## Self-imaging of three-dimensional images by pulsed wave fields

Optics Express, Vol. 10, Issue 8, pp. 360-369 (2002)

http://dx.doi.org/10.1364/OE.10.000360

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### Abstract

Recently, the classical Talbot effect (self-imaging of optical wave fields) has attracted a renewed interest, as the concept has been generalized to the domain of pulsed wave fields by several authors. In this paper we discuss the self-imaging of three-dimensional images. We construct pulsed wave fields that can be used as self-imaging “pixels” of a three-dimensional image and show that their superpositions reproduce the spatial separated copies of its initial three-dimensional intensity distribution at specific time intervals. The derived wave fields will be shown to be directly related to the fundamental localized wave solutions of the homogeneous scalar wave equation – focus wave modes. Our discussion is illustrated by some spectacular numerical simulations. We also propose a general idea for the optical generation of the derived wave fields. The results will be compared to the work, published so far on the subject.

© 2002 Optical Society of America

## 1. Introduction

1. W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. **57**, 772–778 (1967). [CrossRef]

1. W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. **57**, 772–778 (1967). [CrossRef]

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

## 2. Monochromatic self-imaging

*k*=

^{2}*ω*is the wave number and

^{2}/c^{2}*A*(

*k,θ,ϕ*) is the angular spectrum of plane waves of the wave field in the spherical coordinates. Defining the self-imaging condition of the wave field (2) as [1–5

1. W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. **57**, 772–778 (1967). [CrossRef]

*q*is an integer and

*ψ*is an arbitrary phase factor. The relation (4) implies, that a monochromatic wave field (2) periodically reconstructs its initial transversal amplitude distribution if only its angular spectrum of plane waves is sampled so that the condition

*k*is the

_{z}*z*component of the wave vector, is satisfied for every plane wave component of the wave field.

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

*J*is the

_{n}*n*-th order Bessel function. Applying the condition (5) and including only the axially symmetric terms in the summation, we get the following expression for the general cylindrically symmetric, monochromatic, self-imaging wave field:

*a*=

_{q}*A*[

*k*, arccos(2

*πq/kd*)] and

*ψ*=

*0*is assumed. The wave field

*J*[

_{0}*kρ*sin

*θ*]exp[

*ikz*cos

*θ*-

*iωt*] in Eqs.(6) and (7) is a special case of the so-called nondiffracting beams – the zeroth-order Bessel beam [6

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

## 3. Self-imaging of pulsed wave fields

*θ*(

*k*) is the wavelength-dependent angle between the direction of propagation of the plane waves and the optical axis (

*z*-axis) and

*A*(

*k*) is the frequency spectrum. The wave field (9) can be recognized as a quite general form of a class of solutions of the linear homogeneous wave equation – the angularly dispersed plane wave pulses, also referred to as the tilted pulses (see, e.g., Refs. [14–17

14. A. A. Maznev, T. F. Crimmins, and K.A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. **23**, 1378–1380 (1998). [CrossRef]

14. A. A. Maznev, T. F. Crimmins, and K.A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. **23**, 1378–1380 (1998). [CrossRef]

*θ*with the optical axis, we observe an increasing (or decreasing) difference between the pulse front and the phase fronts, thus, the phase fronts move forward (or backward) inside the pulse envelope along the optical axis [see Fig. 3(b)]. In this section we introduce the polychromatic self-imaging in terms of tilted pulses. The spatial localization will be considered in next section.

*θ*(

*k*) in Eq. (9) can be specified from the condition, that the envelope of the pulse along the optical axis should not change in the course of propagation. Also, this property should hold for the ultrashort optical pulses, the frequency spectrum of which spans the entire visible and near-infrared spectrum. Let us choose the function as

*β*are constants. In this case the group velocity

*v*of the pulse along the optical axis,

^{g}*A*(

*k*) in Eq. (9). For the phase velocity of such pulse along the optical axis we obtain

*k*is the mean (or carrier) frequency of the wave field.

_{0}*θ*

^{(T)}(

*k*,

*β*) in Eq. (10) can be interpreted as the definition of the support of angular spectrum of plane waves of the tilted pulse, i.e. the definition of a volume in the

*k*-space where the plane wave components of the wave field have non-zero amplitudes. In our case the support of angular spectrum is a line in

*k*-

_{x}*k*plane (see Fig. 4(a) for an example). The parameter 2/

_{z}*β*in Eq. (10) has an interpretation as being the wave number of the wave vector component directed perpendicularly to the optical axis on this support.

*θ*

^{(T)}(

*k*,

*β*is determined by the Eq. (10). The expression can be given a readily interpretable form if we approximate the

_{q}*x*and

*z*components of the wave vector by

*k*is the carrier wave number and we have also used the Eq. (12). Substitution relation (16) into Eq. (15) yields

_{0}*i k*(

_{0}*x*sin

*θ*

^{(T)}(

*k*) +

_{0},β_{q}*y*cos

*θ*

^{(T)}(

*k*) –

_{0},β_{q}*c t*)] is the carrier wavelength plane wave component of the tilted pulse (15) and

*C*(

*x, zγ*-

*ct*) and of a term, that is a mathematical equivalent of the superposition of the monochromatic carrier-wavelength plane waves that propagate at angles

*θ*

^{(T)}(

*k*) to the optical axis. According to our general idea, the latter term should be self-imaging in the conventional, monochromatic sense of the term- in this case the product (19) behaves as a pulse that vanishes and reconstructs itself periodically. The condition for such behavior can be written in complete analogy with the Eq. (4) of the Sec. 2:

_{0},β_{q}*β*for the superposition (15):

*θ*

^{(T)}(

*k, β*) the condition

_{q}*n*+ 1) tilted pulses, centered around some carrier spatial frequency

*k*(

_{z}*k*,

_{0}*β*

_{Q}) (see Fig. 5). The spatial amplitude corresponding to such superposition can be expressed as

*k*is the interval between the spatial frequencies (see Ref. [18] for example). For this case the self-imaging distance is

_{z}*d*= 2π/Δ

*k*and the width of the peaks of the resulting function is Δ

_{z}*z*≈ 2π/(2

*n*+1)Δ

*k*. The result of the evaluation of Eq.(24) for a superposition of seven tilted pulses is shown on Fig. 5(b).

_{z}## 4. Self-imaging of the transversal and three-dimensional images by pulsed wave fields.

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

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

4. J. Wagner and Z. Bouchal, “Experimental realization of self-reconstruction of the 2D aperiodic objects,” Opt.Commun. **176**, 309–311 (2000). [CrossRef]

*θ*(

*k*) now denoting the wavelength dependent cone angle of the Bessel beams. Consequently, the on-axis behavior of (27) is identical to that of the tilted pulses in previous section, provided that the function

*θ*(

*k*) is the same in both cases. Thus (1) the on-axis group and phase velocities of the wave field (27) differ and overlapping pulses with different phase velocities give rise to the evolving interference, (2) the support of the angular spectrum of plane waves in Eq. (10) yields a pulsed wave field with non-spreading longitudinal amplitude distribution (in this context the support of the angular spectrum on Fig. 4(a) should be considered a plane section of a three-dimensional axially symmetric surface) and (3) the condition (20) determines the set of constants

*β*that provide the self-imaging of the superposition of the wave fields. As a result, we get a cylindrically symmetric pulsed self-imaging wave field that is essentially the axially symmetric superposition of the self-imaging tilted pulses in Eq. (15):

_{q}19. I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research **19**, 1 (1998). [CrossRef]

10. J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. **3**, 366–373 (2001). [CrossRef]

10. J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. **3**, 366–373 (2001). [CrossRef]

*U*(

*ν*,

*ψ*,

*ω*) is the Fourier spectrum of the spatio-temporal amplitude of the wave field at the input transverse plane

*z*=

*0*,

*ν*and

*ψ*are the radial spatial frequency and polar angle in the Fourier plane respectively,

*m*and

*n*are integers. Such an approach is very useful when the self-imaging of arbitrary transversal (or three-dimensional) images is considered because the inevitable effect of filtering of spatial frequencies of initial image is clearly manifested in the results.

*d*= 2×10

^{-5}

*m*,

*γ*= 1,

*k*= 1×10

_{0}^{7}rad

*m*

^{-1},

*q*= 27, 28, …, 31 with

*β*being determined by Eq. (22) (

_{q}*β*= 7.6×10

_{q}^{5}rad

*m*

^{-1}… 1.3×10

^{5}rad

*m*

^{-1}) and

*θ*

^{(T)}(

*k*

_{0}*β*) by Eq. (24) [

_{q}*θ*

^{(T)}(

*k*,

_{0}*β*) = 13 … 32 deg, note, that the angle is considerably larger than that chosen in the example on Fig. 6 resulting in lower side lobes and different scale along the

_{q}*x*axis]. The frequency spectrum

*A*(

*k*) spans the entire visible spectrum (400 to 800

*nm*). The wave field on Fig. 2(b) is the monochromatic, self-imaging superposition of the Bessel beams with cone angles

*θ*

^{(T)}(

*k*,

_{0}*β*) – the cylindrically symmetric generalization of the second term in the product in Eq. (19).

_{q}19. I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research **19**, 1 (1998). [CrossRef]

21. K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A **17**, 1785–1790 (2000). [CrossRef]

21. K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A **17**, 1785–1790 (2000). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

1. | W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. |

2. | J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. |

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

4. | J. Wagner and Z. Bouchal, “Experimental realization of self-reconstruction of the 2D aperiodic objects,” Opt.Commun. |

5. | R. Piestun, Y. Y. Schechner, and J. Shamir, “Self-imaging with finite energy,” Opt. Lett. |

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

7. | R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A |

8. | Z. Bouchal and M. Bertolotti, “Self-reconstruction of wave packets due to spatio-temporal couplings,” J. Mod. Opt. |

9. | Z. Bouchal, “Self-reconstruction ability of wave field” Proc. SPIE, |

10. | J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. |

11. | H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. |

12. | J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber grating,” Opt. Lett. |

13. | P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Physics |

14. | A. A. Maznev, T. F. Crimmins, and K.A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. |

15. | Zs. Bor and B. Rácz, “Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,” Opt. Commun. |

16. | P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. |

17. | P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. |

18. | O. Svelto, |

19. | I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research |

20. | P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. |

21. | K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A |

22. | K. Reivelt and P. Saari, “Optical generation of focus wave modes: errata,” J. Opt. Soc. Am. A |

23. | K. Reivelt and P. Saari, “Optically realizable localized wave solutions of homogeneous scalar wave equation,” Phys. Rev. E (to be published) [accepted for publication]. |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(320.2250) Ultrafast optics : Femtosecond phenomena

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 14, 2002

Revised Manuscript: April 5, 2002

Published: April 22, 2002

**Citation**

Kaido Reivelt, "Self-imaging of three-dimensional images by pulsed wave fields," Opt. Express **10**, 360-369 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-8-360

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### References

- W. D. Montgomery, �Self-Imaging Objects of Infinite Aperture," J. Opt. Soc. Am. 57, 772-778 (1967). [CrossRef]
- J. Turunen and A. T. Friberg, �Self-imaging and propagation-invariance in electromagnetic fields,� Pure Appl. Opt. 2, 51-60 (1993). [CrossRef]
- Z. Bouchal and J. Wagner, �Self-reconstruction effect in free propagation of wavefield,� Opt. Commun. 176, 299-307 (2000). [CrossRef]
- J. Wagner and Z. Bouchal, �Experimental realization of self-reconstruction of the 2D aperiodic objects,� Opt.Commun. 176, 309-311 (2000). [CrossRef]
- R. Piestun, Y. Y. Schechner and J. Shamir, �Self-imaging with finite energy,� Opt. Lett. 22, 200-202 (1997). [CrossRef] [PubMed]
- J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, �Diffraction-free beams,� Phys. Rev. Lett. 58, 1499�1501 (1987). [CrossRef] [PubMed]
- R. Piestun and J. Shamir, �Generalized propagation-invariant wave fields,� J. Opt. Soc. Am. A 15, 3039-3044 (1998). [CrossRef]
- Z. Bouchal and M. Bertolotti, �Self-reconstruction of wave packets due to spatio-temporal couplings,� J. Mod. Opt. 47, 1455 - 1467 (2000). [CrossRef]
- Z. Bouchal, �Self-reconstruction ability of wave field,� Proc. SPIE, vol. 4356, 217-224, (2001). [CrossRef]
- J. Salo and M. M. Salomaa, �Diffraction-free pulses at arbitrary speeds,� J. Opt. A: Pure Appl. Opt. 3, 366-373 (2001). [CrossRef]
- H. Wang, C. Zhou, L. Jianlang and L. Liu, �Talbot effect of a grating under ultrashort pulsed-laser illumination,� Microwave and Opt. Technol. Lett. 25, 184-187 (2000) [CrossRef]
- J. Aza�a and M. A. Muriel, �Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber grating,� Opt. Lett. 24, 1672-1674 (1999). [CrossRef]
- P. Saari, H. S�najalg, �Pulsed Bessel beams,� Laser Phys. 7, 32-39 (1997).
- A. A. Maznev, T. F. Crimmins and K.A. Nelson, �How to make femtosecond pulses overlap,� Opt. Lett. 23, 1378-1380 (1998). [CrossRef]
- Zs. Bor and B. R�cz, �Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,� Opt. Commun. 54, 165-170 (1985). [CrossRef]
- P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius and A. Piskarskas, �Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,� Phys. Rev. Lett. 81, 570-573 (1998). [CrossRef]
- P. Saari, J. Aaviksoo, A. Freiberg, K. Timpmann, �Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,� Opt. Commun. 39, 94-98 (1981). [CrossRef]
- O. Svelto, Principles of Lasers (3rd ed. Plenum Press 1989).
- I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, �Two fundamental representations of localized pulse solutions of the scalar wave equation,� Progr. In Electromagn. Research 19, 1 (1998). [CrossRef]
- P. Saari and K. Reivelt, �Evidence of X-shaped propagation-invariant localized light waves,� Phys. Rev. Lett. 79, 4135-4138 (1997). [CrossRef]
- K. Reivelt and P. Saari, �Optical generation of focus wave modes,� J. Opt. Soc. Am. A 17, 1785-1790 (2000). [CrossRef]
- K. Reivelt and P. Saari, �Optical generation of focus wave modes: errata,� J. Opt. Soc. Am. A 18, 2026-2026, (2001). [CrossRef]
- K. Reivelt and P. Saari, �Optically realizable localized wave solutions of homogeneous scalar wave equation,� Phys. Rev. E (to be published) [accepted for publication].

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