2 f ) 2 ] ,
P = C q = + C q ,
E p ( r , θ ) = E p exp ( i Φ 0 ) m 1 = + m 2 = + ( i ) + 1 J m 1 ( Φ 1 ) J m 2 ( Φ 2 ) exp ( i θ ) × k p a 2 f ( + 2 ) ! ( k p a r 2 f ) F 1 2 [ + 2 2 , + 4 2 , + 1 ; ( k p a r 2 f ) 2 ] ,

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