Abstract

Interferometric methods of vortex generation involve the interference of three or more plane waves. We show that spherical wave interference can produce vortex lattices similar to the one produced in the three-beam interference of plane waves. Three spherical waves of the same curvature are made to interfere in a shear interferometer introduced in a Mach–Zehnder configuration.

© 2007 Optical Society of America

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References

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2007 (2)

2006 (4)

R. M. Jenkins, J. Banerjee, and A. R. Davies, "The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides," J. Opt. A: Pure Appl. Opt. 3, 527-532 (2006).
[CrossRef]

C. S. Guo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-marte, "Spiral interferogram analysis," J. Opt. Soc. Am. A 23, 1400-1410 (2006).
[CrossRef]

K. O'Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four, and five plane waves," Opt. Express 14, 3039-3044 (2006).
[CrossRef] [PubMed]

2005 (1)

S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral interferometery," Opt. Lett. 15, 1953-1958 (2005).

2004 (6)

G. Gibson, J. Courtail, and M. J. Padgett, "Free space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
[CrossRef] [PubMed]

M. Berry, M. Dennis, and M. Soskin, "The plurality of optical singularities," J. Opt. A: Pure Appl. Opt. 6, S155-S156 (2004).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtail, and M. J. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[CrossRef] [PubMed]

M. D. Levenson, T. Ebihura, G. Dai, Y. Morikawa, N. Hyashi, and S. M. Tan, "Optical vortex mask via levels," J. Microlithogr., Microfab., Microsyst. 3, 293-304 (2004).
[CrossRef]

K. Ladavac and D. G. Grier, "Microoptomechanical pump assembled and driven by holographic optical vortices," Opt. Express 12, 1144-1149 (2004).
[CrossRef] [PubMed]

J. Masajada, "Small rotation-angle measurement with optical vortex interferometer," Opt. Commun. 239, 373-381 (2004).
[CrossRef]

2001 (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photon," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

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

2000 (1)

I. Freund, "Optical vortex trajectories," Opt. Commun. 181, 19-23 (2000).
[CrossRef]

1999 (1)

J. Scheuer and M. Orenstein, "Optical vortices crystals: spontaneous generation in nonlinear semiconductor micro-cavities," Science 285, 230-233 (1999).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-R3745 (1996).
[CrossRef] [PubMed]

1995 (1)

I. V. Basisity, M. S. Soskin, and M. V. Vanetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

1994 (1)

P. Senthilkumaran, K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Array generation using double wedge plate interferometer," J. Mod. Opt. 41, 481-489 (1994).
[CrossRef]

1993 (1)

I. Freund, N. Shvartsman, and V. Freiliker, "Optical dislocation network in highly random media," Opt. Commun. 101, 247-264 (1993).
[CrossRef]

1992 (1)

1987 (1)

K. W. Nicholls and J. F. Nye, "Three beam model for studying dislocation in wave pulse," J. Phys. A: Math. Gen. 20, 4673-4696 (1987).
[CrossRef]

1983 (1)

1974 (1)

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Appl. Opt. (2)

J. Microlithogr., Microfab., Microsyst. (1)

M. D. Levenson, T. Ebihura, G. Dai, Y. Morikawa, N. Hyashi, and S. M. Tan, "Optical vortex mask via levels," J. Microlithogr., Microfab., Microsyst. 3, 293-304 (2004).
[CrossRef]

J. Mod. Opt. (1)

P. Senthilkumaran, K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Array generation using double wedge plate interferometer," J. Mod. Opt. 41, 481-489 (1994).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

R. M. Jenkins, J. Banerjee, and A. R. Davies, "The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides," J. Opt. A: Pure Appl. Opt. 3, 527-532 (2006).
[CrossRef]

M. Berry, M. Dennis, and M. Soskin, "The plurality of optical singularities," J. Opt. A: Pure Appl. Opt. 6, S155-S156 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. A: Math. Gen. (1)

K. W. Nicholls and J. F. Nye, "Three beam model for studying dislocation in wave pulse," J. Phys. A: Math. Gen. 20, 4673-4696 (1987).
[CrossRef]

Nature (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photon," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

J. Leach, M. R. Dennis, J. Courtail, and M. J. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[CrossRef] [PubMed]

Opt. Commun. (6)

I. V. Basisity, M. S. Soskin, and M. V. Vanetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

I. Freund, "Optical vortex trajectories," Opt. Commun. 181, 19-23 (2000).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freiliker, "Optical dislocation network in highly random media," Opt. Commun. 101, 247-264 (1993).
[CrossRef]

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

C. S. Guo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

J. Masajada, "Small rotation-angle measurement with optical vortex interferometer," Opt. Commun. 239, 373-381 (2004).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (1)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, "Second harmonic generation and the orbital angular momentum of light," Phys. Rev. A 54, R3742-R3745 (1996).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Science (1)

J. Scheuer and M. Orenstein, "Optical vortices crystals: spontaneous generation in nonlinear semiconductor micro-cavities," Science 285, 230-233 (1999).
[CrossRef] [PubMed]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (5)

Fig. 1
Fig. 1

(a) Experimental setup for Mach–Zehnder interferometer configuration with a shear plate inducted. (b) Experimental setup for Mach–Zehnder interferometer configuration when mirror M2 in (a) is replaced with the Michelson interferometer.

Fig. 2
Fig. 2

(a) Interferogram recorded in the Mach–Zehnder interferometer configuration of Fig. 1(a) with a shear plate inducted and three beams interfering. The same pattern can be formed if the beam from M3 is blocked in the configuration of Fig. 1(b). (b) Formation of a fork fringe pattern when a fourth beam is added, indicating the presence of vortex dipole arrays. The configuration in Fig. 1(b) is used.

Fig. 3
Fig. 3

Simulation result for amplitude distribution is shown on a gray scale, going from low (black) to high (white) amplitude distribution. Phase of the field: the gray scale varies from white representing a phase of zero to black representing (2π). (a) Phase distribution of field U ˜ 1 . (b) Phase distribution of field U ˜ 2 . (c) Phase distribution of field U ˜ 3 . (d) Resultant amplitude of the interference field U ˜ 1 + U ˜ 2 + U ˜ 3 . (e) Phase distribution of the resultant field of (d). (f) Fork fringe formation when fourth beam is added. (g) Phase contour map of the resultant field. Phase contours start from vortices. (h) Zero crossing of real and imaginary parts of the wave function for locating vortices. (i) Phase distribution of the spherical wave with large curvature. (j) Resultant amplitude distribution of three spherical waves with large curvature for vortex array generation. (k) Fork fringe formation when a fourth beam is added to the field corresponding to (j). (l) Phase distribution of the resultant field corresponding to the field shown in (j).

Fig. 4
Fig. 4

Fig. 3. Continued

Fig. 5
Fig. 5

Fig. 3. Continued.

Equations (12)

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

U ˜ 1 = a 1   exp ( i k D ( x 2 + y 2 ) ) ,
U ˜ 2 = a 1   exp ( i k D ( ( x + Δ x ) 2 + y 2 ) ) ,
U ˜ 3 = a 2   exp ( i ( k D ( x 2 + y 2 ) + 2 π ν y ) ) .
e i k z z   exp [ i k 2 z ( x 2 + y 2 ) ] = A   exp [ i k 2 z ( x 2 + y 2 ) ] ,
U ˜ r = a 1   exp ( i k D ( x 2 + y 2 ) ) + a 1   exp ( i k D ( ( x + Δ x ) 2 + y 2 ) ) + a 2   exp ( i ( k D ( x 2 + y 2 ) + 2 π ν y ) ) ,
I = a 2 [ 3 + 2 ( cos   β + cos   γ + cos ( β γ ) ) ] ,
I = | a + a   exp ( i 2 π μ x ) + a   exp ( i 2 π ν y ) | 2 = a 2 [ 3 + 2 ( cos ( 2 π μ x ) + cos ( 2 π ν y ) + cos ( 2 π ( μ x ν y ) ) ) ] ,
U ˜ 3 = a 2   exp ( i ( k D ( x 2 + y 2 ) + 2 π ν y ) ) ,
U ˜ 4 = a 2   exp ( i ( k D ( x 2 + y 2 ) + 2 π μ 0 x ) ) .
| 1 + exp ( i 2 π μ x ) | 2 = | exp ( i k D ( x 2 + y 2 ) ) + exp ( i k D ( ( x + Δ x ) 2 + y 2 ) ) | 2 ,
| 1 + exp ( i 2 π ν y ) | 2 = | exp ( i k D ( x 2 + y 2 ) ) + exp ( i ( k D ( x 2 + y 2 ) + 2 π ν y ) ) | 2 ,
| exp ( i 2 π μ x ) + exp ( i 2 π ν y ) | 2 = | exp ( i k D ( ( x + Δ x ) 2 + y 2 ) ) + exp ( i ( k D ( x 2 + y 2 ) + 2 π ν y ) ) | 2 .

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