Abstract

We propose a simple technique to scale the abruptly autofocusing beams in the direct space by introducing a scaling factor in the phase. Analytical formulas are deduced based on optical caustics, explicitly revealing how the scaling factor controls location, peak intensity, and size of the focal spot. We demonstrate that the multiplication of a scaling factor on the phase is equivalent to the axial-scaling transformation under the paraxial approximation. Further numerical and experimental results confirm theoretical predictions. In addition, amplitude modulation using phase-only holograms is used to maintain the peak intensity level of the focal spots.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

M. V. Berry, “Stable and unstable Airy-related caustics and beams,” J. Opt. 19(5), 055601 (2017).

2016 (3)

2015 (1)

2014 (1)

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

2013 (4)

2012 (3)

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85, 023828 (2012).

M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37(24), 5175–5177 (2012).
[PubMed]

J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express 20(12), 13302–13310 (2012).
[PubMed]

2011 (6)

2010 (1)

1999 (1)

Alonso, M. A.

Arnold, A. S.

Bandres, M. A.

Bent, N.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38(18), 3546–3549 (2013).
[PubMed]

Berry, M. V.

M. V. Berry, “Stable and unstable Airy-related caustics and beams,” J. Opt. 19(5), 055601 (2017).

Bolduc, E.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38(18), 3546–3549 (2013).
[PubMed]

Boyd, R. W.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38(18), 3546–3549 (2013).
[PubMed]

Campos, J.

Chen, Z.

Chremmos, I.

Chremmos, I. D.

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85, 023828 (2012).

Christodoulides, D. N.

Clark, T. W.

Cottrell, D. M.

Couairon, A.

C. Xie, R. Giust, V. Jukna, L. Furfaro, M. Jacquot, P. A. Lacourt, L. Froehly, J. Dudley, A. Couairon, and F. Courvoisier, “Light trajectory in Bessel-Gauss vortex beams,” J. Opt. Soc. Am. A 32(7), 1313–1316 (2015).
[PubMed]

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[PubMed]

Courvoisier, F.

Davis, J. A.

Dudley, J.

Dudley, J. M.

Efremidis, N. K.

Farsari, M.

Franke-Arnold, S.

Froehly, L.

Furfaro, L.

Giovannini, D.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

Giust, R.

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[PubMed]

Huang, K.

Jacquot, M.

Jiang, Y.

Jukna, V.

Karimi, E.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38(18), 3546–3549 (2013).
[PubMed]

Lacourt, P. A.

Lu, X.

Manousidaki, M.

Mathis, A.

Miatto, F. M.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

Mills, M. S.

Moreno, I.

Offer, R. F.

Ouadghiri-Idrissi, I.

Padgett, M. J.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

Panagiotopoulos, P.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[PubMed]

Papazoglou, D. G.

Prakash, J.

Radwell, N.

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[PubMed]

Sand, D.

Santamato, E.

Segev, M.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[PubMed]

Tzortzakis, S.

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[PubMed]

Xie, C.

Yzuel, M. J.

Zhang, P.

Zhang, Z.

Zinn, J. M.

Appl. Opt. (2)

J. Opt. (1)

M. V. Berry, “Stable and unstable Airy-related caustics and beams,” J. Opt. 19(5), 055601 (2017).

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

Nat. Commun. (1)

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
[PubMed]

Opt. Express (5)

Opt. Lett. (7)

Optica (1)

Phys. Rev. A (2)

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85, 023828 (2012).

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).

Phys. Rev. Lett. (1)

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[PubMed]

Other (1)

J. Turunen and A. T. Friberg, “Propagation-Invariant Optical Fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009).

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Figures (6)

Fig. 1
Fig. 1 The scaled AAF beam: (a) geometrical schematic; the benchmark beam profiles (b) at the plane z0c/5 away from the input plane and (c) at the focal plane.
Fig. 2
Fig. 2 Schematic of the experimental setup. HWP: half-wave plate; Pol: polarizer; SLM: spatial light modulator; M1 and M2: mirrors; L1 and L2: lenses; MO1 and MO2: microscope objectives. Inset (top-left): the phase mask of AAF beams accelerating along a n = 2 polynomial curve without a linear phase.
Fig. 3
Fig. 3 Intensity distribution as a function of propagation distance of the AAF beams accelerating along different polynomial curves: (a) n = 2, (b) n = 3; (c) n = 4. (Top row) numerical results; (Centre row) Experimental results; (Bottom row) Comparison of the numerical and experimental on-axis intensity distributions.
Fig. 4
Fig. 4 Simulations (top row) and experimental results (bottom row) of the axial maximum intensity profiles in the direct space after the 4f system for AAF beams along different curves: (a) n = 2; (b) n = 3; (c) n = 4.
Fig. 5
Fig. 5 Peak intensities for different scaling factors: (a) without and (b) with the compensation. The intensity levels for different polynomial curves are shifted to be separated from each other.
Fig. 6
Fig. 6 Comparison of the theoretical and measured FWHMs for the uncompensated and compensated AAF beams along different polynomial curves: (a) n = 2, (b) n = 3 and (c) n = 4.

Equations (9)

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r= r 0 a ( sz ) n n=2,3,4,
z c = 1 s ( r 0 a ) 1 n
ϕ( r )=sk n 2 ( 2n1 )( n1 ) [ ( n1 )a ] 1 n ( r r 0 ) 2n1 n
{ u F ( r=0, z f ) s 2 3 ( r 0 ) 4n2 3n a 2 3n n 4 3 [ 2( n1 ) ] 1 3 πA( ρ c )Ai( 1 )exp( i φ F ) φ F =s ( r 0 ) 2n1 n a 1 n n 2 ( 1 2 k n1 2n1 )
I F ( s )= s 4 3 I F0
ρ F ( s )= ρ F0 /s
u( m,n,z=0 )= s 2/3 A inc ( m,n )exp[ isϕ( m,n ) ]
ψ( m,n )=M( m,n )Mod[ F( m,n )+ ϕ ref ( m,n ),2π ]
{ M( m,n )=1+ sin c 1 ( s 2/3 ) π F( m,n )=sϕ( m,n )πM( m,n )

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