## Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves

Optics Express, Vol. 12, Issue 17, pp. 4001-4006 (2004)

http://dx.doi.org/10.1364/OPEX.12.004001

Acrobat PDF (994 KB)

### Abstract

In this paper it is shown how one can use Bessel beams to obtain a *stationary* localized wave field with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0≤*z*≤*L* of the propagation axis. This intensity envelope remains static, i.e., with velocity *υ*=0; and because of this we call “Frozen Waves” the new solutions to the wave equations (and, in particular, to the Maxwell equations). These solutions can be used in many different and interesting applications, such as optical tweezers, atom guides, optical or acoustic bistouries, various important medical purposes, etc.

© 2004 Optical Society of America

## 1. Introduction

1. For a review, see: E. Recami, M. Zamboni-Rached, K.Z. Nóbrega, C.A. Dartora, and H.E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE Journal of Selected Topics in Quantum Electronics **9**, 59–73 (2003); and references therein. [CrossRef]

1. For a review, see: E. Recami, M. Zamboni-Rached, K.Z. Nóbrega, C.A. Dartora, and H.E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE Journal of Selected Topics in Quantum Electronics **9**, 59–73 (2003); and references therein. [CrossRef]

*z*≤

*L*, where

*z*is the propagation axis and

*L*can be much greater than the wavelength λ of the monochromatic light which is being used. Inside such a space interval, we can construct a

*stationary*envelope with many different shapes, including one or more high-intensity peaks (with distances between them much larger than λ). This intensity envelope remains static, i.e., with velocity

*υ*=0; and because of this we call “Frozen Waves” such new solutions to the wave equations (and, in particular, to the Maxwell equations).

## 2. The mathematical methodology

*ω*,

*k*and

_{ρ}*β*are the angular frequency, the transverse and the longitudinal wave numbers, respectively. We also impose the conditions

*ω*/

*β*≥

*c*) to ensure forward propagation only, as well as a physical behavior of the Bessel function.

*N*+1 Bessel beams with the same frequency

*ω*

_{0}, but with different (and still unknown) longitudinal wave numbers

*β*:

_{n}*A*are constant coefficients. For each

_{n}*n*, the parameters

*ω*

_{0},

*k*and

_{ρn}*β*must satisfy Eq. (2), and, because of conditions (3), when considering

_{n}*ω*

_{0}>0, we must have

*β*and of the coefficients

_{n}*A*in order to reproduce approximately, inside the interval 0≤

_{n}*z*≤

*L*(on the axis

*ρ*=0), a chosen longitudinal intensity pattern that we call |

*F*(

*z*)|

^{2}. In other words, we want to have

*β*=2

_{n}*πn/L*, thus obtaining a truncated Fourier series, expected to represent the desired pattern

*F*(

*z*). Superpositions of Bessel beams with

*β*=2

_{n}*πn/L*have been actually used in some works to obtain a large set of

*transverse*amplitude profiles[2

2. Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. **176**, 299–307 (2000). [CrossRef]

3. Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. **27**, 1376–1378 (2002). [CrossRef]

*β*(when

_{n}*n*<0), which implies backwards propagating components (since

*ω*

_{0}>0); 2) In the cases when

*L*≫λ

_{0}, which are of our interest here, the main terms of the series would correspond to very small values of

*β*, which results in a very short field depth of the corresponding Bessel beams(when generated by finite apertures), impeding the creation of the desired envelopes far form the source.

_{n}*β*, which allows forward propagation components only, and a good depth of field. This problem can be solved by putting

_{n}*Q*>0 is a value to be chosen (as we shall see) according to the given experimental situation, and the desired degree of transverse field localization. Due to Eq. (5), we get

*n*, that we call

*N*, once

*Q*,

*L*and

*ω*

_{0}have been chosen.

*F*(

*z*), in the interval 0≤

*z*≤

*L*, Eq. (4) should be rewritten as:

*Q*,

*L*and

*ω*

_{0}are chosen.

*ρ*≠0, the wave field Ψ(

*ρ*,

*z*,

*t*) becomes

*A*will yield the amplitudes and the relative phases of each Bessel beam in the superposition.

_{n}*high*field concentration around

*ρ*=0.

## 3. Some examples

_{0}=0.632

*µ*m, that is, with

*ω*

_{0}=2.98 10

^{15}Hz), whose longitudinal pattern (along its

*z*-axis) in the range 0≤

*z*≤

*L*is given by the function

*l*

_{1}=

*L*/10,

*l*

_{2}=3

*L*/10,

*l*

_{3}=4

*L*/10,

*l*

_{4}=6

*L*/10,

*l*

_{5}=7

*L*/10 and

*l*

_{6}=9

*L*/10. In other words, the desired longitudinal shape, in the range 0≤

*z*≤

*L*, is a parabolic function for

*l*

_{1}≤

*z*≤

*l*

_{2}, a unitary step function for

*l*

_{3}≤

*z*≤

*l*

_{4}, and again a parabola in the interval

*l*

_{5}≤

*z*≤

*l*

_{6}, it being zero elsewhere (in the interval 0≤

*z*≤

*L*). In this example, let us put

*L*=0.5m.

*A*, which appear in the superposition Eq. (11), by inserting Eq. (13) into Eq. (10). Let us choose, for instance,

_{n}*Q*=0.9998

*ω*

_{0}/

*c*: This choice allows the maximum value

*N*=158 of

*n*, as one can infer from Eq. (8). Let us specify that, in such a case, one is not obliged to use just

*N*=158, but one can adopt for

*N*any values

*smaller*than it; more generally, any value smaller than that calculated via Eq. (8). Of course, using the maximum value allowed for

*N*, one will get a better result.

*N*=20. In Fig.1(a) we compare the intensity of the desired longitudinal function

*F*(

*z*) with that of the Frozen Wave (FW), Ψ(

*ρ*=0,

*z*,

*t*), obtained from Eq. (9) by using the mentioned value

*N*=20.

*N*=20. Obviously, the use of higher values for

*N*will improve the approximation.

*Q*one will get FWs with higher

*transverse*width (for the same number of terms in the series in Eq. (11)), because of the fact that the Bessel beams in Eq. (11) will possess a larger transverse wave number, and consequently higher transverse concentrations. We can verify this expectation by considering, for instance, a desired longitudinal pattern, in the range 0≤

*z*≤

*L*, given by the function

*l*

_{1}=

*L*/2-Δ

*L*and

*l*

_{2}=

*L*/2+Δ

*L*. Such a function has a parabolic shape, with the peak centered at

*L*/2 and a width of 2Δ

*L*. By adopting λ

_{0}=0.632

*µ*m (that is,

*ω*

_{0}=2.98 10

^{15}Hz), let us use the superposition Eq. (11) with two different values of

*Q*: we shall obtain two different FWs that, in spite of having the same longitudinal intensity pattern, will have different transverse localizations. Namely, let us consider

*L*=0.5m and Δ

*L*=

*L*/50, and the two values

*Q*=0.99996

*ω*

_{0}/

*c*and

*Q*=0.99980

*ω*

_{0}/

*c*. In both cases the coefficients

*A*will be the same, calculated from Eq. (10), on using this time the value

_{n}*N*=30 in the superposition Eq. (11). The results are shown in Figs. (2a) and (2b). One can observe that both FWs have the (same) longitudinal intensity pattern, but the one with the smaller

*Q*is endowed with the higher transverse localization.

## 4. Generation of Frozen Waves

*an array*of such apparatuses to generate a sum of them, with the appropriate longitudinal wave numbers and amplitudes/phases (as required by Eq. (11)), thus producing the desired FW. Here, it is worthwhile to notice that we shall be able to generate the desired FW in the the range 0≤

*z*≤

*L*if all Bessel beams entering the superposition Eq. (11) are able to reach the distance

*L*resisting the diffraction effects. We can guarantee this if

*L*≤

*Z*

_{min}, where

*Z*

_{min}is the field depth of the Bessel beam with the smallest longitudinal wave number

*β*

_{n}_{=-N}=

*Q*-2

*πN*/

*L*, that is, with the shortest depth of field. In such a way, once we have the values of

*L*,

*ω*

_{0},

*Q*,

*N*, from Eq. (15) and the above considerations it results that the radius

*R*of the finite aperture has to be

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

*A*of the fundamental superposition Eq. (11), can generate the desired FW. These questions will be analyzed in more detail elsewhere.

_{n}## 5. Conclusions

*stationary*localized wave fields, with high transverse localization, whose longitudinal intensity pattern can assume any desired shape within a chosen space interval 0≤

*z*≤

*L*. The produced envelope remains static, i.e., with velocity

*v*=0, and because of this we have called Frozen Waves such news solutions.

## Acknowledgements

## Footnotes

** | Patent pending. |

* | Once a value for Q has been chosen. |

## References and links

1. | For a review, see: E. Recami, M. Zamboni-Rached, K.Z. Nóbrega, C.A. Dartora, and H.E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE Journal of Selected Topics in Quantum Electronics |

2. | Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. |

3. | Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. |

4. | J. Rosen and A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. |

5. | R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. |

6. | J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(260.1960) Physical optics : Diffraction theory

(350.7420) Other areas of optics : Waves

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 27, 2004

Revised Manuscript: August 10, 2004

Published: August 23, 2004

**Citation**

Michel Zamboni-Rached, "Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves," Opt. Express **12**, 4001-4006 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-4001

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### References

- For a review, see: E.Recami, M.Zamboni-Rached, K.Z.Nóbrega, C.A.Dartora, and H.E.Hernández-Figueroa, �??On the localized superluminal solutions to the Maxwell equations,�?? IEEE Journal of Selected Topics in Quantum Electronics 9, 59-73 (2003); and references therein. [CrossRef]
- Z. Bouchal and J. Wagner, �??Self-reconstruction effect in free propagation wavefield,�?? Opt. Commun. 176, 299-307 (2000). [CrossRef]
- Z. Bouchal, �??Controlled spatial shaping of nondiffracting patterns and arrays,�?? Opt. Lett. 27, 1376-1378 (2002). [CrossRef]
- J. Rosen and A. Yariv, �??Synthesis of an arbitrary axial field profile by computer-generated holograms,�?? Opt. Lett. 19, 843-845 (1994). [CrossRef] [PubMed]
- R. Piestun, B. Spektor and J. Shamir, �??Unconventional light distributions in three-dimensional domains,�?? J. Mod. Opt. 43, 1495-1507 (1996). [CrossRef]
- J. Durnin, J. J. Miceli and J. H. Eberly, �??Diffraction-free beams,�?? Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]

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