Abstract

Accelerating beams are wave packets that preserve their shape while propagating along curved trajectories. Recent constructions of nonparaxial accelerating beams cannot span more than a semicircle. Here, we present a ray based analysis for nonparaxial accelerating fields and pulses in two dimensions. We also develop a simple geometric procedure for finding mirror shapes that convert collimated fields or fields emanating from a point source into accelerating fields tracing circular caustics that extend well beyond a semicircle.

© 2014 Optical Society of America

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References

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  1. M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
    [CrossRef]
  2. G. A. Siviloglou, D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef] [PubMed]
  3. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009).
    [CrossRef] [PubMed]
  4. M. V. Berry, N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [CrossRef]
  5. I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (2012).
    [CrossRef]
  6. F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. 37, 1736–1738 (2012).
    [CrossRef] [PubMed]
  7. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
    [CrossRef] [PubMed]
  8. M. A. Bandres, B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. 15, 013054 (2013).
    [CrossRef]
  9. P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
    [CrossRef] [PubMed]
  10. M. A. Alonso, M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37, 5175–5177 (2012).
    [CrossRef] [PubMed]
  11. A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. 38, 2218–2220 (2013).
    [CrossRef] [PubMed]
  12. M. A. Bandres, M. A. Alonso, I. Kaminer, M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
    [CrossRef] [PubMed]
  13. Y. Kaganovsky, E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010).
    [CrossRef] [PubMed]
  14. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19, 16455–16465 (2011).
    [CrossRef] [PubMed]
  15. S. Vo, K. Fuerschbach, K. Thompson, M. A. Alonso, J. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. A 27, 2574–2582 (2010).
    [CrossRef]
  16. Y. Kaganovsky, E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A 29, 671–688 (2012).
    [CrossRef]
  17. M. A. Alonso, M. A. Bandres, “Generation of nonparaxial accelerating fields through mirrors. II: Three dimensions,”, submitted.
  18. Yu. A. Kravtsov, Yu. A. Orlov, Caustics, Catastrophes and Wave Fields, 2 (Springer, 1999), p. 21.
  19. M. A. Bandres, M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009).
    [CrossRef]
  20. M. P. Do Carmo, Differential Geometry of Curves and Surfaces, (Prentice Hall, 1976), pp. 16–22.
  21. R. Winston, J. C. Miñano, P. Benítez, Nonimaging Optics (Elsevier, 2005), pp. 47–49.
  22. M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University Press, 1999), pp. 484–498.
  23. M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog., Oxf. 57, 43–64 (1969).

2013

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

M. A. Bandres, B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. 15, 013054 (2013).
[CrossRef]

M. A. Bandres, M. A. Alonso, I. Kaminer, M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
[CrossRef] [PubMed]

A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. 38, 2218–2220 (2013).
[CrossRef] [PubMed]

2012

Y. Kaganovsky, E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A 29, 671–688 (2012).
[CrossRef]

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. 37, 1736–1738 (2012).
[CrossRef] [PubMed]

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

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

2011

2010

2009

2007

1979

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

1969

M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog., Oxf. 57, 43–64 (1969).

Aleahmad, P.

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Alonso, M. A.

Balazs, N. L.

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

Bandres, M. A.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

M. A. Bandres, B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. 15, 013054 (2013).
[CrossRef]

M. A. Bandres, M. A. Alonso, I. Kaminer, M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
[CrossRef] [PubMed]

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

M. A. Bandres, M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009).
[CrossRef]

M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009).
[CrossRef] [PubMed]

M. A. Alonso, M. A. Bandres, “Generation of nonparaxial accelerating fields through mirrors. II: Three dimensions,”, submitted.

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Benítez, P.

R. Winston, J. C. Miñano, P. Benítez, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

Berry, M. V.

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

M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog., Oxf. 57, 43–64 (1969).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University Press, 1999), pp. 484–498.

Cannan, D.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Chen, Z.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Christodoulides, D. N.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

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

Courvoisier, F.

Do Carmo, M. P.

M. P. Do Carmo, Differential Geometry of Curves and Surfaces, (Prentice Hall, 1976), pp. 16–22.

Dudley, J. M.

Froehly, L.

Fuerschbach, K.

Furfaro, L.

Giust, R.

Greenfield, E.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

Guizar-Sicairos, M.

Heyman, E.

Hu, Y.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Jacquot, M.

Kaganovsky, Y.

Kaminer, I.

M. A. Bandres, M. A. Alonso, I. Kaminer, M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
[CrossRef] [PubMed]

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Kravtsov, Yu. A.

Yu. A. Kravtsov, Yu. A. Orlov, Caustics, Catastrophes and Wave Fields, 2 (Springer, 1999), p. 21.

Lacourt, P. A.

Li, T.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Mathis, A.

Mills, M.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

Mills, M. S.

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Miñano, J. C.

R. Winston, J. C. Miñano, P. Benítez, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

Miri, M.A.

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Morandotti, R.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Orlov, Yu. A.

Yu. A. Kravtsov, Yu. A. Orlov, Caustics, Catastrophes and Wave Fields, 2 (Springer, 1999), p. 21.

Rodríguez-Lara, B. M.

M. A. Bandres, B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. 15, 013054 (2013).
[CrossRef]

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

Rolland, J.

Segev, M.

M. A. Bandres, M. A. Alonso, I. Kaminer, M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
[CrossRef] [PubMed]

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Siviloglou, G. A.

Thompson, K.

Vo, S.

Winston, R.

R. Winston, J. C. Miñano, P. Benítez, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University Press, 1999), pp. 484–498.

Yin, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Zhang, P.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Zhang, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Am. J. Phys.

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

J. Opt. Soc. Am. A

New J. Phys.

M. A. Bandres, B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. 15, 013054 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photon. News

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News 24, 30–37 (2013).
[CrossRef]

Phys. Rev. Lett.

P. Aleahmad, M.A. Miri, M. S. Mills, I. Kaminer, M. Segev, D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef] [PubMed]

Sci. Prog., Oxf.

M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog., Oxf. 57, 43–64 (1969).

Other

M. A. Alonso, M. A. Bandres, “Generation of nonparaxial accelerating fields through mirrors. II: Three dimensions,”, submitted.

Yu. A. Kravtsov, Yu. A. Orlov, Caustics, Catastrophes and Wave Fields, 2 (Springer, 1999), p. 21.

M. P. Do Carmo, Differential Geometry of Curves and Surfaces, (Prentice Hall, 1976), pp. 16–22.

R. Winston, J. C. Miñano, P. Benítez, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

M. Born, E. Wolf, Principles of Optics, 7 (Cambridge University Press, 1999), pp. 484–498.

Supplementary Material (5)

» Media 1: MOV (951 KB)     
» Media 2: MOV (316 KB)     
» Media 3: MOV (519 KB)     
» Media 4: MOV (5240 KB)     
» Media 5: MOV (679 KB)     

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Figures (7)

Fig. 1
Fig. 1

(a Media 1)) In the paraxial regime, ray families (black lines) that form a parabolic caustic (orange line) preserve their structure under a “paraxial rotation” (a linear shear x, zxpz, z) followed by a displacement. (b Media 2) In the nonparaxial regime, ray families forming a circular caustic preserve their structure under rotations around the circle’s center.

Fig. 2
Fig. 2

(a, Media 3) Construction for drawing mirror shapes that focus a collimated beam onto a caustic of a given shape. (b) Illustration of the mirror shape (black curve) that reflects rays (blue lines) to form a circular caustic (orange). Media 4 shows the propagation of a wavefront or the crest of a pulse.

Fig. 3
Fig. 3

Apodization of the collimated incident field’s intensity needed to achieve intensity uniformity along the caustic, for several values of T/R.

Fig. 4
Fig. 4

Parallel incident rays (yellow), and after one (green) and two (orange) reflections, for (a) the same size mirror as in Fig. 2 (T = 6.2R), and (b) a significantly larger segment of the same mirror. In (b) the ray highlighted in blue, whose first incidence is at x = X(ϕc) = −27.64R, retroreflects at the second reflection. This ray is the boundary between those that cross the caustic a second time and those that don’t: rays incident at x < X(ϕc) cross the caustic but diverge away from each other, so that their disruption to the caustic pattern is not too significant. The segment corresponding to the box in (b) is expanded in (c).

Fig. 5
Fig. 5

Intensity around the caustic for (a) the field reflected by the mirror, and (b) the total field (reflected plus incident), for a mirror with kR = 80, T = 6.2R and an incident field leading to an angular spectrum after reflection with amplitude |A(ϕ)| = exp[−(1 − cosϕ)10/1.710].

Fig. 6
Fig. 6

Intensity around the caustic for a Gaussian pulse reflected by a mirror with k0R = 80, k0w = 0.2, T = 6.2R at several times with a time separation of R/c. Media 5 shows the continuous evolution of the pulse.

Fig. 7
Fig. 7

Mirror shapes that focus collimated beams into fields with elliptic caustics of different orientations. The density of the incident rays reflects the intensity apodization needed to achieve a field resembling a Mathieu field.

Equations (13)

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

[ X ( ϕ ) , Z ( ϕ ) ] = [ R ( R ϕ + T ) sin ϕ 1 + cos ϕ , ( R ϕ + T ) cos ϕ R sin ϕ 1 + cos ϕ ] ,
| U inc [ X ( ϕ ) ] | 2 | d ϕ d X | = 1 + cos ϕ R ( ϕ + sin ϕ ) + T ,
U ref ( r ) = k 2 π | A ( ϕ ) | exp { i k [ R ϕ + r u ( ϕ ) ] } d ϕ ,
| A ( ϕ ) | = | U inc [ X ( ϕ ) ] | R ( ϕ + sin ϕ ) + T 1 + cos ϕ ,
2 R ( π ϕ c ) = T + 2 R cos ϕ c tan ϕ c 2 .
U ref ( r ; t ) = | A ( ϕ ) | P [ t r u ( ϕ ) / c R ϕ ] d ϕ ,
P ( t ) = exp ( c 2 t 2 2 w 2 + i k 0 c t ) ,
U ref ( r , φ , t ) [ a Ai ( h ) + b Ai ( h ) ] exp [ Ω ( ϕ 1 ) + Ω ( ϕ 2 ) 2 ] ,
Ω ( ϕ ) = [ r cos ( ϕ φ ) + R ϕ c t i k 0 w 2 ] 2 2 w 2 k 0 2 w 2 2 ,
h = { 3 [ Ω ( ϕ 1 ) Ω ( ϕ 2 ) ] 4 i } 2 / 3 ,
a = i π [ 2 i k 0 h Ω ( ϕ 1 ) A ( ϕ 1 ) + 2 i k 0 h Ω ( ϕ 2 ) A ( ϕ 2 ) ] ,
b = i π [ 2 i k 0 h Ω ( ϕ 1 ) A ( ϕ 1 ) 2 i k 0 h Ω ( ϕ 2 ) A ( ϕ 2 ) ] ,
ϕ 1 , 2 = φ + π 2 ± arccos ( R r ) .

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