Abstract

We theoretically study the properties of the focal region of abruptly autofocusing and, their variant, autodefocusing ring-Airy beams, under the action of a conical phase gradient. By expanding the analysis of 1D Airy beams to cylindrically symmetric Airy beams, we derive analytic formulas for the position and dimensions of the focus. Our analysis covers in a unified way both beam types, and numerical simulations over a broad parameters range are in excellent agreement with our theoretical predictions. Our results allow the tailoring of the focal region both in size and in position by tuning the initial parameters.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36, 3675–3677 (2011).
    [Crossref] [PubMed]
  2. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36, 1890–1892 (2011).
    [Crossref] [PubMed]
  3. 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).
    [Crossref] [PubMed]
  4. K. Liu, A. D. Koulouklidis, D. G. Papazoglou, S. Tzortzakis, and X.-C. Zhang, “Enhanced terahertz wave emission from air-plasma tailored by abruptly autofocusing laser beams,” Optica 3, 605–608 (2016).
    [Crossref]
  5. H. Deng, Y. Yuan, and L. Yuan, “Annular arrayed-waveguide fiber for autofocusing Airy-like beams,” Opt. Lett. 41, 824–827 (2016).
    [Crossref] [PubMed]
  6. Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20, 18579–18584 (2012).
    [Crossref] [PubMed]
  7. J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express 20, 13302–13310 (2012).
    [Crossref] [PubMed]
  8. H. Chi-Young, K. Kyoung-Youm, and L. Byoungho, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. 4, 174–180 (2012).
    [Crossref]
  9. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010).
    [Crossref] [PubMed]
  10. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36, 1842–1844 (2011).
    [Crossref] [PubMed]
  11. M. Manousidaki, D. G. Papazoglou, M. Farsari, and S. Tzortzakis, “Abruptly autofocusing beams enable advanced multiscale photo-polymerization,” Optica 3, 525–530 (2016).
    [Crossref]
  12. D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Janus waves,” Opt. Lett. 41, 4656–4659 (2016).
    [Crossref] [PubMed]
  13. J. Zhang, Y. Li, Z. Tian, and D. Lei, “Controllable autofocusing properties of conical circular Airy beams,” Opt. Commun. 391, 116–120 (2017).
    [Crossref]
  14. C.-Y. Hwang, K.-Y. Kim, and B. Lee, “Bessel-like beam generation by superposing multiple Airy beams,” Opt. Express 19, 7356–7364 (2011).
    [Crossref] [PubMed]
  15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  16. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [Crossref] [PubMed]
  17. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011).
    [Crossref] [PubMed]
  18. M. Born and E. Wolf, Principles of Optics (Elsevier, 1980), 6th ed.
  19. 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).
    [Crossref]
  20. R.-S. Penciu, K. G. Makris, and N. K. Efremidis, “Nonparaxial abruptly autofocusing beams,” Opt. Lett. 41, 1042–1045 (2016).
    [Crossref] [PubMed]
  21. Y. Zhang, M. R. Belić, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22, 7160–7171 (2014).
    [Crossref] [PubMed]
  22. J. H. McLeod, “The Axicon: A new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [Crossref]
  23. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]

2017 (1)

J. Zhang, Y. Li, Z. Tian, and D. Lei, “Controllable autofocusing properties of conical circular Airy beams,” Opt. Commun. 391, 116–120 (2017).
[Crossref]

2016 (5)

2014 (1)

2013 (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).
[Crossref] [PubMed]

2012 (4)

Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20, 18579–18584 (2012).
[Crossref] [PubMed]

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

H. Chi-Young, K. Kyoung-Youm, and L. Byoungho, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. 4, 174–180 (2012).
[Crossref]

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).
[Crossref]

2011 (5)

2010 (1)

2008 (1)

2007 (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1954 (1)

Belic, M. R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Elsevier, 1980), 6th ed.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Byoungho, L.

H. Chi-Young, K. Kyoung-Youm, and L. Byoungho, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. 4, 174–180 (2012).
[Crossref]

Chen, H.

Chen, Z.

Chi-Young, H.

H. Chi-Young, K. Kyoung-Youm, and L. Byoungho, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. 4, 174–180 (2012).
[Crossref]

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).
[Crossref]

Christodoulides, D. N.

Cottrell, D. M.

Couairon, A.

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).
[Crossref] [PubMed]

Davis, J. A.

Deng, H.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Efremidis, N. K.

Farsari, M.

Fedorov, V. Y.

Huang, K.

Hwang, C.-Y.

Jiang, Y.

Kim, K.-Y.

Koulouklidis, A. D.

Kyoung-Youm, K.

H. Chi-Young, K. Kyoung-Youm, and L. Byoungho, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. 4, 174–180 (2012).
[Crossref]

Lee, B.

Lei, D.

J. Zhang, Y. Li, Z. Tian, and D. Lei, “Controllable autofocusing properties of conical circular Airy beams,” Opt. Commun. 391, 116–120 (2017).
[Crossref]

Li, C.

Li, Y.

Liu, K.

Lu, X.

Makris, K. G.

Manousidaki, M.

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Mills, M. S.

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).
[Crossref] [PubMed]

Papazoglou, D. G.

Penciu, R.-S.

Prakash, J.

Sand, D.

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Tian, Z.

J. Zhang, Y. Li, Z. Tian, and D. Lei, “Controllable autofocusing properties of conical circular Airy beams,” Opt. Commun. 391, 116–120 (2017).
[Crossref]

Tzortzakis, S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Elsevier, 1980), 6th ed.

Yuan, L.

Yuan, Y.

Zhang, J.

J. Zhang, Y. Li, Z. Tian, and D. Lei, “Controllable autofocusing properties of conical circular Airy beams,” Opt. Commun. 391, 116–120 (2017).
[Crossref]

Zhang, P.

Zhang, X.-C.

Zhang, Y.

Zhang, Z.

Zheng, H.

IEEE Photon. J. (1)

H. Chi-Young, K. Kyoung-Youm, and L. Byoungho, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. 4, 174–180 (2012).
[Crossref]

J. Opt. Soc. Am. (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).
[Crossref] [PubMed]

Opt. Commun. (1)

J. Zhang, Y. Li, Z. Tian, and D. Lei, “Controllable autofocusing properties of conical circular Airy beams,” Opt. Commun. 391, 116–120 (2017).
[Crossref]

Opt. Express (4)

Opt. Lett. (9)

R.-S. Penciu, K. G. Makris, and N. K. Efremidis, “Nonparaxial abruptly autofocusing beams,” Opt. Lett. 41, 1042–1045 (2016).
[Crossref] [PubMed]

H. Deng, Y. Yuan, and L. Yuan, “Annular arrayed-waveguide fiber for autofocusing Airy-like beams,” Opt. Lett. 41, 824–827 (2016).
[Crossref] [PubMed]

D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Janus waves,” Opt. Lett. 41, 4656–4659 (2016).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[Crossref] [PubMed]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011).
[Crossref] [PubMed]

I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36, 3675–3677 (2011).
[Crossref] [PubMed]

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36, 1890–1892 (2011).
[Crossref] [PubMed]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010).
[Crossref] [PubMed]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36, 1842–1844 (2011).
[Crossref] [PubMed]

Optica (2)

Phys. Rev. A (1)

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).
[Crossref]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics (Elsevier, 1980), 6th ed.

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Figures (11)

Fig. 1
Fig. 1 Typical intensity profiles (normalized values) of (a) ring-Airy (b) inverse ring-Airy (r0=4 mm, w=200 µm), (c) I(r, z = 0) intensity profiles for rotationally symmetric Airy beams of decreasing radius. b=1 (ring-Airy), b=−1 (inverse ring-Airy). Arrows show the direction which the radius is increasing.
Fig. 2
Fig. 2 Intensity I(x, z) profiles of rotationally symmetric Airy beams and peak intensity along z with different conical phase gradients at the input plane (z=0) θi=1,2,3=30 mrad, 20 mrad, 10 mrad (r0=1.0 mm and w=100 µm, u0=1) (a) ring-Airy (b) inverse ring-Airy. Insets show the position and the dimensions of the focus.
Fig. 3
Fig. 3 Comparison of intensity I(x, z) along propagation for autofocusing colliding 1D Airy beams (1st row) with with radially symmetric Airy beams (2nd row). 3d row: Normalized intensities along the z propagation axis I(0, z)/Imax. Insets: I(x, z) intensity profiles (normalized, false colors), (1D) colliding 1D Airy beams (2D) radially symmetric Airy beams. (Numerical simulation parameters: λ=0.8 µm, α = 0.05 (i) x0 = −2.0 mm, w=100 µm, θ=10 mrad (ii) r0=2.0 mm, w=100 µm, θ=10 mrad (iv) x0=2.0 mm, w=100 µm, θ=10 mrad (v) r0=2.0 mm, w=100 µm, θ=10 mrad).
Fig. 4
Fig. 4 Focal distance fAi, of abruptly autofocusing ring-Airy beams normalized over zAi as a function, (a) of the shape factor s ( θ ˜ = 0 ), (b) of the normalized cone angle θ ˜ ( s = 21 ).
Fig. 5
Fig. 5 Ring-Airy focal spot length normalized over the focal distance fAi as a function (a) of the shape factor s ( θ ˜ = 0 ), (b) of normalized cone angle θ ˜ ( s = 21 ).
Fig. 6
Fig. 6 Focal spot width wAi of ring-Airy, normalized over the width parameter w as a function (a) of the shape factor s ( θ ˜ = 0 ) and (b) of the normalized cone angle θ ˜ ( s = 21 ).
Fig. 7
Fig. 7 Aspect ratio ΔfAi/w of the focal region of a ring-Airy as a function of the radius and width parameters (ro, w). (a) θ = 0, (b) θ = 5 mrad.
Fig. 8
Fig. 8 Position of the focus fAi of an autodefocusing inverse ring-Airy beam, normalized over zAi. (a) as a function of the shape factor s ( θ ˜ = 7.854 rad ), (b) as a function of the normalized cone angle θ ˜ ( s = 19 ).
Fig. 9
Fig. 9 Inverse ring-Airy focal spot length ΔfAi normalized over fAi as a function (a) of the shape factor s ( θ ˜ = 7.854 rad ). (b) of the normalized cone angle θ ˜ ( s = 29 ).
Fig. 10
Fig. 10 Focus width wAi of an abruptly autodefocusing inverse ring-Airy, normalized over the width parameter w as a function (a) of the shape factor s ( θ ˜ = 10.21 rad ). (b) of the normalized cone angle θ ˜ ( s = 19 ).
Fig. 11
Fig. 11 Aspect ratio ΔfAi/w of the focal region of an inverse ring-Airy as a function of the radius and width parameters (ro, w). (a) θ =10 mrad, (b) θ = 15 mrad.

Equations (6)

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u ( r , 0 ) = u o A i ( b ρ ) e b a ρ e i sin θ k r
U 1 D ( x , z ; θ ) = Φ ( x , z , θ ) + Φ ( x , z , θ ) , Φ ( x , z , θ ) = A i ( b x 0 + x w z 2 4 k 2 w 4 + i a z k w 4 b θ z w ) × e b a x 0 + x θ z w a z 2 4 k 2 w 4 + i ( b x o + x + b a 2 w 2 k w 3 z z 3 12 k 3 w 6 + θ k ( x 0 + x ) b θ z 2 2 k w 3 θ 2 k z 2 ) ,
f A i 1 D 4 b z A i [ 2 θ ˜ ± ( 4 θ ˜ 2 + b x ˜ ) 1 / 2 ] ,
f A i z A i = 4 b [ 2 θ ˜ ± ( 4 θ ˜ 2 + b s ) 1 / 2 ] ,
Δ f A i f A i 42 b 1 50 [ ( 4 θ ˜ 2 + b s ) 2 θ ˜ ( 4 θ ˜ 2 + b s ) 1 / 2 ] 1 .
w A i w C ( 1 1 2 2 b θ ˜ 2 + b s ) ( 4 4 θ ˜ 2 + b s + w 2 z A i 2 ) 1 / 2 ,

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