Abstract

We engineer an intensity square lattice using the Fraunhofer diffraction of a Laguerre-Gauss beam by a square aperture. We verify numerically and experimentally that a perfect optical intensity lattice takes place only for even values of the topological charge. We explain the origin of this behavior based on the decomposition of the patterns. We also study the evolution of the lattice formation by observing the transition from one order to the next of the orbital angular momentum varying the topological charge in fractional steps.

© 2011 OSA

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  1. L. M. Pismen, Vortices in Nonlinear Fields: From Liquid Crystals to Superfluids, from Non-Equilibrium Patterns to Cosmic Strings (Oxford University Press, Oxford, 1999).
  2. Y. S. Kivshar, and G. P. Agrawal, Optical solitons: from fibers to photonic crystals (Academic Press, Amsterdam; Boston, 2003).
  3. V. Tikhonenko, J. Christou, and B. Lutherdaves, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B 12(11), 2046–2052 (1995).
    [CrossRef]
  4. V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
    [CrossRef] [PubMed]
  5. D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23(18), 1444–1446 (1998).
    [CrossRef] [PubMed]
  6. J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292(5516), 476–479 (2001).
    [CrossRef] [PubMed]
  7. J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express 15(8), 5196–5207 (2007).
    [CrossRef] [PubMed]
  8. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
    [CrossRef]
  9. J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207(1-6), 85–93 (2002).
    [CrossRef]
  10. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
    [CrossRef] [PubMed]
  11. R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79(4), 043809 (2009).
    [CrossRef]
  12. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
    [CrossRef] [PubMed]
  13. G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  15. J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,”, J. Opt. Soc. Am. B 61, 1023–1028 (1971).
  16. J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
    [CrossRef]
  17. U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
    [CrossRef] [PubMed]
  18. Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011).
    [CrossRef] [PubMed]

2011 (1)

2010 (2)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
[CrossRef] [PubMed]

2009 (2)

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[CrossRef]

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79(4), 043809 (2009).
[CrossRef]

2008 (1)

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[CrossRef] [PubMed]

2007 (1)

2005 (1)

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

2002 (1)

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207(1-6), 85–93 (2002).
[CrossRef]

2001 (2)

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292(5516), 476–479 (2001).
[CrossRef] [PubMed]

1998 (1)

1996 (1)

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

1995 (1)

1971 (1)

J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,”, J. Opt. Soc. Am. B 61, 1023–1028 (1971).

Abo-Shaeer, J. R.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292(5516), 476–479 (2001).
[CrossRef] [PubMed]

Beijersbergen, M. W.

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[CrossRef]

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[CrossRef] [PubMed]

Berkhout, G. C. G.

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[CrossRef]

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[CrossRef] [PubMed]

Chávez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Lutherdaves, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B 12(11), 2046–2052 (1995).
[CrossRef]

Cojocaru, C.

Courtial, J.

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Couteau, C.

U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
[CrossRef] [PubMed]

Dennis, M.

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Dubik, B.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

Ferreira, Q. S.

Fonseca, E. J. S.

Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011).
[CrossRef] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Hickmann, J. M.

Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011).
[CrossRef] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Jennewein, T.

U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
[CrossRef] [PubMed]

Jesus-Silva, A. J.

Jones, A. L.

J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,”, J. Opt. Soc. Am. B 61, 1023–1028 (1971).

Ketterle, W.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292(5516), 476–479 (2001).
[CrossRef] [PubMed]

Kirk, J. P.

J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,”, J. Opt. Soc. Am. B 61, 1023–1028 (1971).

Laflamme, R.

U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
[CrossRef] [PubMed]

Leach, J.

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Leniec, M.

Lutherdaves, B.

Luther-Davies, B.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

Martorell, J.

Masajada, J.

J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express 15(8), 5196–5207 (2007).
[CrossRef] [PubMed]

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207(1-6), 85–93 (2002).
[CrossRef]

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

Padgett, M.

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Petrov, D. V.

Popiolek-Masajada, A.

J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express 15(8), 5196–5207 (2007).
[CrossRef] [PubMed]

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207(1-6), 85–93 (2002).
[CrossRef]

Raman, C.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292(5516), 476–479 (2001).
[CrossRef] [PubMed]

Schoonover, R. W.

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79(4), 043809 (2009).
[CrossRef]

Sinha, U.

U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
[CrossRef] [PubMed]

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Lutherdaves, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B 12(11), 2046–2052 (1995).
[CrossRef]

Torner, L.

Torres, J. P.

Vilaseca, R.

Visser, T. D.

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79(4), 043809 (2009).
[CrossRef]

Vogels, J. M.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292(5516), 476–479 (2001).
[CrossRef] [PubMed]

Weihs, G.

U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
[CrossRef] [PubMed]

Wieliczka, D. M.

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207(1-6), 85–93 (2002).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

New J. Phys. (1)

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Opt. Commun. (2)

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. 207(1-6), 85–93 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79(4), 043809 (2009).
[CrossRef]

Phys. Rev. Lett. (3)

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[CrossRef] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996).
[CrossRef] [PubMed]

Science (2)

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science 292(5516), 476–479 (2001).
[CrossRef] [PubMed]

U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science 329(5990), 418–421 (2010).
[CrossRef] [PubMed]

Other (3)

L. M. Pismen, Vortices in Nonlinear Fields: From Liquid Crystals to Superfluids, from Non-Equilibrium Patterns to Cosmic Strings (Oxford University Press, Oxford, 1999).

Y. S. Kivshar, and G. P. Agrawal, Optical solitons: from fibers to photonic crystals (Academic Press, Amsterdam; Boston, 2003).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

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Figures (7)

Fig. 1
Fig. 1

Diffraction patterns corresponding to the numerical results of Eq. (1).

Fig. 2
Fig. 2

Phase patterns corresponding to the diffraction patterns in Fig. 1, for m = 5 (a) and m = 6 (b). The red dashed square was used only to highlight the center of the pattern.

Fig. 3
Fig. 3

(a) Experimental setup; (b) A beam inside of the square aperture (top) and the phase diagram (bottom) for m = 3. In the figure F is a density neutral filter; fi are lenses; SLM is the spatial light modulator; and SF is a spatial filter.

Fig. 4
Fig. 4

Diffraction patterns corresponding to experimental results for integer topological charges.

Fig. 5
Fig. 5

Diffraction patterns produced by combinations of slits. Each slit combination is shown at the top.

Fig. 6
Fig. 6

Superimposed patterns, P AC (intensity) and P BD (intensity contours).

Fig. 7
Fig. 7

Diffraction patterns corresponding to the experimental results for the fractional topological charges

Equations (2)

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

E d ( k )= + τ( r ) E i ( r ) e i k r d r ,
P ABCD = P AB + P AC + P AD + P BC + P BD + P CD 2 P A 2 P B 2 P C 2 P D ,

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