Abstract

Theoretically formulated in the 1970s within the context of nonrelativistic quantum mechanics, Airy beams have been experimentally realized for the first time only recently, paving the way to innovative optical techniques. While their remarkable features, a non-diffracting property and a transverse shift of the intensity maximum during propagation, are currently theoretically described from the wave optics viewpoint, here their exact relation to rays and geometric wavefront aberrations is revealed using a wavefront family that includes two-dimensional Airy beams. Several members of this family are computationally and experimentally implemented here. The lateral shift of Airy beams during propagation is presented in the context of the three-dimensional caustic representation. This new description allows re-emphasizing the use of “Airy-like” beams in computational imaging for depth of focus extension.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. V. Berry and N. L. Balazs, “Non spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [CrossRef]
  2. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008).
    [CrossRef] [PubMed]
  3. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2010).
    [CrossRef]
  4. G. Siviloglou and D. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef] [PubMed]
  5. G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1–213901-4 (2007).
    [CrossRef]
  6. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
    [CrossRef]
  7. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
    [CrossRef] [PubMed]
  8. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009).
    [CrossRef] [PubMed]
  9. J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
    [CrossRef]
  10. H. T. Dai, X. W. Sun, D. Luo, and Y. J. Liu, “Airy beams generated by a binary phase element made of polymer-dispersed liquid crystals,” Opt. Express 17, 19365–19370 (2009).
    [CrossRef] [PubMed]
  11. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Non linear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
    [CrossRef]
  12. H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950).
  13. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27, 1490–1504 (2010).
    [CrossRef]
  14. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001).
    [CrossRef]
  15. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
    [CrossRef]
  16. A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479 (2004).
    [CrossRef] [PubMed]
  17. Y. Takahashi and S. Komatsu, “Optimized free-form phase mask for extension of depth of field in wavefront-coded imaging,” Opt. Lett. 33, 1515–1517 (2008).
    [CrossRef] [PubMed]
  18. K. Chu, N. George, and W. Chi, “Extending the depth of field through unbalanced optical path difference,” Appl. Opt. 47, 6895–6903 (2008).
    [CrossRef] [PubMed]
  19. N. Caron and Y. Sheng, “Polynomial phase masks for extending the depth of field of a microscope,” Appl. Opt. 47, E39–E43 (2008).
    [CrossRef] [PubMed]
  20. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  21. S. Tucker, W. T. Cathey, and E. R. Dowski, “Extended depth of field and aberration control for inexpensive digital microscope systems,” Opt. Express 4, 467–474 (1999).
    [CrossRef] [PubMed]
  22. W. T. Cathey and E. R. Dowski, “New paradigm for imaging system,” Appl. Opt. 41, 6080–6092 (2002).
    [CrossRef] [PubMed]
  23. J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
    [CrossRef]
  24. F. Yan, L.-G. Zheng, and X.-J. Zhang, “Design of an off-axis three-mirror anastigmatic optical system with wavefront coding technology,” Opt. Eng. 47, 063001-1–063001-10 (2008).
    [CrossRef]
  25. T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903-1–742903-8 (2009).
    [CrossRef]
  26. K. Kubala, E. R. Dowski, and W. T. Cathey, “Reducing complexity in computational imaging systems,” Opt. Express 11, 2102–2108 (2003).
    [CrossRef] [PubMed]

2010 (2)

2009 (6)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef] [PubMed]

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009).
[CrossRef] [PubMed]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
[CrossRef]

H. T. Dai, X. W. Sun, D. Luo, and Y. J. Liu, “Airy beams generated by a binary phase element made of polymer-dispersed liquid crystals,” Opt. Express 17, 19365–19370 (2009).
[CrossRef] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Non linear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903-1–742903-8 (2009).
[CrossRef]

2008 (6)

2007 (2)

G. Siviloglou and D. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef] [PubMed]

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1–213901-4 (2007).
[CrossRef]

2004 (3)

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474–3479 (2004).
[CrossRef] [PubMed]

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

1999 (1)

1995 (1)

1979 (1)

M. V. Berry and N. L. Balazs, “Non spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

1950 (1)

H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950).

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Non linear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Non spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Non spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Broky, J.

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1–213901-4 (2007).
[CrossRef]

Bustin, N.

T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903-1–742903-8 (2009).
[CrossRef]

Caron, N.

Castro, A.

Cathey, W. T.

Chi, W.

Christodoulides, D.

Christodoulides, D. N.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef] [PubMed]

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1–213901-4 (2007).
[CrossRef]

Chu, K.

Cizmar, T.

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
[CrossRef]

Cottrell, D. M.

Dai, H. T.

Davis, J. A.

Dholakia, K.

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Dogariu, A.

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1–213901-4 (2007).
[CrossRef]

Dowski, E. R.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Non linear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Non linear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

George, N.

Harvey, A. R.

T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903-1–742903-8 (2009).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950).

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef] [PubMed]

Komatsu, S.

Kubala, K.

Liu, Y. J.

Luo, D.

Mazilu, M.

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

McAuley, K. P.

Mitry, M. J.

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef] [PubMed]

Morris, J. E.

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
[CrossRef]

Narayanswamy, R.

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

Ojeda-Castañeda, J.

Pauca, V. P.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

Plemmons, B. J.

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

Plemmons, R. J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef] [PubMed]

Prasad, S.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

Ruiz, I.

Setty, H.

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

Sheng, Y.

Siviloglou, G.

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1–213901-4 (2007).
[CrossRef]

G. Siviloglou and D. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef] [PubMed]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef] [PubMed]

Sun, X. W.

Takahashi, Y.

Thompson, K. P.

Torgesen, T.

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

Torgesen, T. C.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

Tucker, S.

Van der Gracht, J.

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

Vettenburg, T.

T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903-1–742903-8 (2009).
[CrossRef]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Non linear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

Wood, A.

T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903-1–742903-8 (2009).
[CrossRef]

Yan, F.

F. Yan, L.-G. Zheng, and X.-J. Zhang, “Design of an off-axis three-mirror anastigmatic optical system with wavefront coding technology,” Opt. Eng. 47, 063001-1–063001-10 (2008).
[CrossRef]

Zhang, X.-J.

F. Yan, L.-G. Zheng, and X.-J. Zhang, “Design of an off-axis three-mirror anastigmatic optical system with wavefront coding technology,” Opt. Eng. 47, 063001-1–063001-10 (2008).
[CrossRef]

Zheng, L.-G.

F. Yan, L.-G. Zheng, and X.-J. Zhang, “Design of an off-axis three-mirror anastigmatic optical system with wavefront coding technology,” Opt. Eng. 47, 063001-1–063001-10 (2008).
[CrossRef]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Non spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Appl. Opt. (6)

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

Nat. Photonics (2)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Non linear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Opt. Eng. (1)

F. Yan, L.-G. Zheng, and X.-J. Zhang, “Design of an off-axis three-mirror anastigmatic optical system with wavefront coding technology,” Opt. Eng. 47, 063001-1–063001-10 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1–213901-4 (2007).
[CrossRef]

Proc. SPIE (4)

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Supercontinuum Airy beams,” Proc. SPIE 7430, 74300W-1–74300W-9 (2009).
[CrossRef]

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgesen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration—corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

J. Van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, B. J. Plemmons, S. Prasad, and T. Torgesen, “Iris recognition with enhanced depth-of-field image acquisition,” Proc. SPIE 5438, 120–129 (2004).
[CrossRef]

T. Vettenburg, A. Wood, N. Bustin, and A. R. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903-1–742903-8 (2009).
[CrossRef]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef] [PubMed]

Other (1)

H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950).

Supplementary Material (10)

» Media 1: MOV (527 KB)     
» Media 2: MOV (440 KB)     
» Media 3: MOV (376 KB)     
» Media 4: MOV (353 KB)     
» Media 5: MOV (328 KB)     
» Media 6: MOV (337 KB)     
» Media 7: MOV (342 KB)     
» Media 8: MOV (379 KB)     
» Media 9: MOV (414 KB)     
» Media 10: MOV (508 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Generation of a “cubic phase” beam continuum: (a) CODE V® computational setup, (b) corresponding experimental setup.

Fig. 2
Fig. 2

Comparison of (top) an Airy ( η = 0 ) and (bottom) a Seidel coma-aberrated ( η = 1 ) beam created using the computational model. (a) Wavefronts (in fringes) at exit pupil, (b) PSFs (at Fourier plane), (c), longitudinal propagation 2D profiles.

Fig. 3
Fig. 3

Parameterization of the evolution from Airy ( η = 0 ) to comatic ( η = 1 ) beams using pupil zones images of a “cubic phase” beam continuum parameterized by η (see also Media 1, Media 2, Media 3, Media 4, Media 5, Media 6, Media 7, Media 8, Media 9, Media 10): ray trace intercepts in the Fourier plane for different η values from left to right, then top to bottom.

Fig. 4
Fig. 4

(a) Computational and (b) experimental generation of the ‘‘cubic phase’’ beam continuum, highlighting the Seidel coma and Zernike trefoil combination that specifically produces an Airy beam. In (a) and (b), from top to bottom: pure Zernike trefoil-aberrated beam ( η = 3 ) , Airy ( η = 0 ) and pure Seidel coma ( η = + 1 ) . From left to right: through-focus PSFs, and 2D beam propagation longitudinal plot.

Fig. 5
Fig. 5

Airy beam longitudinal structure: shift of intensity maximum and invariance. Top row shows 2D longitudinal beam propagation profiles at the y = x plane, and bottom row shows the caustic surfaces. (a) η = 3 , (b) η = 0 , and (c) η = 1 .

Equations (42)

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

W Airy_ 2 D = K Airy ( x 3 + y 3 ) ,
W coma_ 45 ° = K coma 2 ( x 3 + y 3 + x y 2 + x 2 y ) ,
W = K [ x 3 + y 3 + η ( x y 2 + x 2 y ) ] ,
ϵ y ( ρ , φ , η ) = K ρ 2 [ ( 3 + η ) + 2 η cos 2 φ ] ,
ϵ x ( ρ , φ , η ) = K ρ 2 ( 3 η ) sin 2 φ ,
W = K ρ 3 3 + η 4 ( cos φ + sin φ ) + 1 η 4 ( cos 3 φ sin 3 φ ) ,
W = K ρ 3 ( A 7 Z 7 + A 8 Z 8 + A 10 Z 10 + A 11 Z 11 + A 2 Z 2 + A 3 Z 3 ) ,
Y 1 ( z ) = a 3 6 2 K ( 1 + η ) ( z f ) 2 ,
Y 2 ( z ) = a 3 ( 3 5 η ) 6 2 K ( 3 η ) 2 ( z f ) 2 ,
Θ ( x , y ) = α ( x 3 + y 3 ) + β ( x 2 y + x y 2 ) .
P ( r , θ ) = i a i r b i cos ( w i θ + ϕ i ) ,
x = x + y 2 , y = y x 2 .
W = K 2 { y ( y 2 + 3 x 2 ) + η [ y ( y 2 2 x 2 ) ] } .
ϵ x ( x , y , η ) = 1 NA W ( x , y , η ) x = 1 NA 2 K x y ( 3 η ) ,
ϵ y ( x , y , η ) = 1 NA W ( x , y , η ) y = 1 NA K 2 [ 3 ( 1 + η ) y 2 + ( 3 η ) x 2 ] ,
ϵ y ( ρ , φ , η ) = 1 NA K 2 ρ 2 [ ( 3 + η ) + 2 η cos 2 φ ] ,
ϵ x ( ρ , φ , η ) = 1 NA K 2 ρ 2 ( 3 η ) sin 2 φ .
W = K ρ 3 { ( sin φ ) 3 + ( cos φ ) 3 + η [ sin φ ( cos φ ) 2 + ( sin φ ) 2 y ] cos φ } .
( sin φ ) 3 = 3 sin φ sin 3 φ 4 ,
( cos φ ) 3 = 3 cos φ + cos 3 φ 4 .
Z 2 = ρ cos φ ,
Z 3 = ρ sin φ ,
Z 7 = 3 ρ 3 cos φ 2 cos φ ,
Z 8 = 3 ρ 3 sin φ 2 sin φ ,
Z 10 = ρ 3 cos 3 φ ,
Z 11 = ρ 3 sin 3 φ .
W = K { ρ 3 [ 3 A 7 cos φ + 3 A 8 sin φ + A 10 cos 3 φ + A 11 sin 3 φ ] + ρ [ ( A 2 2 A 7 ) cos φ + ( A 3 2 A 8 ) sin φ ] } .
3 A 7 = 3 A 8 = 3 + η 4 ,
A 10 = 3 A 11 = 1 η 4 ,
A 2 2 A 7 = 0 ,
A 3 2 A 8 = 0 ,
A 7 = A 8 = 3 + η 12 ,
A 2 = 2 A 7 ,
A 3 = 2 A 8 .
W = K 2 y [ ( 3 η ) x 2 + ( 1 + η ) y 2 ] .
X ( x , y , z ) = f a x [ W + ( x 2 + y 2 ) a 2 2 f 2 z ] = 2 f a K 2 ( 3 η ) x y a f x z ,
Y ( x , y , z ) = f a y [ W + ( x 2 + y 2 ) a 2 2 f 2 z ] = f a K 2 [ ( 3 η ) x 2 + 3 ( 1 + η ) y 2 ] + a f y z .
X x Y y X y Y x = 0 .
x = 0 or
y = a 2 2 K ( 3 η ) f 2 z .
Y 1 ( y , z ) = 3 ( 1 + η ) f a K 2 y 2 + a f y z .
Y 2 ( x , z ) = ( 3 η ) f a K 2 x 2 + ( 3 5 η ) a 3 2 2 K ( 3 η ) 2 f 3 z 2 .

Metrics