Abstract

We demonstrate how the unusual mathematics of transfinite numbers, in particular, a nearly perfect realization of Hilbert’s famous hotel paradox, manifests in the propagation of light through fractional vortex plates. It is shown how a fractional vortex plate can be used, in principle, to create any number of “open rooms,” i.e., topological charges, simultaneously. Fractional vortex plates are therefore demonstrated to create a singularity of topological charge, in which the vortex state is completely undefined and in fact arbitrary. These results hint that transfinite mathematics is much more common and important to optical systems than previously imagined.

© 2016 Optical Society of America

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References

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  1. G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Dover, 1955).
  2. J. Breuer, Introduction to the Theory of Sets (Dover, 2006).
  3. G. Gamow, One Two Three… Infinity (Dover, 1947).
  4. D. Oi, V. Potoček, and J. Jeffers, Phys. Rev. Lett. 110, 210504 (2013).
    [Crossref]
  5. V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
    [Crossref]
  6. M. Berry, J. Opt. A 6, 259 (2004).
    [Crossref]
  7. J. Leach, E. Yao, and M. Padgett, New J. Phys. 6, 71 (2004).
    [Crossref]
  8. M. Soskin and M. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.
  9. M. Dennis, K. O’Holleran, and M. Padgett, in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.
  10. M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Commun. 112, 321 (1994).
    [Crossref]

2015 (1)

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

2013 (1)

D. Oi, V. Potoček, and J. Jeffers, Phys. Rev. Lett. 110, 210504 (2013).
[Crossref]

2004 (2)

M. Berry, J. Opt. A 6, 259 (2004).
[Crossref]

J. Leach, E. Yao, and M. Padgett, New J. Phys. 6, 71 (2004).
[Crossref]

1994 (1)

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Commun. 112, 321 (1994).
[Crossref]

Beijersbergen, M.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Commun. 112, 321 (1994).
[Crossref]

Berry, M.

M. Berry, J. Opt. A 6, 259 (2004).
[Crossref]

Boyd, R.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

Breuer, J.

J. Breuer, Introduction to the Theory of Sets (Dover, 2006).

Cantor, G.

G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Dover, 1955).

Coerwinkel, R.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Commun. 112, 321 (1994).
[Crossref]

Dennis, M.

M. Dennis, K. O’Holleran, and M. Padgett, in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

Gamow, G.

G. Gamow, One Two Three… Infinity (Dover, 1947).

Jeffers, J.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

D. Oi, V. Potoček, and J. Jeffers, Phys. Rev. Lett. 110, 210504 (2013).
[Crossref]

Kristensen, M.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Commun. 112, 321 (1994).
[Crossref]

Leach, J.

J. Leach, E. Yao, and M. Padgett, New J. Phys. 6, 71 (2004).
[Crossref]

Liapis, A.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

Magaña-Loaiza, O.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

Miatto, F.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

Mirhosseini, M.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

O’Holleran, K.

M. Dennis, K. O’Holleran, and M. Padgett, in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

Oi, D.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

D. Oi, V. Potoček, and J. Jeffers, Phys. Rev. Lett. 110, 210504 (2013).
[Crossref]

Padgett, M.

J. Leach, E. Yao, and M. Padgett, New J. Phys. 6, 71 (2004).
[Crossref]

M. Dennis, K. O’Holleran, and M. Padgett, in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

Potocek, V.

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

D. Oi, V. Potoček, and J. Jeffers, Phys. Rev. Lett. 110, 210504 (2013).
[Crossref]

Soskin, M.

M. Soskin and M. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

Vasnetsov, M.

M. Soskin and M. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

Woerdman, J.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Commun. 112, 321 (1994).
[Crossref]

Yao, E.

J. Leach, E. Yao, and M. Padgett, New J. Phys. 6, 71 (2004).
[Crossref]

J. Opt. A (1)

M. Berry, J. Opt. A 6, 259 (2004).
[Crossref]

New J. Phys. (1)

J. Leach, E. Yao, and M. Padgett, New J. Phys. 6, 71 (2004).
[Crossref]

Opt. Commun. (1)

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, Opt. Commun. 112, 321 (1994).
[Crossref]

Phys. Rev. Lett. (2)

D. Oi, V. Potoček, and J. Jeffers, Phys. Rev. Lett. 110, 210504 (2013).
[Crossref]

V. Potoček, F. Miatto, M. Mirhosseini, O. Magaña-Loaiza, A. Liapis, D. Oi, R. Boyd, and J. Jeffers, Phys. Rev. Lett. 115, 160505 (2015).
[Crossref]

Other (5)

G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Dover, 1955).

J. Breuer, Introduction to the Theory of Sets (Dover, 2006).

G. Gamow, One Two Three… Infinity (Dover, 1947).

M. Soskin and M. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

M. Dennis, K. O’Holleran, and M. Padgett, in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 293–363.

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Figures (7)

Fig. 1.
Fig. 1. Phase of Laguerre–Gauss beams in the waist plane, for orders (a)  n = 0 , m = 1 ; (b)  n = 2 , m = 2 ; and (c)  n = 0 , m = 3 .
Fig. 2.
Fig. 2. Illustration of a spiral phase plate.
Fig. 3.
Fig. 3. Evolution of the field transmitted through a fractional plate, for (a)  α = 4.4 , (b)  α = 4.47 , (c)  α = 4.5 , (d)  α = 4.55 , (e)  α = 4.65 , and (f)  α = 4.995 . For convenience, adjacent charges that were created together are shown in the same color.
Fig. 4.
Fig. 4. Topological charge as a function of α , calculated by numerically evaluating the integral of Eq. (1) at ξ 2 + η 2 = 20 .
Fig. 5.
Fig. 5. Illustration of a multiramp phase plate, with m = 5 .
Fig. 6.
Fig. 6. Illustrating a jump of topological charge greater than 1 for m = 5 . (a) The case α = 1.95 . (b) The case α = 3.2 . (c) The topological charge as a function of α .
Fig. 7.
Fig. 7. Illustrating the “vortex hotel” chain for a beam at different propagation distances, with (a)  z = 0.5    m , (b)  z = 0.7    m , (c) and z = 0.9    m , with w 0 = 1    mm , λ = 500    nm . Here α = 4.47 .

Equations (8)

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t 1 2 π C ψ ( r ) · d r .
t ( ϕ ) = exp [ i α ϕ ] ,
U n ( ρ , ϕ , z ) = exp [ i k z ] exp [ ± i n ϕ ] exp [ i k ρ 2 / 4 z ] × π 8 ( i ) n / 2 k ρ 2 z × [ J ( n 1 ) / 2 ( k ρ 2 / 4 z ) i J ( n + 1 ) / 2 ( k ρ 2 / 4 z ) ] ,
exp [ i α ϕ ] = exp [ i π α ] sin ( π α ) π n = exp [ i n ϕ ] α n .
U α ( r ) = exp [ i π α ] sin ( π α ) π n = U n ( r ) α n .
t ( ϕ ) = exp [ i α ( ϕ 2 π k / m ) ] , 2 π k / m ϕ < 2 π ( k + 1 ) / m ,
c q m = sin ( α π / m ) π ( α / m q ) exp [ i α π / m ] .
U n ( ρ , ϕ , z ) = R exp [ i k z ± i n ϕ + i k ρ 2 / 4 z ( 1 R / 2 ) ] × π 8 ( i ) n / 2 k ρ 2 R z × [ J ( n 1 ) / 2 ( k ρ 2 R / 4 z ) i J ( n + 1 ) / 2 ( k ρ 2 R / 4 z ) ] ,

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