Abstract

We explore the diffraction and propagation of Laguerre-Gaussian beams of varying azimuthal index past a circular obstacle both experimentally and numerically. When the beam and obstacle centers are aligned the famous spot of Arago, which arises for zero azimuthal index, is replaced for non-zero azimuthal indices by a dark spot of Arago, a simple consequence of the conserved phase singularity at the beam center. We explore how the dark spot of Arago behaves as the beam and obstacle centers are progressively misaligned, and find that the central dark spot may break into several dark spots of Arago for higher incident azimuthal index beams.

© 2007 Optical Society of America

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References

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  1. J. E. Harvey, and J. L. Forgham, "The Spot of Arago - New Relevance for an Old Phenomenon," Am. J. Phys. 52, 243-247 (1984).
    [CrossRef]
  2. A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, and O. Quintero, "An experimental apparatus for white light imaging by means of a spherical obstacle," Am. J. Phys. 70, 169-172 (2002).
    [CrossRef]
  3. D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, "Direct measurement of the central fringe velocity in Young-type experiments," Phys. Lett. A 295, 78-80 (2002).
    [CrossRef]
  4. K. Uno, M. Suzuki, and K. Fujii, "Experimental analysis of classical Arago point with white-light laser," Jpn. J. Appl. Phys., Part 2 40, L872-L874 (2001).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (Roberts & Co, 2005).
  6. M. Born, and E. Wolf, in Principles of Optics (Pergamon, Oxford, 1990).
  7. E. A. Hovenac, "Fresnel Diffraction by Spherical Obstacles," Am. J. Phys. 57, 79-84 (1989).
    [CrossRef]
  8. D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, "Applications of spatial light modulators in atom optics," Opt. Express 11, 158-166 (2003).
    [CrossRef] [PubMed]
  9. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
    [CrossRef]
  10. M. R. Dennis, "Rows of optical vortices from elliptically perturbing a high-order beam," Opt. Lett. 31, 1325-1327 (2006).
    [CrossRef] [PubMed]
  11. J. T. Malos, R. Dykstra, M. Vaupel, and C. O. Weiss, "Vortex streets in a cavity with higher-order standing waves," Opt. Lett. 22, 1056-1058 (1997).
    [CrossRef] [PubMed]
  12. G. Molina-Terriza, L. Torner, and D. V. Petrov, "Vortex streets in walking parametric wave mixing," Opt. Lett. 24, 899-901 (1999).
    [CrossRef]
  13. G. Molina-Terriza, D. V. Petrov, J. Recolons, and L. Torner, "Observation of optical vortex streets in walking second-harmonic generation," Opt. Lett. 27, 625-627 (2002).
    [CrossRef]

2006 (1)

2004 (1)

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

2003 (1)

2002 (3)

G. Molina-Terriza, D. V. Petrov, J. Recolons, and L. Torner, "Observation of optical vortex streets in walking second-harmonic generation," Opt. Lett. 27, 625-627 (2002).
[CrossRef]

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, and O. Quintero, "An experimental apparatus for white light imaging by means of a spherical obstacle," Am. J. Phys. 70, 169-172 (2002).
[CrossRef]

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, "Direct measurement of the central fringe velocity in Young-type experiments," Phys. Lett. A 295, 78-80 (2002).
[CrossRef]

2001 (1)

K. Uno, M. Suzuki, and K. Fujii, "Experimental analysis of classical Arago point with white-light laser," Jpn. J. Appl. Phys., Part 2 40, L872-L874 (2001).
[CrossRef]

1999 (1)

1997 (1)

1989 (1)

E. A. Hovenac, "Fresnel Diffraction by Spherical Obstacles," Am. J. Phys. 57, 79-84 (1989).
[CrossRef]

1984 (1)

J. E. Harvey, and J. L. Forgham, "The Spot of Arago - New Relevance for an Old Phenomenon," Am. J. Phys. 52, 243-247 (1984).
[CrossRef]

Am. J. Phys. (3)

J. E. Harvey, and J. L. Forgham, "The Spot of Arago - New Relevance for an Old Phenomenon," Am. J. Phys. 52, 243-247 (1984).
[CrossRef]

A. Kolodziejczyk, Z. Jaroszewicz, R. Henao, and O. Quintero, "An experimental apparatus for white light imaging by means of a spherical obstacle," Am. J. Phys. 70, 169-172 (2002).
[CrossRef]

E. A. Hovenac, "Fresnel Diffraction by Spherical Obstacles," Am. J. Phys. 57, 79-84 (1989).
[CrossRef]

Jpn. J. Appl. Phys (1)

K. Uno, M. Suzuki, and K. Fujii, "Experimental analysis of classical Arago point with white-light laser," Jpn. J. Appl. Phys., Part 2 40, L872-L874 (2001).
[CrossRef]

New J. Phys. (1)

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, "Direct measurement of the central fringe velocity in Young-type experiments," Phys. Lett. A 295, 78-80 (2002).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Co, 2005).

M. Born, and E. Wolf, in Principles of Optics (Pergamon, Oxford, 1990).

Supplementary Material (5)

» Media 1: GIF (46 KB)     
» Media 2: GIF (46 KB)     
» Media 3: GIF (82 KB)     
» Media 4: GIF (439 KB)     
» Media 5: GIF (174 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Experimental setup used to investigate the dark spot of Arago with a monochromatic light source. The obstacle (a ball bearing with a diameter of 2R=2.38 mm) was mounted on a translation stage to move it across the Laguerre-Gaussian (LG) beam. The inset on the top left is a cross section of the LG beam taken at the dashed line to the left of the obstacle. The lighter areas in the figure show the regions of diffraction. For the interference patterns in Fig. 8 an imaging lens (f=25mm) was used to magnify the spot on the CCD camera.

Fig. 2.
Fig. 2.

Arago Spot for (a) a Gaussian beam i.e. l=0, (b) a LG beam with l=1, and (c) interferogram showing the characteristic fork in the interference pattern resulting from the azimuthal 2π phase term.

Fig. 3.
Fig. 3.

Comparison of beam profiles of the dark Arago spot for l=1: (a) measured and (b) calculated.

Fig. 4.
Fig. 4.

Intensity profiles for various displacements D (as marked in the left column) of the ball bearing relative to the center of the beam for a LG beam with azimuthal index l=1 (left: calculated and right: measured). [Media 1]

Fig. 5.
Fig. 5.

(a). Statistical uncertainties in the angular position Δθ and azimuthal index Δl versus displacement D, and (b) boundaries of the range of azimuthal indices present in the transmitted beam <l>±Δl versus displacement for an l=1 beam.

Fig. 6.
Fig. 6.

Intensity profiles for various displacements D (as marked in the left column) of the obstacle relative to the center of the beam for a LG beam with azimuthal index l=3: Numerical (left) and experimental (right).

Fig. 7.
Fig. 7.

(a). Statistical uncertainties in the angular position Δθ and azimuthal index Δl versus displacement D, and (b) the boundaries of the range of azimuthal indices present in the transmitted beam <l>±Δl versus displacement for an l=3 beam.

Fig. 8.
Fig. 8.

Interferogram of the Arago spot from a l=3 LG beam when the obstacle is displaced. Subfigure (a) shows the fork of the centered obstacle [Media 2]. Subfigure (b) shows the splitting up of the l=3 into three l=1 vortices when the obstacle is displaced by D=250 microns [Media 3]. Subfigure (c) shows that the individual l=1 vortices move further apart when the displacement (here D=500 microns) increases [Media 4].

Fig. 9.
Fig. 9.

Calculated phases for l=3 beam when (a) obstacle is centered, (b) obstacle displaced by 250 microns, (c) obstacle is displaced by 500 microns and (d) obstacle displaced by 1 mm. In (d) we can see the appearance of vortex streets. The dashed white line in subfigure (d) marks the area where the interferogram of Fig. 10 was taken.

Fig. 10.
Fig. 10.

Interferogram of the l=3 beam when the obstacle is displaced by D=1mm, taken in the region marked as dashed white square in Fig. 9(d). We find arrays of quasi-aligned vortices of index unity with alternating helicity, also known as vortex streets. [Media 5]

Equations (5)

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ε z = ic 2 ω 2 ε ,
ε ( x , y , z = 0 ) = ε in ( x + iy ) l e ( x 2 + y 2 ) w 0 2 t ( x , y , D ) ,
t ( x , y , D ) = 1 exp ( [ ( x D ) 2 + y 2 ) R 2 ] m ) ,
< O m > = π π d θ 0 rdr ε * ( x , y , z ) O ̂ m ε ( x , y , z ) π π d θ 0 rdr ε ( x , y , z ) 2 ,
η e ( x 2 + y 2 ) w 0 2 + ( x + iy ) e ( x 2 + y 2 ) w 0 2 = ( { x x 0 } + i { y y 0 } ) e ( x 2 + y 2 ) w 0 2 ,

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