## Three-dimensional accelerating electromagnetic waves |

Optics Express, Vol. 21, Issue 12, pp. 13917-13929 (2013)

http://dx.doi.org/10.1364/OE.21.013917

Acrobat PDF (3046 KB)

### Abstract

We present a general theory of three-dimensional non-paraxial spatially-accelerating waves of the Maxwell equations. These waves constitute a two-dimensional structure exhibiting shape-invariant propagation along semicircular trajectories. We provide classification and characterization of possible shapes of such beams, expressed through the angular spectra of parabolic, oblate and prolate spheroidal fields. Our results facilitate the design of accelerating beams with novel structures, broadening scope and potential applications of accelerating beams.

© 2013 OSA

## 1. Introduction

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

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

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

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

3. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip **9**, 1334–1336 (2009) [CrossRef] [PubMed] .

4. 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] .

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

6. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature **494**, 331–335 (2013) [CrossRef] [PubMed] .

7. A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett. **107**, 116802 (2011) [CrossRef] [PubMed] .

8. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. **101**, 071110 (2012) [CrossRef] .

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

9. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics **4**, 103–106 (2010) [CrossRef] .

11. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express **19**, 23132–23139 (2011) [CrossRef] [PubMed] .

11. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express **19**, 23132–23139 (2011) [CrossRef] [PubMed] .

12. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. **106**, 213903 (2011) [CrossRef] [PubMed] .

12. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. **106**, 213903 (2011) [CrossRef] [PubMed] .

15. R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express **19**, 23706–23715 (2011) [CrossRef] [PubMed] .

14. Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett. **37**, 3201–3203 (2012) [CrossRef] [PubMed] .

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

16. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. **33**, 1678–1680 (2008) [CrossRef] [PubMed] .

17. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express **16**, 12866–12871 (2008) [CrossRef] [PubMed] .

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

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

20. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**, 163901 (2012) [CrossRef] [PubMed] .

20. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**, 163901 (2012) [CrossRef] [PubMed] .

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

22. I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News **23**, 26–26 (2012) [CrossRef] .

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

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

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

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

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

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

27. I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express **20**, 18827–18835 (2012) [CrossRef] [PubMed] .

28. P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. **37**, 2820–2822 (2012) [CrossRef] [PubMed] .

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

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

## 2. Three-dimensional nonparaxial accelerating waves

*∂*

*+*

_{xx}*∂*

*+*

_{yy}*∂*

*+*

_{zz}*k*

^{2})

*ψ*= 0 where

*k*is the wavenumber. In free space, the solution of the Helmholtz equation can be described in terms of plane waves through its angular spectral function

*A*(

*θ*,

*ϕ*) as where

**= (sin**

*u**θ*sin

*ϕ*, cos

*θ*, sin

*θ*cos

*ϕ*) is a unit vector that runs over the unit sphere, and dΩ = sin

*θ*d

*θ*d

*ϕ*is the solid angle measure on the sphere.

*y*-axis and therefore they will have defined angular momentum

*J*

*= −*

_{y}*i*(

*z∂*

*−*

_{x}*x∂*

*) along this axis. This operator acts on the spectral function as*

_{z}*J*

*= −*

_{y}*i∂*

*; hence, the spectral function of a rotationally symmetric solution must satisfy −*

_{ϕ}*i∂*

_{ϕ}*A*=

*mA*. In this way, any rotationally symmetric wave must have a spectral function of the form

*A*(

*θ*,

*ϕ*) =

*g*(

*θ*)exp(

*imϕ*), where

*m*is a positive integer and

*g*(

*θ*) is any complex function in the interval [0,

*π*].

*g*(

*θ*). As we can see in Fig. 1 the semicircular propagation path has a radius

*m/k*. It is possible to double the angle of bending (from 90° to 180°) by propagating these waves from

*z*< 0. In this case the waves have a bending angle opposite to the direction of bending and depict full semicircles. Moreover, notice that the propagation characteristics are independent of

*g*(

*θ*), and that

*g*(

*θ*) only controls the shape of the transverse profile. As a consequence, on one hand, if we superpose accelerating waves with different values of

*m*, they will interfere during propagation, leading to families of periodic self-accelerating waves [20

20. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**, 163901 (2012) [CrossRef] [PubMed] .

22. I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News **23**, 26–26 (2012) [CrossRef] .

*m*will propagate with the same propagation constant, hence they will maintain their relative phase as in the initial plane, and preserve their nondiffractive behavior.

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

*ℓ*(

*k*

*) on the real line can be mapped to an accelerating beam. This is in direct analogy to our function*

_{y}*g*(

*θ*) of the nonparaxial case. While in the paraxial case the bending (i.e., transverse acceleration) is controlled by an overall scale parameter, in the nonparaxial case it is controlled by

*m*as described previously.

*g*(

*θ*) can generate an accelerating wave, it is not straightforward to visualize (ab initio) the features of the transverse profile that that function generates. For this reason, we propose to use the

*g*(

*θ*) functions associated with rotationally-symmetric separable solutions of the Helmholtz equation. As it is known [29], there are only four rotationally symmetric separable solutions to this equation, corresponding to the spherical, parabolic, prolate spheroidal and oblate spheroidal coordinate systems, depicted in Fig. 2. The advantages of borrowing the spectral function of these solutions is that we can create complete families of nonparaxial accelerating waves and readily characterize their transverse structures.

*x*> 0 half space. An alternative approach that would allow preserving the closed-form expressions while suppressing backward propagating components is that of performing imaginary displacements on the separable solutions, as discussed in [26

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

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

**109**, 203902 (2012) [CrossRef] [PubMed] .

*m*our waves can be approximated by those of [25

**109**, 203902 (2012) [CrossRef] [PubMed] .

## 3. Parabolic accelerating waves

*β*is a continuous “translation” parameter of the waves. In this case,

*m*can be any positive real number since

*g*(

*θ*) is independent of

*m*. The transverse field distributions at

*z*= 0 of the parabolic accelerating waves are shown in Fig. 3. As one can see, these profiles resemble the ones of the 2D paraxial Airy beams [1

**32**, 979–981 (2007) [CrossRef] [PubMed] .

**99**, 213901 (2007) [CrossRef] .

*y*≈ 0, |

*x*| ≫ 1, as shown in Fig. 2(a). The main lobe of the waves is located near

*x*= −

*m/k*,

*y*=

*β*

*/k*. The fundamental mode is

*β*= 0 and as

*β*increases the waves “translate” in the

*y*-axis. This is consistent with the result of [18

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

*β*≠ 0 there is also a “tilt” in the caustic accompanied by a change in the spacing of the fringes along the caustic sheets. This “tilt” does not change the direction of propagation, thus the acceleration is still horizontal in Fig. 3, and not in the direction to which the intensity pattern points, as it might seem at first. This is analogous to the case of paraxial 2D Airy beams with different scale parameters for each of the constituent Airy functions. As shown in Figs. 3(a) and 3(e), the parabolic accelerating waves present a single intensity main lobe that follows a circular path of radius slightly larger than

*m/k*.

## 4. Prolate spheroidal accelerating waves

*f*, 0),

*m*= 0, 1, 2,...,

*n*= 0, 1, 2,..., and

30. L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E **58**, 6792–6806 (1998) [CrossRef] .

*y*= 0 and

*z*= 0 of the prolate spheroidal accelerating waves are shown in Fig. 4. The waves have a definite parity with respect to the

*y*-axis, which is given by the parity of

*n*. The order

*n*of the waves corresponds to the number of hyperbolic nodal lines at the

*z*= 0 plane, and the width of the waves in the

*y*-axis increases as

*n*increases. As shown in Figs. 4(a) and 4(e), the prolate accelerating waves have two main lobes (or a single lobe for

*n*= 0) that follow parallel circular paths of radius slightly larger than

*m/k*, i.e., the degree

*m*of the waves controls their propagation characteristics.

*f*, let us analyze how the prolate spheroidal coordinate system behaves as a function of

*f.*As

*f*→ 0 the foci coalesce and the prolate spheroidal coordinates tend to spherical ones, while in the other extreme, as

*f*→ ∞ the prolate spheroidal coordinates tend to circular cylindrical ones. Irrespective of the value of

*f*, the intensity of the beams is negligible for

*x*> 0. This limit can be understood as a centrifugal force barrier.

- For
*m*≳*kf*, the prolate accelerating waves resemble the spherical accelerating waves described in [25**109**, 203902 (2012) [CrossRef] [PubMed] .**37**, 5175–5177 (2012) [CrossRef] [PubMed] .**37**, 5175–5177 (2012) [CrossRef] [PubMed] . - For
*m*<*kf*, the waves are located in a coordinate patch that approximates a Cartesian system, hence the prolate accelerating waves take the form*A*(*x*)*H*(*y*), where*A*(*x*) is an accelerating function and*H*(*y*) is a function that retains its form upon propagation and has finite extend. - For
*m*≪*kf*, the prolate spheroidal coordinates tend to the circular cylindrical ones, and the prolate accelerating waves tend to the product of a “half-Bessel” wave [20**108**, 163901 (2012) [CrossRef] [PubMed] .*x*-coordinate times a sine or cosine in the*y*-coordinate.

*ξ*,

*η*,

*ϕ*], are defined as and

*ξ*∈ [0, ∞),

*η*∈ [0,

*π*],

*ϕ*∈ [0, 2

*π*). The caustic cross sections are depicted in Fig. 4(c) and 4(g); by rotating this around the

*y*-axis one gets the caustic surfaces, see Fig. 2(b).

## 5. Oblate spheroidal accelerating waves

*kf*>

*m*. Because

*n*increases, for any

*kf*>

*m*there is a maximum value of

*n*for inner-type waves and for higher

*n*values the waves become outer-type. This transition from inner-type to outer-type as

*n*increases is depicted in middle and bottom rows of Fig. 5.

### 5.1. Outer-type

*m*of the waves controls their propagation characteristics because their two main lobes (or single lobe for

*n*= 0) follows a circular path of radius slightly larger than

*m/k*[see Fig. 5(a)]. The order

*n*gives its parity with respect to the

*y*-axis and corresponds to the number of hyperbolic nodal lines at the

*z*= 0 plane. One of the two cusps that

*n*> 0 oblate waves have, can be suppressed by combining three of these field as in [26

**37**, 5175–5177 (2012) [CrossRef] [PubMed] .

*n*= 0 outer-type waves are very thin (several wavelenghts), even more confined in the

*y*-axis than the parabolic and prolate accelerating waves, cf. Fig. 5(b) and Fig. 3(b), Fig. 4(b); this gives these type of waves a potential advantage in applications.

*x*| ≈

*f*and

*y*≈ 0 the transverse coordinates look like a parabolic system, see Fig. 2(c). Then for

*m*=

*kf*the oblate accelerating waves become the nonparaxial version of the paraxial accelerating parabolic beams in [16

16. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. **33**, 1678–1680 (2008) [CrossRef] [PubMed] .

17. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express **16**, 12866–12871 (2008) [CrossRef] [PubMed] .

16. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. **33**, 1678–1680 (2008) [CrossRef] [PubMed] .

*ξ*,

*η*,

*ϕ*], are defined as and

*ξ*∈ [0, ∞)

*η*∈ [0,

*π*],

*ϕ*∈ [0, 2

*π*). The caustic cross sections are depicted in Fig. 5(d), 5(h), and 5(l); by rotating this around the

*y*-axis one gets the caustic surfaces, which are an oblate spheroid and a hyperboloid of revolution, see Fig. 2(c).

### 5.2. Inner-type

*n*+ 1)/2⌉ hyperpolic stripes that separate two regions of darkness [see Fig. 5(f), 5(g), 5(j) and 5(k)] and therefore their topological structure is different than all the other waves presented in this work. First, the caustic of these waves does not present a cusp. Also, the intensity cross section at the

*y*= 0 plane of the

*n*= 0 inner-type wave only presents a single lobe of several wavelengths width, instead of a long tail of lobes present in all the other accelerating beams, cf. Figs. 5(e), 5(i) and Fig. 5(a). Moreover, the position of the maximum is no longer near

*x*= −

*m/k*but at some

*x*< −

*m/k*. The maximum amplitude remains constant during propagation until it decays very close to 90° of bending; this behavior is completely different than that of other accelerating waves that present a small oscillation of their maximum during propagation - compare Figs. 5(e), 5(i) and Fig. 5(a). Finally, these waves have definite parity with respect to the

*y*-axis, which is given by the parity of

*n*. For example, the waves with

*n*= 2 [see Fig. 5(g)] and

*n*= 3 both form two parabolic stripes, but have opposite parity. If we combine these waves of opposite parity, i.e.,

*ψ*

*±*

_{n}*i*

*ψ*

_{n}_{+1}, where

*n*is even, we can create continuous stripes of light that will also carry momentum along the hyperbolic stripes at a given plane containing the

*y*-axis. The magnitude of this local momentum density is proportional to the intensity of the beam and to the spatial frequency of the stripe pattern of the even and odd constituent waves. Asymptotically, on the field’s tails the momentum is proportional to the wavenumber.

*y*-axis one gets the caustic surfaces which are two hyperboloids of revolution.

## 6. Vector solutions

**Π**

*satisfy the vector Helmholtz equations, i.e., ∇*

_{e,m}^{2}

**Π**

*+*

_{e,m}*k*

^{2}

**Π**

*= 0, one can recover the electromagnetic field components by which are called electric type waves or which are called magnetic type waves. Therefore, we can find electromagnetic accelerating waves with different vector polarizations by setting*

_{e,m}**Π**

*=*

_{e,m}*ψ*

*v̂*, where

*ψ*is any of the scalar accelerating waves presented earlier and

*v̂*is any unit vector of a Cartesian coordinate system.

## 7. Conclusion

## Acknowledgments

## References and links

1. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

2. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

3. | J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip |

4. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

5. | T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics |

6. | N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature |

7. | A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett. |

8. | A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. |

9. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics |

10. | D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. |

11. | I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express |

12. | I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. |

13. | I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. |

14. | Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett. |

15. | R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express |

16. | M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. |

17. | J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express |

18. | M. A. Bandres, “Accelerating beams,” Opt. Lett. |

19. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

20. | I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. |

21. | F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. |

22. | I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News |

23. | M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. |

24. | P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. |

25. | P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. |

26. | M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. |

27. | I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express |

28. | P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. |

29. | C. P. Boyer, E. G. Kalnins, and W. Miller Jr, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J |

30. | L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E |

31. | J. Stratton, |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.5500) Other areas of optics : Propagation

(350.7420) Other areas of optics : Waves

(070.3185) Fourier optics and signal processing : Invariant optical fields

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 18, 2013

Revised Manuscript: May 15, 2013

Manuscript Accepted: May 15, 2013

Published: June 3, 2013

**Citation**

Miguel A. Bandres, Miguel A. Alonso, Ido Kaminer, and Mordechai Segev, "Three-dimensional accelerating electromagnetic waves," Opt. Express **21**, 13917-13929 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-13917

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### References

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