## Generation of nonparaxial accelerating fields through mirrors. I: Two dimensions |

Optics Express, Vol. 22, Issue 6, pp. 7124-7132 (2014)

http://dx.doi.org/10.1364/OE.22.007124

Acrobat PDF (1356 KB)

### 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

## 1. Introduction

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

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

2. G. A. Siviloglou and 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 and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**, 264–267 (1979). [CrossRef]

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

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

## 2. Accelerating beams as beams invariant under rotations

*x*,

*z*). In the paraxial regime, accelerating beams follow parabolic paths. This restriction in shape is easily understood in terms of the ray picture [4

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

*z*plays the role of the main propagation direction, one replaces the standard Euclidean geometry with a “paraxial geometry”. That is, the expression for the field must satisfy the paraxial wave equation

*k*is the wavenumber), and while this equation is invariant under translations in

*x*and

*z*, it is not invariant under rotations. It is instead invariant under “paraxial rotations” (i.e., translations in the transverse Fourier spectrum [19

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

*x*accompanied by phase factors: if

*U*(

*x*,

*z*) is a paraxial solution, then so is

*U*(

*x*−

*pz*,

*z*)exp[i

*k*(

*px*−

*p*

^{2}

*z*/2)], where

*p*is the paraxial “angle” of rotation. That is, the paraxial rotation of a field with intensity

*I*(

*x*,

*z*) = |

*U*(

*x*,

*z*)|

^{2}is a field with intensity

*I*(

*x*−

*pz*,

*z*). In this geometry, the curvature of a caustic (as a function of

*z*) is given simply by its second derivative in

*z*(which is invariant under paraxial rotations

*x*,

*z*→

*x*−

*pz*,

*z*), so the only paths with constant curvature are parabolas whose axes are normal to

*z*. This means that for the main intensity lobe to have constant width, the underlying caustic must be parabolic.

*z*in the paraxial case, radial in the nonparaxial one). This is due to the fact that the complete ray distribution (which is fully determined by the caustic) is invariant under an appropriate combination of transformations. For the paraxial regime, such a transformation combination consists of a displacement and a paraxial rotation, while in the nonparaxial case it is a rotation around the circle’s center (see Fig. 1). As a result, the transverse distribution of all intensity maxima and minima remains also invariant, since the optical path difference between the two rays that cross at a given point outside the caustic depends only on the transverse distance from the point to the caustic.

## 3. Circular caustics and their generation through mirrors

*π*. An easy way to achieve this is to use curved mirrors, which are capable of operating at much larger angles than lenses and spatial light modulators. Let us employ the simple construction depicted in Fig. 2(a), used often in the design of solar concentrators [21]. Cut an outline of the shape of the desired caustic in, say, thick cardboard, and place it on top of a flat piece of paper at the desired position. Then wind a piece of string around this cardboard shape, leaving plenty of extra string. Insert a pin at the desired point source location and tie the end of the string to it. [For an incident collimated field, the distant pin can be replaced by a rod with a sliding ring to which the string is tied, as in Fig. 2(b).] Now stretch the string with a pencil. Slide the pencil, keeping the string stretched, so that it unwinds from (or winds around) the cardboard shape. The resulting curve drawn on the paper is the shape of a mirror that reflects the light coming from a point source at the pin’s position into a caustic with the shape of the cardboard outline. Note that, for a given source position and caustic shape, there is a remaining degree of freedom corresponding to the length of the string. This degree of freedom controls the size of the mirror and affects the relative amplitude of the field at several parts of the caustic, as will be seen later.

*R*(for which the reflected field’s intensity is approximately shape invariant), centered at the origin, and a collimated incident field traveling in the negative

*z*direction (i.e., the source is at

*z*=

**∞**), as shown in Fig. 2(b). In this case, the string-based mental picture translates into the following simple parametric equation for the mirror over the (

*x*,

*z*) plane: where

*ϕ*∈ (−

*π*,

*π*) is the angle between the

*z*axis and the reflected ray, and

*T*is the excess of string with respect to the distance of the source and the circle’s center, so that the point of lowest

*z*for the mirror equals −

*T*/2, and corresponds to

*x*= −

*R*,

*ϕ*= 0. In the limit

*T*≫

*R*, the mirror shape reduces to a parabola

*Z*=

*T*/2 − (

*X*+

*R*)

^{2}/2

*T*. Depending on how large the mirror is (namely, what range in

*ϕ*is used), the caustic can span a significant part of the circle (equal to the used range in

*ϕ*), coming arbitrarily close to closing it, even though the illumination is arriving from only one direction. Since this construction is completely geometrical and ignores the wavelength, there is no restriction of this mirror having to be small or large. Also, there is no “quantization” restriction on the ratio between the caustic perimeter and the wavelength, since the circle is not closed.

*X*in

*ϕ*. Let

*U*

_{inc}(

*x*) be the amplitude of the collimated illuminating field, and

*A*(

*ϕ*) be the angular spectrum of the reflected field. Then, to within a constant factor, conservation of power density requires |

*U*

_{in}(

*X*)|

^{2}d

*X*= |

*A*(

*ϕ*)|

^{2}d

*ϕ*. (Notice that we are using the Debye approximation [22] in which a plane wave is assigned to each ray.) Requiring |

*A*(

*ϕ*)| to be constant over a range of directions implies that

*U*

_{inc}(

*x*) (whose phase is constant) must be given according to i.e., minus the inverse of the derivative of the first element of Eq. (1). The shape of this apodization is shown in Fig. 3 for several values of

*T/R*(plotted parametrically with |

*U*

_{inc}|

^{2}and

*X*as functions of

*ϕ*). For large mirrors (

*T/R*≫ 1), (

*X*+

*R*)/

*T*≈ sin

*ϕ*/(1 + cos

*ϕ*) and

*T/R*. This can be appreciated from Fig. 4(a), where the incident rays are shown in yellow and the rays following a first reflection are shown in green. The amplitude of each component is proportional to the square root of the density of rays. If we fix the angular density of the green rays, the density of yellow rays decreases for increasing

*T/R*, meaning that the relative importance of the incident field decreases. Also shown are the rays following a second reflection (orange) which are seen to stay clear of the caustic. Note that not all rays hit the finite section of the mirror being used a second time, and those that do are separated into two bundles (both shown in orange) after the second reflection: (i) The bundle on the left correspond to rays that were incident on the right-hand side of the mirror and touched the caustic on its lower half after the first reflection. It is easy to see that neither these rays nor their subsequent reflections can ever return to the caustic. (ii) The bundle on the right result from rays that were incident on the left-hand side of the mirror and touched the caustic on its upper half. These rays also stay away from the caustic provided the mirror accepts only incident rays with

*x*≥

*X*(

*ϕ*

_{c}), where

*ϕ*is the solution of Here,

_{c}*ϕ*corresponds to the angle after the first reflection of the ray that goes on to hit the mirror normally on its second reflection, and that therefore retroreflects and touches the caustic a second time. This ray is shown in blue in Figs. 4(b,c) for the case where a significantly larger segment of the mirror is used. Rays incident on the mirror at

_{c}*x*>

*X*(

*ϕ*

_{c}) do not touch the caustic following the second (or any subsequent) reflection. On the other hand, rays incident on the mirror at

*x*<

*X*(

*ϕ*

_{c}) do cross the circular caustic after their second reflection, but they do not form a second caustic that overlaps with the first, and their spacing (and hence their disruption to the caustic pattern) is comparable to that of the incident rays. For the case shown in Fig. 4,

*X*(

*ϕ*

_{c}) = −27.64

*R*, so one would have to use a mirror segment significantly larger than the caustic for multiple-reflected rays to overlap with the caustic. As will be discussed in [17], one can design mirrors in three dimensions for which both the incident and multiply reflected rays stay away from the caustic.

*A*(

*ϕ*)| = exp[−(1 − cos

*ϕ*)

^{10}/1.7

^{10}] (i.e., the angular spectrum’s magnitude is nearly constant for |

*ϕ*| < 0.6

*π*and then tapers down to zero), and (b) the intensity for the same field when we include the incident field before reflection in the case where

*T*= 6.2

*R*. For larger

*T*, the effects caused by the presence of the incident field become less visible, as discussed earlier. The effects of multiple reflections are negligible in the region shown here.

## 4. Pulsed solutions

*P*is an arbitrary pulse form and

*c*is the speed of light. Let us use a (normalized) Gaussian pulse of spatial width

*w*: where

*k*

_{0}is the pulse’s central wavenumber. The reflected pulse can be estimated by using an asymptotic technique known as a uniform approximation [23], in which the integral expression for the field is replaced by an expression involving Airy functions and their derivatives, whose integral expression has the same saddle points as those of the field. The resulting expression is where (

*r*,

*φ*) are the polar coordinates of

**r**, Ai is the Airy function of the first kind, Ai′ is its derivative, and Note that it is important to choose the correct branches for all the square roots and other fractional powers within the different regions in order to get a good approximation. This approximation is valid everywhere except for a small region near the origin (where the true field is negligible), provided |

*A*(

*ϕ*)| does not vary significantly over angular variations of magnitude

*k*

_{0}

*R*. Note also that the monochromatic case, shown in Fig. 5, corresponds to the limit of large spatial width

*w*. For this case, the contribution proportional to Ai is the leading one along regions of the caustic where the amplitude is fairly constant, while the contribution proportional to Ai′ provides corrections near the ends of the caustic corresponding to regions of space where the parts of the field approaching the caustic and departing from it are of significantly unequal magnitudes. For pulsed beams (short spatial width

*w*) both contributions to the field estimate are important.

## 5. Concluding remarks

7. 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]

9. 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]

7. 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]

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

*π*. The density of the incident rays reflects the intensity apodization needed to achieve a field resembling a Mathieu field.

## Acknowledgments

## References and links

1. | M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, “Accelerating optical beams,” Opt. Photon. News |

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

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

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

5. | I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. |

6. | 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. |

7. | 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. |

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

9. | 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. |

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

11. | A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, and J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. |

12. | M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express |

13. | Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express |

14. | L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express |

15. | S. Vo, K. Fuerschbach, K. Thompson, M. A. Alonso, and J. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. A |

16. | Y. Kaganovsky and E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A |

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

18. | Yu. A. Kravtsov and Yu. A. Orlov, |

19. | M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. |

20. | M. P. Do Carmo, |

21. | R. Winston, J. C. Miñano, and P. Benítez, |

22. | M. Born and E. Wolf, |

23. | M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog., Oxf. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(350.5500) Other areas of optics : Propagation

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

(080.4035) Geometric optics : Mirror system design

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 22, 2014

Revised Manuscript: March 12, 2014

Manuscript Accepted: March 12, 2014

Published: March 19, 2014

**Citation**

Miguel A. Alonso and Miguel A. Bandres, "Generation of nonparaxial accelerating fields through mirrors. I: Two dimensions," Opt. Express **22**, 7124-7132 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7124

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### References

- 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]
- G. A. Siviloglou, D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef] [PubMed]
- M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef] [PubMed]
- M. V. Berry, N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979). [CrossRef]
- I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, “Nondiffracting accelerating wave packets of Maxwells equations,” Phys. Rev. Lett. 108, 163901 (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]
- 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]
- M. A. Bandres, B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. 15, 013054 (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]
- M. A. Alonso, M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37, 5175–5177 (2012). [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]
- M. A. Bandres, M. A. Alonso, I. Kaminer, M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013). [CrossRef] [PubMed]
- Y. Kaganovsky, E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- Y. Kaganovsky, E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A 29, 671–688 (2012). [CrossRef]
- 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. A. Bandres, M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009). [CrossRef]
- 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.
- M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog., Oxf. 57, 43–64 (1969).

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